text:
History of Pisa
Pisa:
Tourist map of Pisa
Photos of Pisa
Pisa is a charming city located in central Italy, in Tuscan region. Pisa is situated on the bank of the river Arno and the total population currently living in Pisa, is about 90,000. It is important to say that Pisa was once an independent Republic of Pisa with one of the most powerful fleets in Europe.
The origin of Pisa seems to be blurred. There are different theories about the origins of this city, one being linked to Ligurian region and the other to that of ancient Greece. What is certain is that Pisa at first was an Etruscan colony, than a Roman colony, later part of the Longboard Kingdom and at the end part of the unified Italian Kingdom. Pisa's economic, cultural and artistic growth was mostly expanded in the Middle Ages. In the 11th century already, Pisa became an important fleet power and took major part in trading in the Mediterranean Sea. With its mighty fleet Pisa brought home numerous victories against the ships of the Ottoman Empire. Pisa's influence, at the time, spread to the Northern Africa as well.
Because of its attitudes and political orientation Pisa often entered into conflicts with the Papacy. Even a great scandal involving Pisa occurred when false information was announced that Pisa captured the high priests on their way to Rome and handed them over to Federico I. It is believed that this scandal was deliberately launched in order to damage the Republic of Pisa.
The gradual decline of Pisa began with a defeat against Genoa in a fleet war that occurred in 1284 near Livorno. This battle has been recorded in history as a great battle of Melonie. There is a large monument called the Malonian Tower, constructed as a stark reminder of this great battle and its victims. Over 10,000 men from Pisa were captured in this battle and taken to Genoa prisons. This was a great loss for Pisa and the economy of the city.
Soon afterwards there was a brief economic recovery of Pisa, but the new decline was caused when Emperor Ariga VII (1301-1313) stepped down from the throne. Soon the loss of Sardinia followed, which had a huge financial negative impact on Pisa. After a brief reign over Lucca (a beautiful Tuscan city), Pisa was conquered by Florence (1406). A period of great crisis came and it lasted till influential family Medici started to rule in Florence.
In the sixteenth century Pisa tried to return to the world scene by founding a university and institutions of the order of the knights of St Stephen, who were supposed to be part of the battle against the Turks. In 1848, the university battalion volunteers won a great victory against Austria in the Curtatone city of Italy. Shortly afterwards, Pisa succeeded in obtaining the Kingdom of Sardinia under its wing, which was a base for the creation of the Kingdom of Italy proclaimed in 1861.
Regrettably, Pisa was severely damaged in the bombardment during the Second World War. The reconstruction, however, was rapid and soon after the war was finished, its former glory was restored.
Five Continents Museum
Gallerie dell'Accademia
Peggy Guggenheim Museum
Fishermen's island
Villa Borromeo
Island Madre
Island Bella
Villa Ducale in Stresa
Tourist destinations in Italy:
Photos of Venice Tourist map of Venice
History of Venice
The city of canals and alleyways, numerous islands and bridges, the city of gondolas and wondrous vistas, jaw dropping architecture, postcard - perfect palaces, the city of the carnival and the masks, in a nut shell – a city one can fall in love with quite easily. All of these words seem to describe Venice as one of the most well-known tourist destinations in the world. Venice offers loads to visitors: stunningly beautiful palaces and superb buildings that will take you on a trip through the rich history of this unique city, numerous canals to cross either on foot or by a gondola, a number of islands that still seem to stay hidden to most tourists, as well as a huge number of tourist sights and museums. As the saying goes, even the Venetians get lost in Venice.
Discover Venice
Photos of Rome Tourist map of Rome
The eternal city, a city of glorious past, a city of chequered history, the cradle of Roman Catholic Church - are just some attributes that distinguish Rome from other popular tourist destinations. Tourists will be able to take a peek at the beauty and glory of once the most important city in the world and to enjoy its more modern architectural marvels, mostly built by a number of different popes in quick or slow succession. Don't just visit Colosseum and Saint Peter's Cathedral, Rome has so much more to offer.
Photos of Ravenna Tourist map of Ravenna
History of Ravenna
Ravenna is a city of history, numerous monuments that stand in silent testimony of its turbulent past as well as its fame and glory as the capital city of many an empire and kingdom. The main attractions of Ravenna are certainly its churches and mosaics that all date back to the 5th and 6th century and the early Christian societies in Europe. The scale and splendour of colorful Byzantine mosaics, their historical significance as well as monuments that date back to the Ostrogoths were even recognized by UNESCO who included eight sites in this small city on their World Heritage list. Ravenna is a city which lets its visitors to embark on a journey through history and especially through the history of Byzantine, from the civilization of which only but a faint remnant has survived throughout the world.
Discover Ravenna
Photos of Milan Tourist map of Milan
History of Milan
Milano is a city of fashion, a city fascinated with the idea of beauty, a city of stunningly beautiful Gothic architecture, a city of compelling culture, a city of the young, city of fabulous fairs and architectural accomplishments. All of these attributes seem to describe Milan, which is the business capital of Italy, which oozes its charm to its visitors in a matter of seconds. To a contemporary tourist Milan offers grand, awe inspiring buildings, a multitude of museums, riveting religious buildings, its culture as a mixture of the modern trends and the tradition that goes all the way back to the Roman Empire, glitz and glamour of its shopping malls and different nightlife zones that will appeal to the tastes of all visitors.
Photos of Lake Garda Tourist map of Lake Garda
History of Lake Garda
This is the page dedicated to Lake Garda on which you can find a complete tourist guide and suggestions about what to visit while planning your trip to the lake. Lake Garda is one of three great lakes in the northern Italy. Its lovely little towns on the shores of the lake draw an incredible number of tourists each year, who flock there for its mild and mellow climate, strolls on the lake shores, traditional Italian cuisine and of course to stand in awe before the heritage of this region laid bare for us to gaze at in a host of castles, wondrous villas and picturesque villages and dainty little towns.
Discover Lake Garda
Photos of Pisa Tourist map of Pisa
The city of Pisa rises above the docks of Arno river. Pisa is one of the most important cities in Tuscany and it is well known in the world thanks to the city's famous symbol, the Leaning Tower of Pisa. We could say that Pisa is both, an ancient city, proud of it's middle ages past, when Pisa was one of the leading naval force in Europe, and a modern city, full of positive energy that comes from its numerous students, as Pisa has three of the most important universities in Italy. Even if most famous monument in Pisa is the leaning Tower, it is not the only one in this city. Bell Tower of San Nicola Church and the church of San Michele of Scalzi are the other two leaning towers in Pisa.
Photos of Lake Como Tourist map of Lake Como
History of Lake Como
With its gardens with a riot of colour, wonderful villas and splendid panoramic views Lake Como has become one of the most visited tourist places in Italy. The compellingly beautiful Lake Como seems to have enchanted artists and travelers for centuries: from famous Giuseppe Verdi and Vicenzo Bellini who apparently wrote his opera "Norma" here, to Gioacchino Rossini and a French writer Flaubert.
Discover Lake Como
Tourist map of Lake Maggiore
History of Lake Maggiore
Lake Maggiore represents the most beautiful jewel of Piemonte region in Italy. Located between mountains and valleys, Lago Maggiore is the second largest lake in Italy (after Lago di Garda), covering the surface of 212km². Almost 80% of the lake's surface is located on Italian territory, between regions Piemonte and Lombardy, while the remaining 20% belongs to Switzerland. Here, you will have a chance to enjoy such artistic landscapes that everyone should see at least once in a lifetime. Between the emerald green vegetation and the shining blue sky stand numerous castles, sumptuous palaces and prestigious Italian gardens. These monuments testify about the link between the two noble families, which, for years, marked the history of the lake: families Visconti and Borromeo.
Discover Lake Maggiore
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Wynstones Press began as a printing business 30 years ago, initially located in Gloucestershire, then moving to Stourbridge in the mid 1990s, then onto the Ruskin Glass Centre in 2001. The publishing side of the business began with the producing of Childrens' books and Artists' postcards, and this side of the business developed further during the years, and now includes Books, Christmas Cards, Prints, Folded Cards and Postcards, almost entirely reproduced from original work by different artists.
Printing ceased in 2003, by which time more distribution work was undertaken, selling beautiful Advent Calendars imported from a variety of European countries, and the selling of specialist books published by Mercury Press in the USA. Work is distributed worldwide, most sales being to North America, Europe and Australia.
See www.wynstonespress.com for details of all items available for sale.


Welcome back! So happy you stopped by for day 3's tag. I just love this one! Hope you do too!
Size and cut the shapes from the Easter Grass file and a tag from the Shape Library.
Color the background of the tag using the ink and blender tool. Set aside.
Put the basket together. I glued the vertical splines at the bottom only and did a over/under weave. Make sure the top edge is open and not glued shut.
Color the "grass" using markers. I crumpled mine up a bit and glued to the back inside of the basket. You can't really see it, but it's there for a little bit of filler.
Using the brown/oatmeal ink, add a little bit of color to the edges of the bunny pieces. Color the pads of the feet with a bit of pink. Glue the pads to the feet and add the tail using a piece of the foam pad. Glue the body to the grass, tucking the bottom of the body into the basket. Add a pop up to the feet.
Glue the basket to the tag. Glue a few sequins to the basket and add some seed beads to the sequin centers.
You can see a little more detail in this image -- the weave on the basket is pretty tight and the seed beads are a perfect fill if you're not stitching the sequins to the paper.
The Sentiment was created in eCal3 as a print 2 print. Super easy and the font is Bodini. Just fill the font with the desired color and set the stroke to none. Size the font, make sure it is set to print*; add a rectangle (Fill is white and stroke is set to none and layer to the back). Print and cut according to the step by step instructions in the software.
*Check preview to make sure the sentiment will print and the rectangle cuts.
Hoping you have some more creative time set for this coming week. Don't forget to stop by the FB page for more inspiration and if you'd like to participate, we'd be happy to have you join in the fun too!
just haven't been able to share some of the things I've been making.... until now!
Let's start with this one now the featured post on www.invitebyvoice.com!
I'm so excited to be designing some more sound cards and other projects using their EZ Sound Modules and voice boxes. So stay tuned... more goodies to come.


text:
Ghana's peace, economic growth attract Hungary — Hungarian President
BY: Donald Ato Dapatem
President Janos Ader proposing a toast to President Nana Addo Dankwa Akufo-Addo towards better working relations between Ghana and Hungary, and Mrs Anita Herczegh (extreme right), First Lady of Hungary, at a dinner at the Jubilee House. Picture: Samuel Tei Adano Follow @Graphicgh
Hungary is attracted to Ghana because of its stability, peace and being one of the fastest growing economies in Africa, the President of Hungary, Mr Janos Ader, has stated.
"In terms of political stability, Ghana is one of the most politically stable countries in this region and Africa as a whole. Ghana is one of the countries that has proven and shown the largest economic growth post-COVID-19 and these reasons substantiate our working together to strengthen our relations," he added.
Mr Ader stated this during a state banquet President Nana Addo Dankwa Akufo-Addo held in his honour at the Jubilee House last Thursday night.
The two Presidents had earlier had a tete-a-tete before leading their ministers to deliberate on how to improve on existing ties on education, health, hydrology and sanitation, among others.
Mr Ader, who is accompanied by his wife, Mrs Anita Herczegh, was met at the Jubilee House by President Akufo-Addo and the First Lady, Mrs Rebecca Akufo-Addo, after the all-white-dressed Ghana Navy Officers treated the visiting President to a guard of honour and 21-gun salute.
Mr Ader said other areas of the cooperation which would be given a boost was water management, agriculture, and education.
He noted that during the past last five years, Ghanaian students studying in Hungary had increased by three and a half fold.
He expressed the hope that those students who studied architecture, engineering, medicine, agriculture would put to good use their knowledge for the benefit of the people of Ghana and for the realisation of their aspirations.
President Akufo-Addo acknowledged Hungary's achievements in technology, hydrology and sanitation, Agribusiness and food technology, among others, stressing that these were all relevant to
Ghana in its quest for rapid transformation of its economy.
He expressed the optimism that the visit would lay a strong foundation for a new and mutually beneficial cooperation.
He said he expected the ties between the two nations would be replicated in the relations between the private sector which would enhance exchanges between them.
President Akufo-Addo mentioned the memorandum of understanding (MoU) signed between the two countries in the area of sanitation and water management resources.
That, he said, led to a 50-million-euro facility from the EXIM Bank of Hungary to be granted to one of the leading private sector operators of Ghana, the JUSPON Group of Companies, for the building of waste treatment plants, in Kumasi, Takoradi, Tamale, and Dambai.
President Akufo-Addo said one of the essential pillars of the relationship must be the mutual nurturing of the youth because they were the ones to inherit the legacy of the current cooperative endeavours.
"It is my desire that we pay close attention to educational and cultural exchanges between Ghanaian and Hungarian youth to enable us to deepen the understanding between our peoples and make better use of the strong friendship," he added.
He commended Hungary for the large number of scholarship it had been granting to Ghanaian youth to continue to study in that country.
President Akufo-Addo said as members of the United Nations, the two countries should continue to cooperate effectively to create a safer and more secure world and oppose resolutely the spread of terrorism and violent extremism in Africa, Europe and in the world.

redpajama_set_name: RedPajamaCommonCrawl


Everyday you can get a new, topical joke to claim as your own.
Researchers have found that watching television boosts toodlers' chances of developing attention deficit problems, as well as aggressiveness and obesity. But researchers are quick to mention that these side effects are acceptable if watching TV will keep those little shits occupied.


Red White & Rails – a wine tasting adventure aboard a vintage railcar!
Join the South Coast Railroad Museum for a wine tasting trip on the vintage railcar Acoma! Ride from Santa Barbara to San Luis Obispo and back enjoying a wine tasting from Grassini Family Vineyards, paired with chocolate from the Chocolate Gallery! Trip includes guided historical interpretation as well as live music.
If you are interested in booking, we encourage you to book as soon as possible – seats aboard this Vintage Rail Trip will not last!
We depart Santa Barbara at approximately 9:45am, and from Goleta at approximately 10:00am. Train travels up the Pacific Coast to San Luis Obispo, with an approximate 5:00pm return to Goleta and Santa Barbara.
Cost is $125 per person. Acoma seats just 25 guests total. Book now! Please contact mia@goletadepot.org with any questions!


Tait Communications delivers unified critical communications solutions together with our partners. Our dedication to deliver value to clients across the globe motivates us to work with best-in-class partners. We know that our clients are depending on us to deliver critical communication solutions that enable the operational outcomes they seek.
In the complex critical communications environment, our clients expect and understand the value of great partners and partnerships. We leverage a diverse set of best-of-breed partners to ensure that we can deliver outcomes that add value to your business.
We operate using open systems wherever possible because we understand that industry standard approaches make partnering and collaboration easier and more efficient. And when we are easy to partner with, it means everyone can spend more time and effort creating value for the client. An open standards environment facilitates our ability to co-create and innovate with partners, bringing new capabilities and solutions to market in a more agile fashion.
We are proud of the strong partner relationships we have. Together, we deliver world-class solutions for clients every day.
Strategic technology partners whose products and services support Tait Solutions.
Our ecosystem of partners, combined with our agile processes, enable Tait to be responsive in innovative ways and to create, test and deliver accredited turn-key solutions.
Learn how we provide integrated location solutions that enable our clients to safely and efficiently manage their field operations.
In emergency situations, good communication systems save lives. Read more on how we and our partners deliver seamless mission critical communications within budget.
Our systems enable the retrieval and analysis of audio and data from communications.


text:
Psilopsella uniseriata är en mossdjursart som beskrevs av Canu och Bassler 1927. Psilopsella uniseriata ingår i släktet Psilopsella och familjen Romancheinidae. Inga underarter finns listade i Catalogue of Life.

Källor

Mossdjur
uniseriata

redpajama_set_name: RedPajamaWikipedia


Zend_Http_Client provides an easy interface for preforming Hyper-Text Transfer Protocol (HTTP) requests. Zend_Http_Client supports most simple features expected from an HTTP client, as well as some more complex features such as HTTP authentication and file uploads. Successful requests (and most unsuccessful ones too) return a Zend_Http_Response object, which provides access to the response's headers and body (see this section).
The class constructor optionally accepts a URL as its first parameter (can be either a string or a Zend_Uri_Http object), and an array or Zend_Config object containing configuration options. Both can be left out, and set later using the setUri() and setConfig() methods.
Zend_Http_Client uses Zend_Uri_Http to validate URLs. This means that some special characters like the pipe symbol ('|') or the caret symbol ('^') will not be accepted in the URL by default. This can be modified by setting the 'allow_unwise' option of Zend_Uri to 'TRUE'. See this section for more information.
The constructor and setConfig() method accept an associative array of configuration parameters, or a Zend_Config object. Setting these parameters is optional, as they all have default values.
The request() method takes one optional parameter - the request method. This can be either GET, POST, PUT, HEAD, DELETE, TRACE, OPTIONS or CONNECT as defined by the HTTP protocol . For convenience, these are all defined as class constants: Zend_Http_Client::GET, Zend_Http_Client::POST and so on.
If no method is specified, the method set by the last setMethod() call is used. If setMethod() was never called, the default request method is GET (see the above example).
Adding GET parameters to an HTTP request is quite simple, and can be done either by specifying them as part of the URL, or by using the setParameterGet() method. This method takes the GET parameter's name as its first parameter, and the GET parameter's value as its second parameter. For convenience, the setParameterGet() method can also accept a single associative array of name => value GET variables - which may be more comfortable when several GET parameters need to be set.
While GET parameters can be sent with every request method, POST parameters are only sent in the body of POST requests. Adding POST parameters to a request is very similar to adding GET parameters, and can be done with the setParameterPost() method, which is similar to the setParameterGet() method in structure.
Note that when sending POST requests, you can set both GET and POST parameters. On the other hand, while setting POST parameters for a non-POST request will not trigger and error, it is useless. Unless the request is a POST request, POST parameters are simply ignored.
Zend_Http_Client provides methods of accessing the last request sent and last response received by the client object. Zend_Http_Client->getLastRequest() takes no parameters and returns the last HTTP request sent by the client as a string. Similarly, Zend_Http_Client->getLastResponse() returns the last HTTP response received by the client as a Zend_Http_Response object.
 See RFC 2616 - http://www.w3.org/Protocols/rfc2616/rfc2616.html.


text:
Resilience to Disturbance Despite Limited Dispersal and Self-Recruitment in Tropical Barrel Sponges: Implications for Conservation and Management
Bell, James J and Smith, David and Hannan, Danielle and Haris, Abdul and Jompa, Jamaludin and Thomas, Luke (2014) 'Resilience to Disturbance Despite Limited Dispersal and Self-Recruitment in Tropical Barrel Sponges: Implications for Conservation and Management.' PLoS ONE, 9 (3). e91635-e91635. ISSN 1932-6203
journal.pone.0091635.pdf - Published Version
While estimates of connectivity are important for effective management, few such estimates are available for reef invertebrates other than for corals. Barrel sponges are one of the largest and most conspicuous members of the coral reef fauna across the Indo-Pacific and given their large size, longevity and ability to process large volumes of water, they have a major role in reef functioning. Here we used a panel of microsatellite markers to characterise the genetic structure of two barrel sponge species, Xestospongia testudinaria and a currently undescribed Xestospongia species. We sampled across seven populations in the Wakatobi Marine National Park, SE Sulawesi (Indonesia) spanning a spatial scale of approximately 2 to 70 km, and present the first estimates of demographic connectivity for coral reef sponges. Genetic analyses showed high levels of genetic differentiation between all populations for both species, but contrasting patterns of genetic structuring for the two species. Autocorrelation analyses showed the likely dispersal distances of both species to be in the order of 60 and 140 m for Xestopongia sp. and Xestospongia testudinaria, respectively, which was supported by assignment tests that showed high levels of self-recruitment (>80%). We also found consistently high inbreeding coefficients across all populations for both species. Our study highlights the potential susceptibility of barrel sponges to environmental perturbations because they are generally long-lived, slow growing, have small population sizes and are likely to be reliant on self-recruitment. Surprisingly, despite these features we actually found the highest abundance of both barrel sponge species (although they were generally smaller) at a site that has been severely impacted by humans over the last fifty years. This suggests that barrel sponges exhibit environmental adaptation to declining environmental quality and has important implications for the management and conservation of these important reef species. © 2014 Bell et al.
Animals; Xestospongia; Body Size; Cluster Analysis; Confidence Intervals; Bayes Theorem; Genetics, Population; Conservation of Natural Resources; Tropical Climate; Microsatellite Repeats; Alleles; Geography; Principal Component Analysis; Indonesia; Genetic Variation; Animal Distribution
Faculty of Science and Health > Life Sciences, School of


This month's minihead had a bumper entry with 19 boats racing, including some special guests from Leeds. We had lovely conditions on a sunny March morning! Thanks to the Dev Men for organising the event with help from Victoria who did an incredible job running the kitchen!
Next month's event will take place on Sunday 9th April and will be run by the Junior Squad.


text:
\section{Introduction}\label{Sec:Intro}

This paper deals with Stefan--like problems governed by fractional diffusion equations (FDE). A classical Stefan problem is a problem  where a phase-change occurs, usually linked to  melting (change from solid to liquid) or freezing  (change from liquid to solid). In these problems the diffusion, considered as a heat flow, is expressed in terms of instantaneous local flow of temperature modeled by the Fourier Law. Therefore, the governing equations related to each phase are the well-known heat equations. There is also  a latent heat-type condition at the interface connecting the velocity of the free boundary and the heat flux of the temperatures in both phases known as ``Stefan condition''.
 A vast literature on Stefan problems is given in \cite{Alexiades, Cannon, Crank, Tarzia:biblio, Tarzia}.

For example,  the following is the mathematical formulation for a classical one-dimensional two-phase Stefan problem: \textsl{Find the triple $\left\{ u_1,u_2, s\right\}$ such that they have sufficiently regularity and they verify that:
}\begin{equation}{\label{St-Clasico}}
\begin{array}{llll}
     (i)  &   \frac{\partial}{\partial t}u_2(x,t)=\lambda^2_2\, \frac{\partial^2}{\partial x^2} u_2 (x,t), &   0<x<s(t), \,  0<t<T,  \, \,\\
     (ii)  &   \frac{\partial}{\partial t}u_1(x,t)=\lambda^2_1\, \frac{\partial^2}{\partial x^2} u_1 (x,t), &   x>s(t), \,  0<t<T,  \, \,\\
     (iii) &   u_1(x,0)=U_i, & 0\leq x,\\
       (iv)  &  u_2(0,t)=U_0,   &  0<t\leq T,  \\
         (v) & u_1(s(t),t)=u_2(s(t),t)=U_m, & 0<t\leq T, \\
  (vi) & \rho l \frac{d}{dt}s(t)=k_1\frac{\partial}{\partial x} u_1(s(t),t) -k_2\frac{\partial}{\partial x} u_2 (s(t),t), & 0<t\leq T, \\
	(vii) & s(0)=0,
                                             \end{array}
                                             \end{equation}
where $U_i<U_m<U_0$, $\lambda^2_j=\frac{k_j}{\rho c_j}$, $j=1$ (solid), $j=2$ (liquid) and we have assumed that  the thermophysical properties are constant as well as the free boundary can be represented by an increasing function of time.\\

Problem (\ref{St-Clasico}) is clearly governed by the heat equations $(\ref{St-Clasico}-i)$ and $(\ref{St-Clasico}-ii)$,  and
has a phase-change condition (namely the Stefan condition) given by equation $(\ref{St-Clasico}-vi)$. \\

When the governing  equations $(\ref{St-Clasico}-i)$ and $(\ref{St-Clasico}-ii)$, or the Stefan condition $(\ref{St-Clasico}-vi)$ are replaced by other equations involving fractional derivatives in problems like $(\ref{St-Clasico})$,  we will refer to them as fractional Stefan-like problems.

For example, the heat equation can be replaced by a fractional diffusion equation (FDE), which is closely linked to the study of anomalous diffusion.  A detailed explanation about the relation between anomalous diffusion and randon walk processes can be founded at the work done by Metzler and Klafter  \cite{MK:2000}. As we know, the diffusion equation is connected  to  the Brownian motion, where the mean square displacement (msd) of particles is  proportional to time. However, in Random Walks the msd is proportional to a power of  time. It is also interesting the approach given in \cite{BaFr:2005,GKS,Sax:1994} where it is suggested that anomalous diffusion could be caused by heterogeneities in the domain.

 For the relation between fractional diffusion equations and their applications,  we refer the reader to \cite{FM-Libro, Podlubny, Pskhu-Libro} and references therein where applications to the theory of linear viscoelasticity or thermoelasticity, among other, are presented. \\

In this paper, two approaches leading to subdiffusion are considered. The first one linked to
the mathematical interest as generalized operators which interpolates classical derivatives
(see \cite{Diethelm}), and  the second one related to Fourier's generalization laws (see \cite{Povstenko}).  These two approaches derived in two different formulations for the
FDE. In order to present them, let a function $u=u(x,t)$ be  defined for given  one-dimensional variables $x$ and time $t$.
 A first formulation for the FDE  given in terms of  fractional integrals (see \cite{Fujita:1989}) is given by:
\begin{equation}\label{FDE-IntFrac}
 _0I_t^\alpha u_{xx}(x,t) =  u(x,t) - u(x,0)
\end{equation}
where, $_0I^\alpha_t$ is the fractional integral of  Riemann--Liouville of order  {$\alpha$} in the $t-$variable defined as
\begin{equation*}
_{0}I^{\alpha}_t u(x,t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-\tau)^{\alpha-1} u(x,\tau){\rm d}\tau
\end{equation*}
for every $u$ such that $u(x,\cdot )\in L^1(0,T)$ for every $x>0$.
 Equation \eqref{FDE-IntFrac} is  derived also in \cite{MK:2000}, when a fractal time random walk is considered. As  it can be seen, no partial derivative in time is part of equation (\ref{FDE-IntFrac}),  but differenciating respect on time to both members we get a second formulation for a FDE
\begin{equation}\label{FDE-RL}
  \,_0^{RL}D_t^{1-\alpha}u_{xx}(x,t)= u_{t}(x,t),
\end{equation}
where $_0^{RL}D_t^{1-\alpha}$ is the fractional derivative of Riemann--Liouville in the $t-$variable defined  for every $\alpha \in (0,1)$ as
$$_{0}^{RL}D^{1-\alpha}_t u(x,t)=\frac{\partial }{\partial t}\, _{0}I^{\alpha}_t u(x,t)
= \frac{1}{\Gamma(\alpha)}\frac{\partial }{\partial t}\int^{t}_{0}(t-\tau)^{\alpha-1} u(x,\tau){\rm d}\tau $$
for every $u\in AC_t[0,T]=\left\{ u \;|\;  u(x, \cdot ) \, \text{ is absolutely continuous on }
[0,T] \text{ for every } x\in{\mathbb R}^+\right\}$.\\
Nevertheless,  when discussing about FDE associated to fractional time derivatives,
the reader may retract on the FDE for the Caputo  derivative, that is
\begin{equation}\label{FDE-Caputo}
_{0}^CD^{\alpha}_t u(x,t)= u_{xx}(x,t).
\end{equation}
Here, the partial time derivative has been replaced by a fractional derivative in the sense of Caputo respect on time. The Caputo derivative  $_0^{C}D_t^{\alpha}$ is defined for every $\alpha \in (0,1)$ as
$$\,^C_{0} D^{\alpha}_t \,u(x,t)= \left[ \, _{0}I^{1-\alpha}_t \left( u_t \right)  \right] (x,t) = \frac{1}{\Gamma(1-\alpha)}\displaystyle\int^{t}_{0}(t-\tau)^{-\alpha} u_t(x,\tau){\rm d}\tau$$
 for every $u\in AC_t[0,T]$.

As we said before, in this paper, problems like (\ref{St-Clasico}) governed by equations like (\ref{FDE-RL}) or (\ref{FDE-Caputo}) will be studied. The literature on fractional phase-change problems is rather scant. In \cite{GeVoMiDa:2013} a fractional two-phase moving-boundary problem is approximated by a scale Brownnian motion model for subdiffusion. In \cite{Voller:2014} sharp and diffuse interface models of fractional Stefan problems are discussed. In \cite{RoBoTa:2018} a formulation of a one-phase fractional phase-change problem is given arising a time dependence on the initial extreme of the fractional derivative.  When the starting time considered in the fractional derivative of the governing equation is equal to 0, the mathematical point of view becomes interesting because they admit  self-similar solutions in terms of the Wright functions (see \cite{GoLuMa:2000, JiMi:2009, Pa:2013,RoSa:2013, RoTa:2014}). It is worth noting that this kind of problems are not deduced as in \cite{RoBoTa:2018, VoFaGa:2013}.\\

This paper is a continuation of a previous work \cite{RoTa:2017-TwoDifferent}, related to  fractional   one-phase change problems. In Section \ref{Section:2} some basic definitions and properties on fractional calculus are given. In Section \ref{Section:3}, two fractional two-phase Stefan-like problems are considered, admitting both exact self-similar  solutions.  While the two governing equations are equivalent under certain assumptions for boundary-value-problems, when different ``fractional Stefan conditions'' are considered, the solutions obtained seem to be different. The uniqueness of the self-similar solution for one of the problems is obtained while it is an open problem for the other (see \cite{RoTa:2014}). Finally, numerical examples and graphics of the solutions are presented by
considering a dimensionless model in Section \ref{Section:4}.

\section{Basic definitions and properties}
\label{Section:2}
\begin{proposition}\label{propo frac}\cite{Diethelm} The following properties involving the fractional integrals and derivatives hold:
\begin{enumerate}
\item \label{RL inv a izq de I} The  fractional derivative of  Riemann--Liouville  is a left inverse operator of the fractional integral of Riemann--Liouville of the same order  $\alpha\in {\mathbb R}^+$. If $f \in AC[a,b]$, then
$$^{RL}_{a}D^{\alpha}\,_{a}I^{\alpha}f(t)=f(t)  \quad \text{for every }\, t\in (a,b)$$

\item\label{not inverse} The fractional integral of Riemann--Liouville is not, in general, a left inverse operator of the fractional derivative of Riemann--Liouville.\\

In particular, if $0<\alpha<1$, then
$ _{a}I^{\alpha}(^{RL}_{a}D^{\alpha}f)(t)=f(t) - \dfrac{_{a}I^{1-\alpha}f(a^+)}{\Gamma(\alpha)(t-a)^{1-\alpha}}.$

\item\label{caso part I inv de RL} If there exist some $\phi \, \in L^1(a,b)$ such that $f=\,_aI^\alpha \phi$, then
$$_{a}I^{\alpha}\,^{RL}_{a}D^{\alpha} f(t)=f(t)  \quad \text{for every } \, t\in (a,b).$$

\item\label{relacion RL-C} If  $f\in AC[a,b],$ then
$$ ^{RL}_{a}D^{\alpha}f (t)=\frac{f(a)}{\Gamma(1-\alpha)}(t-a)^{-\alpha}+\, ^C_{a} D^{\alpha}f(t).$$
\end{enumerate}
\end{proposition}

The fractional integral and derivatives of power functions can be easy calculated (see  e.g.  \cite{Podlubny}). In fact, for every $t\geq a$ we have that
\begin{equation}\label{Int-de-pot}
_aI^{\alpha}\left((t-a)^{\beta}\right)=\frac{\Gamma(\beta + 1)}{\Gamma(\beta+\alpha+1)}(t-a)^{\beta+\alpha }, \qquad \text{ for every } \quad  \beta>-1,
\end{equation}
and that
\begin{equation}\label{DRL-de-pot}
_a^{RL}D^{\alpha}\left((t-a)^{\beta}\right)=\begin{cases}\frac{\Gamma(\beta + 1)}{\Gamma(\beta-\alpha+1)}(t-a)^{\beta-\alpha} & \text{ if } \beta \neq \alpha - 1,\\
0 & \text{ if } \beta=\alpha-1.
\end{cases}
\end{equation}
In particular, if $\beta>0$, $_a^{RL}D^{\alpha}\left((t-a)^{\beta}\right)=\,_a^{C}D^{\alpha}\left((t-a)^{\beta}\right)$ due to Proposition \ref{propo frac} item \ref{relacion RL-C} and the Caputo derivative of $(t-a)^{\beta}$ is not defined for $-1<\beta<0$.

\begin{proposition} \cite{Samko} The following limits hold:
\begin{enumerate}
\item If we set $_aI^0=Id$ for the identity operator, then for every $ f \, \in L^1(a,b)$,
$$ \displaystyle\lim_{\alpha\searrow 0}\, _aI^\alpha f(t) =_aI^0f(t)=f(t), \qquad a.e.  $$
\item  For every $f \in AC[a,b]$, we have
$$ \displaystyle\lim_{\alpha\nearrow 1}\,_a^CD^\alpha f(t) = f'(t) \hspace{0.5cm} \text{and} \hspace{0.5cm}  \displaystyle\lim_{\alpha\searrow 1}\,_a^CD^\alpha f(t) = f'(t)-f'(a^+)\qquad \text{ for all } \, t\in (a,b). $$
\item  For every $f \in AC[a,b]$,
$$ \displaystyle\lim_{\alpha\nearrow 1}\,_a^{RL}D^\alpha f(t) = f'(t)\hspace{0.5cm} \text{and} \hspace{0.5cm} \displaystyle\lim_{\alpha\searrow 1}\,_a^{RL}D^\alpha f(t) = f'(t) \quad a.e. $$
\end{enumerate}
\end{proposition}

\begin{definition} For every $x\in {\mathbb R}$ , the \textit{Wright} function is defined as
\begin{equation}\label{W} W(x;\rho;\beta)=\sum^{\infty}_{k=0}\frac{x^{k}}{k!\Gamma(\rho k+\beta)} ,\quad  \rho>-1 \text{ and }  \beta\in {\mathbb R}.\end{equation}
An important particular case of the Wright function is the \textit{Mainardi} function defined by
$$ M_\rho (x)= W(-x,-\rho,1-\rho)=\sum^{\infty}_{n=0}\frac{(-x)^n}{n! \Gamma\left( -\rho n+ 1-\rho \right)}, \quad \,0<\rho<1.  $$
\end{definition}

\begin{proposition}\label{Props W} \cite{Pskhu-Libro, Wr1:1934}
  Let $\alpha > 0$, $\rho \in (0,1)$ and $\beta \in {\mathbb R}$. Then the next assertions follows:
\begin{enumerate}
\item \label{deriv comun} For every $x\in {\mathbb R}$ we have
$$ \frac{\partial}{\partial x} W(x,\rho,\beta) = W(x,\rho,\rho+\beta).$$
\item For every $x>0$ and $c>0$,
\begin{equation}\label{eq pskhu}
_0I^\alpha \left[ x^{\beta-1}W(-cx^{-\rho},-\rho, \beta)\right]=x^{\beta+\alpha-1}W(-cx^{-\rho},-\rho, \beta+\alpha) .
\end{equation}

\end{enumerate}
\end{proposition}
\begin{proposition}  \label{Propo monot Wrights}\cite{RoTa:2017-TwoDifferent,Wr1:1934} For every $\beta \geq 0$, $\rho \in (0,1)$:
\begin{enumerate}
\item \label{W dec} The Wright function  $W\left(-\cdot,-\rho,\beta\right) $ is positive and strictly decreasing in ${\mathbb R}^+$.
\item \label{rel-Wrights} For every $x\geq 0$ the following equality holds
$$\rho x W\left(-x,-\rho,\beta-\rho \right) = W\left(-x,-\rho,\beta- 1\right)+(1-\beta) W\left(-x,-\rho,\beta\right). $$
\item \label{desig-W} If, in addition  $0< \rho\leq \mu < \delta$, then for every $x>0$ the following inequality holds
\begin{equation}
\Gamma(\delta)W(-x, -\rho, \delta)<\Gamma(\mu) W(-x,-\rho, \mu).
\end{equation}

\end{enumerate}
\end{proposition}

            %

\begin{proposition}  \cite{Wr2:1940} \label{Wrights-x->infty}  For every $\beta \geq 0$ and  $\rho \in (0,1)$ the following limit holds
$$ \lim\limits_{x\rightarrow \infty}  W(-x,-\rho, \beta)=0. $$
\end{proposition}
\begin{proposition}\label{convergencia-W}\cite{RoSa:2013, RoTa:2017-TwoDifferent}\label{conv M y W cuando al tiende a 1}Let $x\in {\mathbb R}^+_0$ be. Then  the following limits hold:\\
\begin{equation}\label{limite-M}\lim\limits_{\alpha\nearrow 1}M_{\alpha/2}\left(2x\right)=\lim\limits_{\alpha\nearrow 1}W\left(-2x,-\frac{\alpha}{2},1-\frac{\alpha}{2}\right)= M_{1/2}(2x)=\frac{e^{-x^2}}{\sqrt{\pi}},
\end{equation}

\begin{equation}\label{limite-W}
\lim\limits_{\alpha\nearrow 1}W\left(-2x,-\frac{\alpha}{2},\frac{\alpha}{2}\right)=\frac{e^{-x^2}}{\sqrt{\pi}},
\end{equation}
\begin{equation}\label{limite-erf}\lim\limits_{\alpha\nearrow 1}\left[1-W\left(-2x,-\frac{\alpha}{2}, 1\right)\right]={\rm erf}(x),
\end{equation}
and
\begin{equation}\label{limite-erfc}\lim\limits_{\alpha\nearrow 1}\left[W\left(-2x,-\frac{\alpha}{2}, 1\right)\right]={\rm erfc}(x),
\end{equation}
where $erf(\cdot)$ is the error function defined by ${\rm erf}(x)=\frac{2}{\sqrt{\pi}}\displaystyle\int_0^xe^{-z^2}dz$ and ${\rm erfc}(\cdot)$ is the complementary error function defined by ${\rm erfc}(x)=1-{\rm erf}(x)$. Moreover, the convergence is uniform over compact sets.
\end{proposition}

\begin{proposition} The  fractional initial-boundary-value problems (\ref{Caputo-IC}) and (\ref{RL-IC}) for the quarter plane are equivalent if there exists $\beta>0$ and $\delta >0$ such that  $\beta<\alpha<1$  and $u_{xx}(x,\cdot)$ is an $O(t^{-\beta})$ in $(0,\delta)$:
\begin{equation}{\label{Caputo-IC}}
\begin{array}{llll}
     (i)  &  _0^CD^\al_t u(x,t)= \frac{\partial^2}{\partial x^2} u (x,t), &   0<x, \,  0<t,  \, \,\\
     (ii)   & u(x,0)=u_0(x), & 0\leq x,\\
       (iii)  &  u(0,t)=g(t),   &  0<t,
                                                      \end{array}
                                             \end{equation}

\begin{equation}{\label{RL-IC}}
\begin{array}{llll}
     (i)  &  \frac{\partial}{\partial t} u(x,t)=  \, _0^{RL}D^{1-\al}_t \left( \frac{\partial^2}{\partial x^2} u (x,t)\right) , &   0<x, \,  0<t,  \, \,\\
     (ii)   & u(x,0)=u_0(x), & 0\leq x,\\
       (iii)  &  u(0,t)=g(t),   &  0<t,
                                                      \end{array}
                                             \end{equation}
\end{proposition}
\proof Let $u=u(x,t)$ be a function satisfying equation $(\ref{Caputo-IC}-i)$. Applying $_0^{RL}D^{1-\al}_t$ to both sides and using Proposition \ref{propo frac} item \ref{RL inv a izq de I} we get $(\ref{RL-IC}-i)$. \\
Let now, for the inverse suppose that $u$ satisfies equation $(\ref{RL-IC}-i)$. Applying $_0I^{1-\alpha}_t$ to both sides and using  Proposition \ref{propo frac} item \ref{not inverse} yields that
 \begin{equation}\label{eq1}
 _0^CD^\al_t u(x,t)= \frac{\partial^2}{\partial x^2} u (x,t)-
\frac{\displaystyle\lim\limits_{t\searrow 0}\, _0I^{\alpha}\left( \frac{\partial^2}{\partial x^2} u (x,t)\right)}{\Gamma(1-\alpha)t^{\alpha}}, \quad 0<x, \quad 0<t.
\end{equation}
Now, for every $x$ fixed we have that  $u_{xx}(x,\cdot)$ is an $O(t^{-\beta})$ in $(0,\delta)$, then for $t>0$ small it holds that
\begin{equation}\label{Des-1} -C\tau^{-\beta}\leq u_{xx}(x,\tau)\leq C\tau^{-\beta},\quad 0<\tau\le t< \delta.
\end{equation}
Multiplying by $\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}$ in (\ref{Des-1}),  integrating between 0 and $t$ and applying formula (\ref{Int-de-pot})  yields that

\begin{equation}\label{Des-2} -C\frac{\Gamma(1-\beta) t^{\alpha-\beta}}{\Gamma(\alpha-\beta + 1)}\leq \, _0I^{\alpha}_t \,u_{xx}(x,t) \leq C\frac{\Gamma(1-\beta) t^{\alpha-\beta}}{\Gamma(\alpha-\beta + 1)},\quad \quad t<\delta.
\end{equation}
Taking the limit when $ t$ tends to zero in $(\ref{Des-2})$ and being $\beta<\alpha$ we conclude that
equation $(\ref{Caputo-IC}-i)$ holds as we wanted to see.
\endproof

\begin{remark} Equations (\ref{Caputo-IC}-i) and (\ref{RL-IC}-i) has been treated as equivalent in
literature, as it can be seeing at \cite{FM-Libro,   MK:2000, Povstenko}, but the condition
\begin{equation}\label{cond FDE eq}
\displaystyle\lim\limits_{t\searrow 0}\, _0I^{\alpha}\left( \frac{\partial^2}{\partial x^2} u (x,t)\right)=0
\end{equation}
must be considered and should not be forget it.
\end{remark}

\begin{remark}
It is easy to check that the following functions  verifies equation  (\ref{Caputo-IC}-i) and
(\ref{RL-IC}-i) (we have taken $\lambda=1$ without loss of generality)
\begin{equation}\label{solFDE-1}
w_1(x,t)=x^2+ \frac{2}{\Gamma(\alpha+1)}t^{\alpha}.
\end{equation}
\begin{equation}\label{solFDE-2}
w_2(x,t)=E_\alpha(t^{\alpha})\exp\left\{-x\right\}
\end{equation}
and
\begin{equation}\label{solFDE-3}
w_3(x,t)=W\left(-\frac{x}{t^{\alpha/2}},-\frac{\alpha}{2}, 1\right).
\end{equation}
The condition (\ref{cond FDE eq}) trivially holds  for function $w_1$ and $w_2$ and it is no difficult to check it for  $w_3$ (by differenciating first and using Proposition \ref{Props W} then).
\end{remark}

\section{The Fractional Stefan-like Problems}
\label{Section:3}

In this section, two fractional Stefan-like problems admitting both explicit self-similar solutions will be treated. Before that, some clarification about the used terminology is presented.

We refer to  fractional Stefan problems when the governed equations in such problem are derived  from physical assumptions, like considering memory fluxes.

For example, suppose that a process of  melting of a  semi--infinite slab  ($0\leq x<\infty$) of  some material  is taking place, and  the flux involved is a flux with memory. The melt temperature is $U_m$, and a constant temperature $U_0>U_m$ is imposed on the fixed face $x=0$.
Let  $u_1=u_1(x,t)$ and $u_2=u_2(x,t)$ be the temperatures at the solid and liquid phases respectively. Let $J_1=J_1(x,t)$ and $J_2=J_2(x,t)$ be the respective functions for the fluxes at position $x$ and time $t$ and  let $x=s(t)$ be  the function representing the (unknown) position of the free boundary at time $t$. Suppose further that:\\
$(i)$ All the thermophysical parameters are constants.\\
$(ii)$ The function $s$ is an increasing function and consequently, an invertible function.\\
$(iii)$ $J_1 $ and $J_2$ are fluxes modeling the material with  memory which verifies that  ``\textsl{the  weighted sum of the fluxes back in time at the current time, is proportional to the gradient of temperature}'', that is, the following equations hold
\begin{equation}\label{J_1-elegido}
\nu_{\alpha}\,_{0}I^{1-\alpha}_tJ_1(x,t)=-k_1\frac{\partial u_1}{\partial x}(x,t)
\end{equation}
and
\begin{equation}\label{J_2-elegido}
\nu_{\alpha}\,_{h(x)}I^{1-\alpha}_tJ_2(x,t)=-k_2\frac{\partial u_2}{\partial x}(x,t)
\end{equation}
where the initial time in the fractional integral (\ref{J_2-elegido}) is given by function  $h$ which gives us the time when the phase change occurs. That is,
$$ t=h(x)=s^{-1}(x) \qquad (\text{i.e. } \,\, x=s(t) )$$

 The number $\nu_{\alpha}$ is a parameter with physical dimension (see\eqref{med-nu_al}) such that
\begin{equation}\label{nu-al}
\displaystyle\lim_{\alpha\nearrow 1}\nu_{\alpha}=1, \end{equation}  which has been added in order to preserve the consistency with respect to the units of measure in equations  (\ref{J_1-elegido}) and (\ref{J_2-elegido}). Also, the parameter
\begin{equation}\label{mu-alpha} \mu_\alpha=\frac{1}{\nu_\alpha}
\end{equation}
will be used in the following equations. More details about these parameters are given in Section 4.    \\

Making an analogous reasoning for the two-phase free-boundary problem, than the one made in \cite{RoBoTa:2018} for the one--phase free--boundary problem, the mathematical model for the problem described above is given by

\begin{equation}{\label{Fr-Stefan-Problem}}
\begin{array}{llll}
     (i)  &   \frac{\partial}{\partial t}u_2(x,t)=\lambda_2^2\,\mu_{\alpha_2}\frac{\partial}{\partial x} \left( \,_{h(x)}^{RL}D^{1-\alpha}_t \left( \frac{\partial}{\partial x} u_2 (x,t)\right) \right), &   0<x<s(t), \,  0<t<T,  \, \,\\
     (ii)  &   \frac{\partial}{\partial t}u_1(x,t)=\lambda_1^2\,\mu_{\alpha_1}\frac{\partial}{\partial x} \left( _{0}^{RL}D^{1-\alpha}_t \left( \frac{\partial}{\partial x} u_1 (x,t)\right) \right), &   x>s(t), \,  0<t<T,  \, \,\\
     (iii) &   u_1(x,0)=U_i, & 0\leq x,\\
      (iv) &   u_2(0,t)=U_0,   &  0<t\leq T,  \\
         (v) & u_1(s(t),t)=u_2(s(t),t)=U_m, & 0<t\leq T, \\
  (vi) & \rho l\frac{d}{dt}s(t)= k_1\mu_{\alpha_1}\left. _0^{RL}D^{1-\al}_t \frac{\partial}{\partial x} u_1(x,t)\right|_{(s(t)^+,t)}  & \\
  & \hspace{2.5cm} - k_2\mu_{\alpha_2}\left. _{h(x)}^{RL}D^{1-\alpha}_t \frac{\partial}{\partial x} u_2 (x,t)\right|_{(s(t)^-,t)}, & 0<t\leq T. \\
  s(0)=0
                                             \end{array}
                                             \end{equation}

where $U_i<U_m<U_0$ and $\mu_{\alpha}=\frac{1}{\nu_{\alpha}}$, (note that the parameter $\mu_\alpha$ can be the same in equations \eqref{Fr-Stefan-Problem}$-i$ and \eqref{Fr-Stefan-Problem}$-ii$,
and without loss of generality we will take from now on that $\mu_{\alpha_2}=\mu_{\alpha_1}$).\\
Note that self-similar solutions to problem (\ref{Fr-Stefan-Problem}) had not been yet founded, due to the difficulty imposed by the variable button limit in the fractional derivative for the liquid phaace.
As it was said at the beginning of this section, this paper deals with Stefan-like problems admitting explicit self-similar solutions. These problems come from the assumption of consider the button limit $t_0=0$ in the fractional time derivatives in the Caputo or Riemann--Liouville sense.    \\

\textbf{The Stefan-Like Problem for the Caputo derivative. }The next problem was treated in \cite{RoTa:2014} and can be obtained by replacing all the times derivatives in (\ref{St-Clasico}) by fractional derivatives in the Caputo sense of order $\alpha \in (0,1)$, i.e.

\begin{equation}{\label{St-like-2ph-Caputo}}
\begin{array}{llll}
     (i)  &  _0^CD^\al_t u_2(x,t)=\lambda^2_{\alpha_2}\, \frac{\partial^2}{\partial x^2} u_2 (x,t), &   0<x<s(t), \,  0<t<T,  \, \,\\
     (ii)  &   _0^CD^\al_t u_1(x,t)=\lambda^2_{\alpha_1}\, \frac{\partial^2}{\partial x^2} u_1 (x,t), &   x>s(t), \,  0<t<T,  \, \,\\
     (iii) &   u_1(x,0)=U_i, & 0\leq x,\\
     (iv)  &   u_2(0,t)=U_0,   &  0<t\leq T,  \\
         (v) & u_1(s(t),t)=u_2(s(t),t)=U_m, & 0<t\leq T, \\
  (vi) & \rho l _0^CD^\al_t s(t)=k_{\alpha_1} \frac{\partial}{\partial x} u_1(s(t)^+,t) - k_{\alpha_2}\frac{\partial}{\partial x} u_2 (s(t)^-,t), & 0<t\leq T, \\
(vii) & s(0)=0. &
                                             \end{array}
                                             \end{equation}

where $U_i<U_m<U_0$, $\lambda_{\alpha_i}$  are positive parameters named as ``subdiffusion coefficients'' given by
$  \lambda_{\alpha_i}=\lambda_i\sqrt{\mu_{\alpha}}$ for  $i=1,2,$
and $ k_{\alpha_i}$ are positive parameters named as ``subdiffusion thermal conductivities'' given by $k_{\alpha_i}=k_i \mu_{\alpha}$, $i=1,2$.

\begin{definition}\label{Def sol St} The triple $\{u_1, u_2,s\} $ is a solution to  problem $(\ref{St-like-2ph-Caputo})$  if the following conditions are satisfied

\begin{enumerate}
    \item $u_1$  is continuous in the region $\mathcal{R_T}=\left\{(x,t)\colon 0\leq x \leq s(t), \, 0<t\leq T \right\}$ and at  the point $(0,0) $, $u_1 $ verifies that
       $$ 0\leq \underset{(x,t)\rightarrow (0,0)}{\liminf}u_1(x,t)\leq \underset{(x,t)\rightarrow (0,0)}{\limsup } u_1(x,t)<+\infty .$$
			\item $u_2$  is continuous in the region $\left\{(x,t)\colon  x > s(t), \, 0<t\leq T \right\}$ and at  the point $(0,0) $, $u_2 $ verifies that
       $$ 0\leq \underset{(x,t)\rightarrow (0,0)}{\liminf}u_2(x,t)\leq \underset{(x,t)\rightarrow (0,0)}{\limsup } u_2(x,t)<+\infty .$$

	\item  $u_1\in $ $C((0,\infty) \times(0,T) )\cap C^2_x((0,\infty) \times(0,T))$, such that $u_{1} \in \, AC_t[0,T]$

\item  $u_2\in $ $C((0,\infty) \times(0,T) )\cap C^2_x((0,\infty) \times(0,T))$, such that $u_{2} \in \, AC_t[0,T]$ 	.
  \item $s \in AC[0,T]$.

    \item $u_1$, $u_2$ and $s$ satisfy $(\ref{St-like-2ph-Caputo})$.

		\end{enumerate}
\end{definition}

\begin{theorem}\cite{RoTa:2014} A self-similar solution to poblem (\ref{St-like-2ph-Caputo}) is given by
\begin{equation}\label{sol Caputo-2f}
\left\{\begin{array}{l} u_2(x,t)=U_0-\frac{ U_0-U_m}{1-W\left(-2\xi_\alpha \lambda,-\frac{\alpha}{2},1\right)}\left[1-W\left(-\frac{x}{\lambda_{\alpha_2} t^{\alpha/2}},-\frac{\alpha}{2},1\right)\right] \\
        u_1(x,t)=U_i+\frac{U_m-U_i }{W\left(-2\xi_\alpha,-\frac{\alpha}{2},1\right)}W\left(-\frac{x}{\lambda_{\alpha_1} t^{\alpha/2}},-\frac{\alpha}{2},1\right) \\
				 s(t)=2\xi_\alpha  \lambda_{\alpha_1} t^{\alpha/2} \end{array}\right.
\end{equation}
 where $\xi_\alpha$ is \textbf{a} solution to the equation
\begin{equation}\label{eq xi}
\frac{k_{\alpha_2}(U_0-U_m)\Gamma(1-\frac{\alpha}{2}) }{ \lambda_{\alpha_2}} F_2(2\lambda x)- \frac{k_{\alpha_1}(U_m-U_i) \Gamma(1-\frac{\alpha}{2})}{\lambda_{\alpha_1}}F_1(2x)=\Gamma\left(1+\frac{\alpha}{2}\right)\lambda_{\alpha_1} \rho l 2 x, \, x>0   \end{equation}

where  $ \lambda=\frac{\lambda_{\alpha_1}}{\lambda_{\alpha_2}}=\frac{\lambda_1\sqrt{\mu_\alpha}}{\lambda_2\sqrt{\mu_\alpha}} = \frac{\alpha_1}{\alpha_2}>0$,  and  $F_1:{\mathbb R}^+_0\rightarrow {\mathbb R}$ and $F_2:{\mathbb R}^+_0\rightarrow {\mathbb R}$
are the functions defined by
 \begin{equation}\label{F1-y-F2}
F_1(x)=\frac{M_{\alpha/2}(x)}{W\left(-x,-\frac{\alpha}{2},1\right)}  \quad \text{ and } \quad
F_2(x)=\frac{M_{\alpha/2}(x)}{1-W\left(-x,-\frac{\alpha}{2},1\right)}  .
\end{equation}

\end{theorem}

\bigskip
\begin{note} The uniqueness of solution to equation $(\ref{eq xi})$ is still an open problem. However, the uniqueness of similarity solution will be achived next for the Riemann--Liouville Stefan--like problem.
\end{note}

\noindent \textbf{The Stefan-Like Problem for the Riemann--Liouville derivative.} Consider now the following problem:
\begin{equation}{\label{St-like-2ph-RL}}
\begin{array}{llll}
     (i)  &   \frac{\partial}{\partial t}w_2(x,t)=\lambda^2_{\alpha_2}\,\frac{\partial}{\partial x} \left( _0^{RL}D^{1-\al}_t \left( \frac{\partial}{\partial x} w_2 (x,t)\right) \right), &   0<x<r(t), \,  0<t<T,  \, \,\\
     (ii)  &   \frac{\partial}{\partial t}w_1(x,t)=\lambda^2_{\alpha_1}\,\frac{\partial}{\partial x} \left( _0^{RL}D^{1-\al}_t \left( \frac{\partial}{\partial x} w_1 (x,t)\right) \right), &   x>r(t), \,  0<t<T,  \, \,\\
     (iii) &   w_1(x,0)=U_i, & 0\leq x,\\
       (iv)  &  w_2(0,t)=U_0,   &  0<t\leq T,  \\
         (v) & w_1(r(t),t)=w_2(r(t),t)=U_m, & 0<t\leq T, \\
  (vi) & \rho l \frac{d}{dt}r(t)= \left. k_{\alpha_1} \,_0^{RL}D^{1-\al}_t \frac{\partial}{\partial x} w_1(x,t)\right|_{(r(t)^+,t)}  & \\
 &	\hspace{3cm} -k_{\alpha_2} \left. _0^{RL}D^{1-\al}_t \frac{\partial}{\partial x} w_2 (x,t)\right|_{(r(t)^-,t)}, & 0<t\leq T,\\
			(vii)	& r(0)=0. &\\
                                             \end{array}
                                             \end{equation}
where, as before,  $U_i<U_m<U_0$, $  \lambda_{\alpha_i}=\lambda_i\sqrt{\mu_{\alpha}}$ for  $i=1,2,$ and  $k_{\alpha_i}=k_i \mu_{\alpha}$, $i=1,2$.

\begin{remark}\label{cambio de limites en fr-St-Cond} The expression $\left. k_{\alpha_1} \,_0^{RL}D^{1-\al}_t \frac{\partial}{\partial x} w_1(x,t)\right|_{(r(t)^+,t)} $ is equivalent to
\begin{equation}\label{intercambio-lim-1} \lim\limits_{x\rightarrow r(t)^+} k_{\alpha_1}\,_0^{RL}D^{1-\al}_t\, \frac{\partial}{\partial x} w_1(x,t),  \end{equation}
which should not coincide with
\begin{equation}\label{intercambio-lim-2} k_{\alpha_1}\,_0^{RL}D^{1-\al}_t\, \left( \lim\limits_{x\rightarrow r(t)^+} \frac{\partial}{\partial x} w_1(x,t)\right).  \end{equation}
\end{remark}

\begin{definition}\label{Def sol St} The triple $\{w_1, w_2,r\} $ is a solution of  problem $(\ref{St-like-2ph-RL})$  if the following conditions are satisfied

\begin{enumerate}
    \item $w_1$  is continuous in the region $\mathcal{R_T}=\left\{(x,t)\colon 0\leq x \leq s(t), \, 0<t\leq T \right\}$ and at  the point $(0,0) $, $u_1 $ verifies that
       $$ 0\leq \underset{(x,t)\rightarrow (0,0)}{\liminf}w_1(x,t)\leq \underset{(x,t)\rightarrow (0,0)}{\limsup } w_1(x,t)<+\infty .$$
			\item $w_2$  is continuous in the region $\left\{(x,t)\colon  x > r(t), \, 0<t\leq T \right\}$ and at  the point $(0,0) $, $w_2 $ verifies that
       $$ 0\leq \underset{(x,t)\rightarrow (0,0)}{\liminf}w_2(x,t)\leq \underset{(x,t)\rightarrow (0,0)}{\limsup } w_2(x,t)<+\infty .$$

	\item  $w_1\in $ $C((0,\infty) \times(0,T) )\cap C^2_x((0,\infty) \times(0,T))$, such that $w_{1x} \in
	AC_t(0,T))$.
\item  $w_2\in $ $C((0,\infty) \times(0,T) )\cap C^2_x((0,\infty) \times(0,T))$, such that $w_{2x} \in 	AC_t[0,T]$ 	.
  \item $r \in C^1(0,T)$.
		\item There exist $\left.^{RL}_{0}D^{1-\alpha}_t \frac{\partial}{\partial x} w_2 (x,t)\right|_{(s(t)^+,t)}$
		and $\left.^{RL}_{0}D^{1-\alpha}_t \frac{\partial}{\partial x} w_1 (x,t)\right|_{(r(t)^-,t)}$ for all
		$t \in (0,T]$.
    \item $w_1$, $w_2$ and $s$ satisfy $(\ref{St-like-2ph-RL})$.

		\end{enumerate}
\end{definition}

\begin{theorem} An explicit solution for the two-phase fractional Stefan-like  problem (\ref{St-like-2ph-RL}) is given by
\begin{equation}\label{SOL-St-like-2ph-RL}
\left\{\begin{array}{l} w_2(x,t)=U_0-\frac{U_0-U_m}{1-W\left(-2\eta_\alpha \lambda,-\frac{\alpha}{2},1\right)}\left[1-W\left(-\frac{x}{\lambda_{\alpha_2} t^{\alpha/2}},-\frac{\alpha}{2},1\right)\right]\\
        w_1(x,t)=U_i+\frac{U_m-U_i}{W\left(-2\eta_\alpha,-\frac{\alpha}{2},1\right)}  W\left(-\frac{x}{\lambda_{\alpha_1} t^{\alpha/2}},-\frac{\alpha}{2},1\right) \\
				 r(t)=2\eta_\alpha  \lambda_{\alpha_1} t^{\alpha/2} \end{array}\right.
\end{equation}
 where $\eta_\alpha$ is the unique positive  solution to the equation
\begin{equation}\label{eq eta}
\frac{k_{\alpha_2}(U_0-U_m)}{ \lambda_{\alpha_1}\lambda_{\alpha_2}}G_2(2\lambda x) -\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2 } G_1(2x)
 =\left(\rho l + \frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2} \right)2x,
 \end{equation}
where  $ \lambda=\frac{\lambda_{\alpha_1\sqrt{\mu_\alpha}}}{\lambda_{\alpha_2}\sqrt{\mu_\alpha}}=\frac{\lambda_1}{\lambda_2}>0$, $U_i<U_m<U_0$  and  $G_1:{\mathbb R}^+_0\rightarrow {\mathbb R}$ and $G_2:{\mathbb R}^+_0\rightarrow {\mathbb R}$
are the functions defined by
 \begin{equation}\label{G1-y-G2}
G_1(x)=\frac{W\left(- x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)}{W\left(- x,-\frac{\alpha}{2},1\right)}  \quad \text{ and } \quad      G_2(x)=\frac{2/\alpha W\left(-x,-\frac{\alpha}{2},\frac{\alpha}{2}\right)  }{1- W\left(-x,-\frac{\alpha}{2},1\right)}.
\end{equation}

\end{theorem}
\proof
Let the functions

\begin{equation}{\label{sol-prop} }
\begin{array}{rcl}
w_i\colon  {\mathbb R}^+_0 \times (0,T)& \rightarrow & {\mathbb R} \\
(x,t) & \rightarrow  & w_i(x,t) = A_i + B_i \left[
1- W\left(-\frac{x}{\lambda_{\alpha_i} t^{\alpha/2}},-\frac{\alpha}{2}, 1 \right)\right]
\end{array}
\end{equation}

be the proposed solutions for $i=1,2.$ Rewriting expression  (\ref{eq pskhu}) for the variable $t$  and taking $c=\frac{x}{\lambda_{\alpha_i}}$  gives
\begin{equation}{\label{eq-pskhu-t}}
_0I^\alpha_t\,t^{\beta-1}W\left(-\frac{x}{\lambda_{\alpha_i}}t^{-\rho},-\rho, \beta\right)=t^{\beta+\alpha-1}W\left(-\frac{x}{\lambda_{\alpha_i}}t^{-\rho},-\rho, \beta+\alpha\right).
\end{equation}

Then, by using (\ref{eq-pskhu-t}) for $\beta=1-\frac{\alpha}{2}$ and  Proposition \ref{Props W}  it is easy to check that $w_i$ verifies equations  $(\ref{St-like-2ph-RL}-i)$ and $(\ref{St-like-2ph-RL}-ii)$ respectively for $i=1,2.$\\

From condition $(\ref{St-like-2ph-RL}-v)$ we deduce that $r(t)$ must be proportional to $t^{\alpha/2}$. Therefore we set
\begin{equation}\label{r}
r(t)=2\eta_\alpha \lambda_{\alpha_1} t^{\alpha/2}, \quad t \geq 0
\end{equation}
where $\eta_\alpha$ is a constant to be determined and $\lambda_{\alpha_1}$ was added for simplicity in the next calculations. Now, from conditions $(\ref{St-like-2ph-RL}-iii)$, $(\ref{St-like-2ph-RL}-iv)$ and $(\ref{St-like-2ph-RL}-v)$ it holds that
$$\begin{array}{ll}
 A_1= U_i+\frac{U_m-U_i}{W\left(-2\eta_\alpha,-\frac{\alpha}{2}, 1\right)},
& B_1= -\frac{U_m-U_i}{W\left(-2\eta_\alpha,-\frac{\alpha}{2}, 1\right)} \\
 A_2= U_0,
& B_2= -\frac{U_0-U_m}{1-W\left(-2\eta_\alpha \lambda,-\frac{\alpha}{2}, 1\right)}
\end{array} $$
As before, by considering (\ref{eq-pskhu-t}) for $\beta=1-\frac{\alpha}{2}$ and Proposition \ref{Props W},
it holds that

$$\hspace{-10cm} _0^{RL}D^{1-\al}_t w_{i_x}(x,t)= $$
\begin{equation}\label{DRL-w_x}
 \frac{B_i \alpha/2}{\lambda_{\alpha_1} \lambda_{\alpha_i}  t^{1-\alpha/2} } W\left(-\frac{x}{\lambda_{\alpha_i} t^{\alpha/2}},-\frac{\alpha}{2}, 1+\frac{\alpha}{2} \right)+ \frac{B_i \alpha/2}{\lambda_{\alpha_1} \lambda_{\alpha_i}}\frac{x}{t}W\left(-\frac{x}{\lambda_{\alpha_i} t^{\alpha/2}},-\frac{\alpha}{2}, 1 \right), \quad i=1,2.
\end{equation}

Then  replacing $(\ref{DRL-w_x})$ and $(\ref{r})$ in equation $(\ref{St-like-2ph-RL}-vii)$, and evaluating the limits following (\ref{intercambio-lim-1}) it yields that $\eta_\alpha $ must verify the next equality \\

\begin{equation}
\begin{split} \rho l 2 \eta_\alpha  \lambda_{\alpha_1}  = -\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2}
     \frac{ W\left(-2\eta_\alpha,-\frac{\alpha}{2}, 1+\frac{\alpha}{2} \right)}{ W\left(-2\eta_\alpha,-\frac{\alpha}{2}, 1 \right)} - \frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2}2\eta_\alpha - \\
+\frac{k_{\alpha_2}(U_0-U_m)}{\lambda_{\alpha_1} \lambda_{\alpha_2}} \frac{ W\left(-2\lambda \eta_\alpha,-\frac{\alpha}{2}, 1+\frac{\alpha}{2} \right)}{1-  W\left(-\lambda 2\eta_\alpha,-\frac{\alpha}{2}, 1 \right)} + \frac{k_{\alpha_2}(U_0-U_m)}{\lambda_{\alpha_1}\lambda_{\alpha_2}}\frac{2 \lambda \eta_\alpha W\left(-\lambda 2\eta_\alpha,-\frac{\alpha}{2}, 1 \right)}{1-  W\left(-\lambda_\alpha 2\eta_\alpha,-\frac{\alpha}{2}, 1 \right)}.
\end{split}
\end{equation}
which leads to conclude that $\left\{w_1,w_2,r\right\}$ is a solution to (\ref{St-like-2ph-RL}) if and only if $\eta_\alpha $ is a solution to  the equation

\begin{equation}\label{eq-eta-pre}
\begin{split}
\frac{k_{\alpha_2}(U_0 - U_m)}{\lambda_{\alpha_1} \lambda_{\alpha_2}}\frac{W\left(-\lambda 2 x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)
 + 2\lambda x W\left(-\lambda 2 x,-\frac{\alpha}{2},1\right)}{1-W\left(-\lambda 2 x,-\frac{\alpha}{2},1\right)}
 -\qquad \qquad\qquad\qquad
 \\
\qquad \qquad \qquad -k_{\alpha_1}\frac{U_m - U_i}{\lambda^2_{\alpha_1}}\frac{W\left(-2x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)}{W\left(-2x,-\frac{\alpha}{2},1\right)}
=\left(\rho l +\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2} \right)2x, \quad x>0.
\end{split}
\end{equation}

which, by using Proposition $\ref{Propo monot Wrights}-\ref{rel-Wrights}$ leads to equation (\ref{eq eta}).\\
The next step is to prove that Eq. (\ref{eq eta}) has unique solution. For that purpose
we define function $G$ in ${\mathbb R}^+$ as
$$ G(x)= \frac{k_{\alpha_2}(U_0-U_m)}{ \lambda_{\alpha_1}\lambda_{\alpha_2}}G_2(2 \lambda x) -
          \frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2 } G_1(2x) -\left(\rho l+\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2} \right)2x. $$

Note that $G$ is continuous function such that
\begin{equation}\label{G-1a}
G(0^+)=+\infty. \end{equation}

From Proposition $\ref{Propo monot Wrights}-\ref{desig-W}$ for every $x>0$ we have that
\begin{equation}\label{G2} 0<\frac{W\left(-2x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)}{W\left(-2x,-\frac{\alpha}{2},1\right)}<\frac{1}{\Gamma\left(\frac{\alpha}{2}+1\right)}, \end{equation}
then $G_1$ is bounded. Also, from (\ref{G2}) it holds that
\begin{equation}\label{G-1b}
\begin{split} -\frac{k_{\alpha_1}(U_i-U_m)}{\lambda_{\alpha_1}^2 }\frac{1}{\Gamma\left(\frac{\alpha}{2}+1\right)} + \frac{k_{\alpha_2}(U_0-U_m)}{ \lambda_{\alpha_1}\lambda_{\alpha_2}}G_2(2 \lambda x)-\left(\rho l+\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2} \right)2x< \\
G(x)< \frac{k_{\alpha_2}(U_0-U_m)}{\lambda_{\alpha_1}\lambda_{\alpha_2}}G_2( 2\lambda x)-\left(\rho l+\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2} \right)2x,
\end{split}
\end{equation}
and taking the limit when $x \rightarrow \infty$ in (\ref{G-1b}) and using Proposition \ref{Wrights-x->infty} we obtain that
\begin{equation}\label{G-1c}
G(+\infty)=-\infty. \end{equation}

Finally, consider the function $K\colon {\mathbb R}^+ \rightarrow {\mathbb R}$  defined as
\begin{equation}\label{K}
K(x)=-\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2 }\left[ G_1(2x) +2x\right] -\rho l 2x.
\end{equation}

Applying Proposition \ref{Props W} item \ref{deriv comun} and being $\frac{(U_m-U_i)}{\lambda_{\alpha_1}^2 }>0$ it results that $K$ is a strictly decreasing function. By the other side, from Proposition \ref{Propo monot Wrights} item \ref{W dec}   we have that $G_2$ is a strictly decreasing function. Then it can be concluded that $G$ is a strictly decreasing function. Therefore Eq. (\ref{eq eta}) has a unique positive solution.

\endproof

\begin{remark} The limits described in Remark \ref{cambio de limites en fr-St-Cond} are different if we compute them for the functions $w_1$ and $r$. In fact,  by using the computation made in the previous theorem,  we get\\

\begin{equation}\label{intercambio-lim-3}
\begin{split} \lim\limits_{x\rightarrow r(t)^+} \,_0^{RL}D^{1-\al}_t\, \frac{\partial}{\partial x} w_1(x,t)=\frac{B_1}{\lambda_{\alpha_1}}\left[\frac{\alpha}{2}t^{\alpha/2-1}W\left(-2\eta_\alpha,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right) + \right.\\
\left. \frac{\alpha}{2} 2\eta_\alpha t^{\alpha/2-1}W\left(-2\eta_\alpha,-\frac{\alpha}{2},1\right)  \right].  \end{split}\end{equation}

and from Proposition $\ref{Propo monot Wrights}-\ref{rel-Wrights}$, we have:

\begin{equation}\label{Wr-al-2} W\left(-2\eta_\alpha,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)+ 2\eta_\alpha W\left(-2\eta_\alpha,-\frac{\alpha}{2},1\right) = \frac{2}{\alpha}  W\left(-2\eta_\alpha,-\frac{\alpha}{2},\frac{\alpha}{2}\right). \end{equation}

Then
\begin{equation}\label{intercambio-lim-3'}
\lim\limits_{x\rightarrow r(t)^+} \,_0^{RL}D^{1-\al}_t\, \frac{\partial}{\partial x} w_1(x,t)= \frac{B_1}{\lambda_{\alpha_1}} t^{\alpha/2-1}W\left(-2\eta_\alpha,-\frac{\alpha}{2},\frac{\alpha}{2}\right)
\end{equation}

whereas

\begin{equation}\label{intercambio-lim-4}\, _0^{RL}D^{1-\al}_t\, \left( \lim\limits_{x\rightarrow r(t)^+}\frac{\partial}{\partial x} w_1(x,t)\right)=\frac{B_1}{\lambda_{\alpha_1}} t^{\alpha/2-1} \frac{\Gamma\left(1-\frac{\alpha}{2}\right)}{\Gamma\left(\frac{\alpha}{2}\right)}M_{\alpha/2}(2\eta_\alpha).  \end{equation}

And we know that  $(\ref{intercambio-lim-3'})$ and $(\ref{intercambio-lim-4})$ are different due to  Proposition $\ref{Propo monot Wrights}-\ref{desig-W}$.
\end{remark}

\begin{theorem}\label{differents} If $\lambda =1$,  the explicit solutions (\ref{SOL-St-like-2ph-RL}) to problem (\ref{St-like-2ph-RL}), and  (\ref{sol Caputo-2f}) to  problem (\ref{St-like-2ph-Caputo}) are different.
\end{theorem}
\proof Take $U_i=-1$, $U_m=0$ and $U_0=1$. Let $\left\{u_1, u_2, s\right\} $ be the solution to problem $(\ref{St-like-2ph-Caputo})$. Then $s(t)=2\lambda_{\alpha_1}\xi_\alpha t$ where   $\xi_\alpha$ is a  positive solution to equation
\begin{equation}\label{eq xi-demos-dif-1}
\frac{k_{\alpha_2} \Gamma(1-\frac{\alpha}{2}) }{\lambda_{\alpha_1} \lambda_{\alpha_2}} \frac{M_{\alpha/2}(2 \lambda x)}{1-W\left(-\lambda 2 x,-\frac{\alpha}{2},1\right)}- \frac{k_{\alpha_1} \Gamma(1-\frac{\alpha}{2})}{\lambda^2_{\alpha_1}}\frac{M_{\alpha/2}(2x)}{W\left(-2x,-\frac{\alpha}{2},1\right)}=\Gamma(1+\frac{\alpha}{2}) \rho l 2x.
\end{equation}

By the other side, let $\left\{w_1,w_2,r\right\}$ be the solution to problem $(\ref{St-like-2ph-RL})$. Then $\eta_\alpha$ is the positive solution to equation

\begin{equation}
\frac{k_{\alpha_2} 2/\alpha}{ \lambda_{\alpha_1}\lambda_{\alpha_2}}\frac{ W\left(-2\lambda x,-\frac{\alpha}{2},\frac{\alpha}{2}\right) }{1- W\left(-\lambda 2x,-\frac{\alpha}{2},1\right)} -\frac{k_{\alpha_1}}{\lambda_{\alpha_1}^2 }\frac{W\left(- 2x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)}{W\left(- 2x,-\frac{\alpha}{2},1\right)} =\left(\rho l + \frac{k_{\alpha_1}}{\lambda_{\alpha_1}^2} \right)2x,
 \end{equation}

\noindent or equivalently,

\begin{equation}\label{eq-RL-demos}
\begin{split}
\frac{k_{\alpha_2} }{ \lambda_{\alpha_1}\lambda_{\alpha_2}}\frac{ \Gamma(1+\frac{\alpha}{2})2/\alpha  W\left(-2\lambda x,-\frac{\alpha}{2},\frac{\alpha}{2}\right) }{1- W\left(-2\lambda   x,-\frac{\alpha}{2},1\right)} -\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \\ - \frac{k_{\alpha_1} \Gamma(1+\frac{\alpha}{2})}{\lambda^2_{\alpha_1} }\frac{W\left(- 2x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)+ 2x W\left(- 2x,-\frac{\alpha}{2},1\right)   }{W\left(- 2x,-\frac{\alpha}{2},1\right)}
 =\Gamma(1+\frac{\alpha}{2})\rho l 2x.
\end{split}
 \end{equation}

From Proposition $\ref{Propo monot Wrights}-\ref{rel-Wrights}$, for every $x>0$ we have that
\begin{equation}\label{caso part-igualdad W} W\left(- 2x,-\frac{\alpha}{2},1+\frac{\alpha}{2}\right)+ 2x W\left(- 2x,-\frac{\alpha}{2},1\right) = \frac{2}{\alpha}  W\left(- 2x,-\frac{\alpha}{2},\frac{\alpha}{2}\right). \end{equation}

Then using the fact that the Gamma function verifies that  $\frac{\Gamma(1+\frac{\alpha}{2})}{\frac{\alpha}{2}}=\Gamma(\frac{\alpha}{2})$ and replacing \eqref{caso part-igualdad W} in \eqref{eq-RL-demos} we deduce that  $\eta_\alpha$ is the unique positive solution to the equation

\begin{equation}\label{eq nu-demos-dif-1-bis}
\frac{k_{\alpha_2}}{\lambda_{\alpha_1}\lambda_{\alpha_2}} \frac{ \Gamma(\frac{\alpha}{2})  W\left(- 2\lambda  x,-\frac{\alpha}{2},\frac{\alpha}{2}\right) }{1- W\left(-2\lambda x,-\frac{\alpha}{2},1\right)} -\frac{k_{\alpha_1}}{\lambda^2_{\alpha_1}} \frac{ \Gamma(\frac{\alpha}{2}) W\left(- 2x,-\frac{\alpha}{2},\frac{\alpha}{2}\right)}{W\left(-2 x,-\frac{\alpha}{2},1\right)}= \Gamma(1+\frac{\alpha}{2})  \rho l  2x, \,x>0.
 \end{equation}

If we suppose then that $\xi_\alpha=\eta_\alpha$, it result that there exist $\xi_\alpha >0$ such that

\begin{equation}\label{eq-xi=eta}
\begin{split}
\frac{k_{\alpha_1}}{\lambda^2_{\alpha_1}}  \frac{ \Gamma(\frac{\alpha}{2}) W\left(-2\xi_\alpha,-\frac{\alpha}{2},\frac{\alpha}{2}\right)}{W\left(-\xi_\alpha,-\frac{\alpha}{2},1\right)} - \frac{k_{\alpha_1}}{\lambda^2_{\alpha_1}} \frac{  \Gamma(1-\frac{\alpha}{2}) M_{\alpha/2}(2\xi_\alpha)}{W\left(-\xi_\alpha,-\frac{\alpha}{2},1\right)}=\qquad \qquad\\
 \qquad \qquad=
\frac{k_{\alpha_2}}{\lambda_{\alpha_1}\lambda_{\alpha_2}}  \frac{ \Gamma(\frac{\alpha}{2})  W\left(-\lambda 2\xi_\alpha,-\frac{\alpha}{2},\frac{\alpha}{2}\right) }{1- W\left(-\lambda \xi_\alpha,-\frac{\alpha}{2},1\right)} -  c_2  \frac{ \Gamma(1-\frac{\alpha}{2}) M_{\alpha/2}( \lambda 2\xi_\alpha)}{1-W\left(-\lambda \xi_\alpha,-\frac{\alpha}{2},1\right)}.
\end{split}
\end{equation}

By using the hypothesis that $\lambda =1$, we conclude that

\begin{equation} \label{eq h-demos-dif}
 \frac{\frac{k_{\alpha_1}}{\lambda^2_{\alpha_1}} }{W\left(-\xi_\alpha,-\frac{\alpha}{2},1\right)}=\frac{\frac{k_{\alpha_2}}{\lambda_{\alpha_1}\lambda_{\alpha_2}}}{1-W\left(-\lambda \xi_\alpha,-\frac{\alpha}{2},1\right)},
\end{equation}
which leads to
\begin{equation} \label{demos-dif-2}
W\left(-\xi_\alpha,-\frac{\alpha}{2},1\right)=\frac{1}{1+\frac{k_{\alpha_2}\lambda_{\alpha_2} }{k_{\alpha_1}\lambda_{\alpha_1}}}.
\end{equation}

Replacing $(\ref{demos-dif-2})$ in equation  $(\ref{eq xi-demos-dif-1})$ yields that
$$  \rho l \lambda_{\alpha_1} 2\xi_\alpha=0 $$
which leads to $\xi_\alpha=0,$ contradicting the fact that $\xi_\alpha>0$.
\endproof

\begin{note} It is worth noting that an analogous proof for Theorem \ref{differents} but considering $\lambda\neq 1$ does not holds. In fact, if we define the function $h_\alpha\colon {\mathbb R}^+ \rightarrow {\mathbb R}$ as $$
h_\alpha(x)= \Gamma\left(\frac{\alpha}{2}\right) W\left(-x,-\frac{\alpha}{2},\frac{\alpha}{2}\right)- \Gamma\left(1-\frac{\alpha}{2}\right) M_{\alpha/2}( x)
$$

\begin{center}
\begin{figure}[h]
\includegraphics[width=0.5\columnwidth]{h_functions_alphas} \\
\caption{The function $h_\alpha(x)= \Gamma(\frac{\alpha}{2})
        W\left(-x,-\frac{\alpha}{2},\frac{\alpha}{2}\right)- \Gamma(1-\frac{\alpha}{2}) M_{\alpha/2}( x)$ for different values of $\alpha$,}
\label{F:h-alpha}
\end{figure}
\end{center}

\noindent then equality (\ref{eq-xi=eta}) can be expressed as
\begin{equation} \label{eq h-demos-dif}
\frac{k_{\alpha_2}}{\lambda_{\alpha_1}\lambda_{\alpha_2}}
 \frac{h_\alpha(\lambda 2\xi_\alpha)}{1-W\left(-\lambda 2\xi_\alpha,-\frac{\alpha}{2},1\right)}=
\frac{k_{\alpha_1}}{\lambda^2_{\alpha_1}}
  \frac{h_\alpha(2\xi_\alpha)}{W\left(- 2\xi_\alpha,-\frac{\alpha}{2},1\right)}.
\end{equation}
  If $\lambda \neq 1$, it is not possible to cancel the espressions $h_\alpha(\lambda 2\xi_\alpha)$ and
  $h_\alpha(2\xi_\alpha)$ in equation (\ref{eq h-demos-dif}). Moreover the graphics in Figure \ref{F:h-quotient2-alpha} lead us to suppose that there exists a positive solution to equation
	\begin{equation} \label{eq h-demos-dif-2}
\frac{k_{\alpha_2}}{\lambda_{\alpha_1}\lambda_{\alpha_2}} \frac{h_\alpha(\lambda x)}{1-W\left(-\lambda x,-\frac{\alpha}{2},1\right)}=
\frac{k_{\alpha_1}}{\lambda^2_{\alpha_1}}\frac{h_\alpha(x)}{W\left(- x,-\frac{\alpha}{2},1\right)}, \, x>0,
\end{equation}
then, it is not possible to get a contradiction like \eqref{demos-dif-2}.\\

\begin{center}
\begin{figure}[h]
\begin{tabular}{cc}
\includegraphics[width=0.5\columnwidth]{h_cociente_alphas}
 &
\includegraphics[width=0.5\columnwidth]{h_cociente2_alphas-lambda2} \\
\end{tabular}
\caption{The left and right quotients of equation (\ref{eq h-demos-dif-2})  for different values of $\alpha$}
\label{F:h-quotient2-alpha}
\end{figure}
\end{center}

However, if we take different values of $\lambda$ (which are different to 1) and the parameters $\xi_\alpha$ and $\eta_\alpha$ are estimated numerically for different values of $\alpha$, we show that they are different and converging both to the same value when $\alpha\nearrow 1$. Numerical examples will be given in the next section.

\end{note}

\begin{theorem} The explicit solution (\ref{SOL-St-like-2ph-RL}) to problem (\ref{St-like-2ph-RL})   converges, when $\alpha\nearrow 1$, to the unique solution to the classical Stefan problem given by
\begin{equation}{\label{Stefan-Clasico}}
\begin{array}{llll}
     (i)  &   \frac{\partial}{\partial t}u_2(x,t)=\lambda^2_2\, \frac{\partial^2}{\partial x^2} u_2 (x,t), &   0<x<s(t), \,  0<t<T,  \, \,\\
     (ii)  &   \frac{\partial}{\partial t}u_1(x,t)=\lambda^2_1\, \frac{\partial^2}{\partial x^2} u_1 (x,t), &   x>s(t), \,  0<t<T,  \, \,\\
     (iii) &   u_1(x,0)=U_i, & 0\leq x,\\
       (iv)  &  u_2(0,t)=U_0,   &  0<t\leq T,  \\
         (v) & u_1(s(t),t)=u_2(s(t),t)=U_m, & 0<t\leq T, \\
  (vi) & \frac{d}{dt}s(t)=k_1\frac{\partial}{\partial x} u_1(s(t),t) -k_2\frac{\partial}{\partial x} u_2 (s(t),t), & 0<t\leq T, \\
   (vii) & s(0)=0
                                             \end{array}
                                             \end{equation}

\end{theorem}

\proof The unique solution to problem (\ref{Stefan-Clasico}) is the Neumann solution given in \cite{Weber:1901},

\begin{equation}
\left\{\begin{array}{l}
z_2(x,t)=U_0-(U_0-U_m)\frac{\mbox{erf\,}\left(\frac{x}{2\lambda_2\sqrt{t}}\right)}{\mbox{erf\,} \left(\nu_1\lambda\right)} \\
 z_1(x,t)=
     U_i+(U_m-U_i)\frac{\mbox{erfc\,} \left(\frac{x}{2\lambda_1\sqrt{t}}\right)}{\mbox{erfc\,} \left(\nu_1\right)} \\

w(t)= 2\eta_1 \lambda_1 \sqrt{t}
 \end{array}\right.
\end{equation}

\noindent where $ \eta_1$  is the unique  solution to the equation
  \begin{equation}\label{conv 4} \frac{k_2(U_0-U_m)}{ \lambda_1 \lambda_2}\frac{\exp\left\{ -\lambda^2 x^2 \right\}}{\sqrt{\pi} \mbox{erf\,}\left( \lambda x \right)}- \frac{k_1(U_m-U_i)}{\lambda_1^2}\frac{\exp \left\{-x^2\right\}}{\sqrt{\pi}\mbox{erfc\,} \left(x\right)} = \rho l x, \quad x>0. \end{equation}

Reasoning like in  the previous theorem we can state that the solution  to problem (\ref{St-like-2ph-RL}) is given by (\ref{SOL-St-like-2ph-RL})  where $\eta_\alpha$ is the unique positive  solution to the equation\\

\begin{equation}\label{eq-eta-nueva}
\frac{k_{\alpha_2}(U_0-U_m)}{ \lambda_{\alpha_1}\lambda_{\alpha_2} \alpha}\frac{ W\left(-2\lambda  x,-\frac{\alpha}{2},\frac{\alpha}{2}  \right)}{1-W\left(-2\lambda  x,-\frac{\alpha}{2},1  \right)} -\frac{k_{\alpha_1}(U_m-U_i)}{\lambda_{\alpha_1}^2 \alpha } \frac{ W\left(-2 x,-\frac{\alpha}{2},\frac{\alpha}{2}  \right)}{W\left(-2x,-\frac{\alpha}{2},1  \right)}
 = \rho l x,   \quad x>0.
 \end{equation}

Clearly, if we take $\alpha=1$ in equation (\ref{eq-eta-nueva}) we recover equation (\ref{conv 4}). So, let the sequence  $\left\{ \eta_\alpha\right\}_\alpha$ be, where $\eta_\alpha$ is the unique positive solution to equation $(\ref{eq-eta-nueva})$ for each $0<\alpha<1$. \\
Defining the functions
$$ f_\alpha(x)=\frac{k_{\alpha_2}(U_0-U_m)}{\rho l \lambda_{\alpha_1}\lambda_{\alpha_2} \alpha}\frac{ W\left(-2\lambda  x,-\frac{\alpha}{2},\frac{\alpha}{2}  \right)}{1-W\left(-2\lambda  x,-\frac{\alpha}{2},1  \right)} -\frac{k_{\alpha_1}(U_m-U_i)}{\rho l \lambda_{\alpha_1}^2 \alpha } \frac{ W\left(-2 x,-\frac{\alpha}{2},\frac{\alpha}{2}  \right)}{W\left(-2x,-\frac{\alpha}{2},1  \right)}
$$
for every $x \in {\mathbb R}^+$ and $0<\alpha \leq 1$, it holds that
 $f_\alpha(\eta_\alpha)=\eta_\alpha$ for every $\alpha \in (0,1]$.  \\

From \cite{Tar:1981} we know that $f_1$ is a strictely decreassing function in ${\mathbb R}^+$. Taking a close interval $[a,b]\subset {\mathbb R}^+$ such that $\eta_1 \in [a,b]$, using the uniform convergence over compact sets  of all the positive functions given in Proposition \ref{convergencia-W}  and proceding like in \cite[Theorem 2]{RoTa:2017-TwoDifferent} we can state that
\begin{equation}\label{conv eta}
\lim\limits_{\alpha\nearrow 1}\eta_\alpha=\eta_1.
\end{equation}

Finally,   by taking the limit when $\alpha \nearrow 1$ in solution (\ref{SOL-St-like-2ph-RL}) by applying Proposition \ref{convergencia-W}, the thesis holds.
\endproof

\begin{remark} By using the same technique described before, we can improve the result given in \cite[Theorem 3.3]{RoTa:2014} by considering the functions $g_\alpha$ defined  in ${\mathbb R}^+$ by
\begin{equation*}
\begin{split} g_\alpha(x)=\frac{k_{\alpha_2}(U_0-U_m)}{\rho l \lambda_{\alpha_1}\lambda_{\alpha_2} }\frac{\Gamma(1-\alpha/2)}{\Gamma(1+\alpha/2)}\frac{ M_{\alpha/2}\left(-2\lambda  x \right)}{1-W\left(-2\lambda  x,-\frac{\alpha}{2},1  \right)}\\
 -\frac{k_{\alpha_1}(U_m-U_i)}{\rho l \lambda_{\alpha_1}^2 \alpha }\frac{\Gamma(1-\alpha/2)}{\Gamma(1+\alpha/2)} \frac{ M_{\alpha/2}\left(-2 x\right)}{W\left(-2x,-\frac{\alpha}{2},1  \right)}
\end{split}
\end{equation*}

and a sequence $\{\xi_\alpha\}_\alpha$  were $\xi_\alpha$ is a solution to
equation $g_\alpha(x)=x$, $x>0$.
\end{remark}

\section{The dimesionless problems and numerical results}
\label{Section:4}

In the aim to give different graphics of the solutions obtained in  Section 3, the  problems  (\ref{St-like-2ph-Caputo}) and (\ref{St-like-2ph-RL}) will be rewritten in their dimensionless form. \\

First, we give the following table exhibiting the usual heat conduction  physical dimensions related to this work.
Let us write $\textbf{T}$ for temperature, $\textbf{t}$ for time, $\textbf{m}$ for mass and $\textbf{X}$ for position.

\begin{equation}\label{medidas}
\begin{array}{ccc}
u_1, u_2, w_1, w_2 & \textsl{temperatures} & [\textbf{T}]\\
k_1,k_2 & \textsl{thermal conductivities} & \left[\frac{{\textbf{m X}}}{{\textbf{Tt}^3}}\right] \\
\rho & \textsl{mass density} &  \left[\frac{{\textbf {m }}}{{\textbf {X}^3}}\right]  \\
c_1, c_2 & \textsl{specific heats} &   \left[\frac{\textbf{X}^2 }{\textbf{T}\textbf{t}^2}\right]\\
\lambda_i^2=\frac{k_i}{\rho c}, \, i=1,2 & \textsl{ diffusion coefficients} &  \left[\frac{\textbf{X}^2}{\textbf{t}}\right]\\
l & \textsl{latent heat per  unit  mass} & \left[\frac{\textbf{X}^2 }{\textbf{t}^2}\right]
\end{array}
\end{equation}

\begin{proposition}\label{frac-dim} For every $\alpha \in (0,1)$ it holds that
\begin{enumerate}
\item $\left[_0I^{\alpha} f \right]=[f]{\textbf{t}^\alpha}$ for every
  $f=f(t) \in L^1(0,T)$.
\item $\left[_0^{RL}D^{\alpha} f \right]=\dfrac{[f]}{\textbf{t}^\alpha}$ for every $f=f(t) \in AC[0,T]$.
 \item   $\left[_0^{C}D^{\alpha} f \right]=\dfrac{[f]}{\textbf{t}^\alpha}$ for every $f=f(t) \in AC[0,T]$.
\end{enumerate}
\end{proposition}

Recall that the parameters $\nu_\alpha$ and $\mu_\alpha$ given in (\ref{nu-al})  where added  to preserve the consistency with respect to the units of measure in equations  (\ref{J_1-elegido}) and (\ref{J_2-elegido}). That is, being $ \left[ J \right]= \left[ k u_x \right]=\frac{\textbf{m}}{\textbf{t}^3}$ and using Proposition \ref{frac-dim}, it holds that
 \begin{equation}\label{med-IalJ}
 \left[ _0I^{1-\alpha}_tJ(x,t) \right]=\left[\frac{1}{\Gamma(1-\alpha)} \int_{0}^t\frac{J(x,\tau)}{(t-\tau)^\alpha}{\rm d}\tau \right]=\frac{\textbf{m}}{\textbf{t}^{2+\alpha}}.
 \end{equation}

Then, replacing (\ref{med-IalJ}) in (\ref{J_1-elegido}) one gets

\begin{equation}\label{med-nu_al}
\left[\nu_\alpha\right]=\frac{\left[k\frac{\partial u}{\partial x}\right]}{\left[_{h(x)}I^{1-\alpha}_tJ\right])}=\frac{1}{\textbf{t}^{1-\alpha}}.
\end{equation}

Therefore, \begin{equation}\label{med-mu_al}
\left[\mu_\alpha\right]=\textbf{t}^{1-\alpha}.
\end{equation}

\begin{proposition}\label{adim-fcns} Let $x_0$ be a characteristic position and let $U^*$ be a characteristic temperature. Then, if the following rescaling variable are considered
\begin{equation}\label{variable-change}
y=\frac{x}{x_0}, \qquad \tau=\frac{\lambda_1^2}{x_0^2}t \quad  \text{and} \qquad \tilde{w}=\frac{w}{U^*},
\end{equation}
it holds that
\begin{equation}\label{non-dim-1}
_0I^\alpha_t(w_{x}(x,t))=\frac{U^*x_0}{\lambda_1^2}\left(\frac{\lambda_1^2}{x_0^2}\right)^{1-\alpha}\,_0I^\alpha_\tau (\tilde{w}_{y}(y,\tau)),
\end{equation}
\begin{equation}\label{non-dim-2}
_0I^\alpha_t(w_{xx}(x,t))=\frac{U^*}{\lambda_1^2}\left(\frac{\lambda_1^2}{x_0^2}\right)^{1-\alpha}\,_0I^\alpha_\tau (\tilde{w}_{yy}(y,\tau))
\end{equation}
and
\begin{equation}\label{non-dim-3}
_0^{RL}D^{1-\alpha}_t(w_{xx}(x,t))=\frac{U^*}{x_0^2}\left(\frac{\lambda_1^2}{x_0^2}\right)^{1-\alpha}\,_0^{RL}D^{1-\alpha}_\tau (\tilde{w}_{yy}(y,\tau)).
\end{equation}
\end{proposition}

\proof We prove here equation (\ref{non-dim-1}). By considering the rescaling (\ref{variable-change}), we have
\begin{equation} \tilde{w}(y,\tau)=\frac{w(x(y),t(\tau))}{U^*}.\end{equation}

Then
$$ _0I^\alpha_t(w_{x}(x,t))= \frac{1}{\Gamma(\alpha)}\int_0^t \frac{w_x(x,z)}{(t-z)^{1-\alpha}}dz= \frac{U^*}{\Gamma(\alpha)}\int_0^t \frac{\frac{1}{x_0}\tilde{w}_y(y,\tau(z))}{(t-z)^{1-\alpha}}dz=$$
$$  = \frac{U^*}{\Gamma(\alpha)}\int_0^{\frac{\lambda_1^2}{x_0^2}t} \frac{\tilde{w}_y(y,v)}{\left(\frac{x_0^2}{\lambda_1^2}\right)^{1-\alpha}(\frac{\lambda_1^2}{x_0^2}t-v)^{1-\alpha}}\frac{x_0}{\lambda_1^2}dv=  \frac{U^*}{x_0}\left(\frac{x_0^2}{\lambda_1^2}\right)^{\alpha}\,_0I^\alpha_\tau (\tilde{w}_{y}(y,\tau)).
  $$

\endproof

Now, let us consider problems problems  (\ref{St-like-2ph-Caputo}) and (\ref{St-like-2ph-RL}). By using Proposition \ref{adim-fcns} it is easy  to state that the  governing equation $(\ref{St-like-2ph-RL}-i)$ is equivalent to the following equation

\begin{equation}\label{equiv-2}
\frac{\partial}{\partial \tau}\tilde{w}_2(y,\tau)= \lambda^2 \mu_\alpha \left(\frac{\lambda_1^2}{x_0^2} \right)^{1-\alpha}\, _0^{RL}D^{1-\alpha}_\tau  \tilde{w}_{2yy} (y,\tau).
\end{equation}

Note that $\mu_\alpha=\left(\frac{x_0^2 }{\lambda_1^2} \right)^{1-\alpha}$ is an admissible parameter because $[\mu_\alpha]=\textbf{t}^{1-\alpha}$ and that $\lim\limits_{\alpha\nearrow 1}\mu_\alpha=1$.
Then, the parameter $\mu_\alpha \left(\frac{\lambda_1^2}{x_0^2} \right)^{1-\alpha}$ in equation (\ref{equiv-2}) can be omitted.  \\

Analogously, transforming the governing equations, the Stefan conditions and the initial and boundary data in problems (\ref{St-like-2ph-Caputo})  and (\ref{St-like-2ph-RL}), and by taking $U_m=0$ and  $U^*=|U_i|$,   it follows that  the non-dimensional   associated form are given by

 \begin{equation}{\label{adim-St-like-2ph-Caputo}}
\begin{array}{llll}
     (i)  &  \,_0^{C}D^{\alpha}_\tau \tilde{u}_2(y,\tau)=\lambda^2 \tilde{u}_{2yy}(y,\tau), &   0<y<\tilde{s}(\tau), \,  0<\tau<\tilde{T},  \, \,\\
     (ii)  &   \,_0^{C}D^{\alpha}_\tau \tilde{u}_1(y,\tau)= \tilde{u}_{2yy}(y,\tau), &   y>\tilde{s}(\tau), \,  0<\tau<\tilde{T},  \, \,\\
     (iii) &   \tilde{u}_1(y,0)=-1, & 0\leq x,\\
     (iv)  &   \tilde{u}_2(0,\tau)=\frac{U_0}{|U_i|},   &  0<\tau\leq \tilde{T},  \\
         (v) & \tilde{u}_1(\tilde{s}(\tau),\tau)=\tilde{u}_1(\tilde{s}(\tau),\tau)=0, & 0<\tau\leq \tilde{T}, \\
  (vi) &  \,_0^{C}D^{\alpha}_\tau \tilde{s}(\tau)= {\rm{Ste}} \left[ \tilde{u}_{1y}(\tilde{s}(\tau)^+,\tau) - \frac{k_2}{k_1} \tilde{u}_{2y} (\tilde{s}(\tau)^-,\tau)\right], & 0<\tau\leq \tilde{T}, \\
(vii) & \tilde{s}(0)=0. &
                                             \end{array}
                                             \end{equation}
and
\begin{equation}{\label{adim-St-like-2ph-RL}}
\begin{array}{llll}
     (i)  &   \tilde{w}_{2\tau}(y,\tau)=\lambda^2 \,_0^{RL}D^{1-\alpha}_\tau {w}_{2yy} (y,\tau), &   0<y<\tilde{r}(\tau), \,  0<\tilde{t}<\tilde{T},  \, \,\\
     (ii)  &   \tilde{w}_{1\tau}(y,\tau)= \,_0^{RL}D^{1-\alpha}_\tau {w}_{1yy} (y,\tau), &   y>\tilde{r}(\tau), \,  0<\tau<\tilde{T},  \, \,\\
     (iii) &   \tilde{w}_1(y,0)=-1, & 0\leq y,\\
       (iv)  &  \tilde{w}_2(0,t)=\frac{U_0}{|U_i|},   &  0<\tau\leq \tilde{T},  \\
         (v) & \tilde{w}_1(\tilde{r}(\tau),\tau)=\tilde{w}_2(\tilde{r}(\tau),\tau)=0, & 0<\tau\leq \tilde{T}, \\
  (vi) &  \frac{d}{dt}\tilde{r}(\tau)= {\rm{Ste}} \left[ \left. \,_0^{RL}D^{1-\alpha}_\tau w_{1y}(y,\tau)\right|_{(\tilde{r}(\tau)^+,\tau)} \right.  & \\
 &	\hspace{3cm} \left. -\frac{k_2}{k_1} \left. \,_0^{RL}D^{1-\alpha}_\tau \tilde{w}_{2y} (y,\tau)\right|_{(\tilde{r}(\tau)^-,\tau)}\right], & 0<\tau\leq \tilde{T},\\
 			(vii)	& \tilde{r}(0)=0. &\\

                                             \end{array}
                                             \end{equation}

where  $\lambda=\frac{\lambda_2}{\lambda_1}$ and ${\rm{Ste}}=\frac{|U_i|c_1}{l}$ is the non-dimensional Stefan number.

\vspace{1cm}

In the following table there are different tests, i.e. sets of parameters for $\lambda$,
$\frac{k_2}{k_1}$, $U=\frac{U_0}{|U_i|}$ and  $Ste$. For each test in Table 1 a correpondig graphic of the comparison  between the $\xi_{\alpha}$ and $\eta_{\alpha}$ is given in Figure 3.

\begin{table}[h!]
\centering
\begin{tabular}{|c |c | c| c| c|}
\hline
       & $\lambda$ & $\frac{k_2}{k_1}$ & $U=\frac{U_0}{|U_i|}$  &  $Ste$ \\
\hline
Test 1 & 0.5 & 0.5 & 1.0  &  0.5 \\
Test 2 & 2.0 & 2.0 & 1.0  &  0.5 \\
Test 3 & 0.5 & 0.5 & 1.0  &  1.2 \\
Test 4 & 2.0 & 2.0 & 1.0  &  1.2 \\
\hline
\end{tabular}
\caption{Different set of tests}
\end{table}

\newpage

\begin{figure}[h]
\label{xi_al-vs-eta_al}
\begin{tabular}{cc}
\includegraphics[width=0.45\columnwidth]{xi_vs_eta_for_Test1_alphas} &
\includegraphics[width=0.45\columnwidth]{xi_vs_eta_for_Test4_alphas} \\
\includegraphics[width=0.45\columnwidth]{xi_vs_eta_for_Test3_alphas} &
\includegraphics[width=0.45\columnwidth]{xi_vs_eta_for_Test6_alphas} \\
\label{xi-vs-eta}
\end{tabular}
\caption{$\xi_\alpha$ vs. $\eta_\alpha$ for different values of $\alpha$}
\end{figure}

 At the end, we present  in Figures 4 and  5  some color maps of temperature   for  tests 2 and  3, respectively. Three values of $\alpha$ are considered and as it is expected from Theorem 4, both solutions approach themselves when $\alpha \nearrow 1$.

\begin {figure}[h]
\label{C_vs_RL_test2}
\caption {Caputos's approach Solutions Vs. Riemann-Liouville's aproach Solutions for Test 2}
\centering
\begin{tabular}{|m{0.5cm}|m{7cm}| m{7cm}|}

\hline
 $\alpha$ & \textbf{Caputo Stefan-Like Pb. } & \textbf{Riemann-Liouville  Stefan-Like Pb. } \\
\hline
\hline
$0.7$ & \includegraphics[width=0.5\columnwidth]{Test2_Caputo_alpha_050New.png} &
\includegraphics[width=0.5\columnwidth]{Test2_RL_alpha_050New.png} \\
$0.8$ & \includegraphics[width=0.5\columnwidth]{Test2_Caputo_alpha_070New.png} &
\includegraphics[width=0.5\columnwidth]{Test2_RL_alpha_070New.png} \\
$0.9$ & \includegraphics[width=0.5\columnwidth]{Test2_Caputo_alpha_090New.png} &
\includegraphics[width=0.5\columnwidth]{Test2_RL_alpha_090New.png}\\
\hline
\end{tabular}
\end{figure}

\begin {figure}[h]
\label{C_vs_RL_test3}
\caption {Caputos's approach Solutions Vs. Riemann-Liouville's aproach Solutions for Test 3}
\centering
\begin{tabular}{|m{0.5cm}|m{7cm}| m{7cm}|}

\hline
 $\alpha$ & \textbf{Caputo Stefan-Like Pb. } & \textbf{Riemann-Liouville  Stefan-Like Pb. } \\
\hline
\hline
$0.7$ & \includegraphics[width=0.5\columnwidth]{Test3_Caputo_alpha_050New.png} &
\includegraphics[width=0.5\columnwidth]{Test3_RL_alpha_050New.png} \\
$0.8$ & \includegraphics[width=0.5\columnwidth]{Test3_Caputo_alpha_070New.png} &
\includegraphics[width=0.5\columnwidth]{Test3_RL_alpha_070New.png} \\
$0.9$ & \includegraphics[width=0.5\columnwidth]{Test3_Caputo_alpha_090New.png} &
\includegraphics[width=0.5\columnwidth]{Test3_RL_alpha_090New.png}\\
\hline
\end{tabular}
\end{figure}

\newpage

\section{Conclusion}
  We have presented two different fractional  two-phase Stefan-like problems  for the Riemann-Liouville and Caputo derivatives  of order $\alpha\in (0,1)$ with the particularity that, if the parameter $\alpha=1$ is replaced in both problems, we recover the same classical Stefan problem. In both cases, explicit solutions in terms of self-similar variables where given.
It was interesting to see that, the role of the different  ``fractional Stefan conditions'' associated to each problem was decisive to conclude that the solutions obtained where different. Also, as it was expected, we have proved that the two different solutions converge to the same triple of limits functions when $\alpha$ tends to 1, where this ``limit solution'' is the classical solution to the classical Stefan problem mentioned before.

\section{Acknowledgements}

\noindent The present work has been sponsored by the Projects PIP N$^\circ$ 0275 from CONICET--Univ. Austral, ANPCyT PICTO Austral 2016 N$^\circ 0090$, Austral N$^\circ 006-$INV$00020$ (Rosario, Argentina) and European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement N$^\circ$ 823731 CONMECH.

redpajama_set_name: RedPajamaArXiv


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Homebrew Con, or the National Homebrewer's Conference as it used to be called, was held in the twin cities of Minneapolis / St Paul in Minnesota this year. This was a new state for me to visit personally so I decided it would make a good road trip even though it was a 15 hour drive from State College, PA.
On day one I called in at Cleveland (OH) to pick up my homebrew buddy then stopped overnight in Toledo. On day two we crossed the state of Indiana and negotiated the Chicago (IL) traffic before traveling onto Milwaukee (WI).
We previously visited Milwaukee on our Wisconsin brewery tour back in 2014 (see these posts – parts 1 & 2) so were familiar with the brewing scene. With it being a Sunday it was not ideal with some breweries being closed but we managed a sampler flight at Water Street Brewing together with visits to some quality craft beer bars.
Prior to hitting the interstates with visited Sprecher Brewing located in Glendale just north of Milwaukee to stock up on cases of their Belgian Tripel and Wee Heavy.
The journey over to Minneapolis was through wave after wave of thunderstorms which made it tough going before encountering rush hour traffic in St Paul.
Arriving in Minneapolis gave us two full days to visit some of the many craft breweries located in the twin cities.
We also sampled their Double IPA which was due to be poured at Pro Night during Homebrew Con.
Next up were stops at Fulton Brewing Company and Modist Brewing Company.
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Our final stop on Tuesday was Inbound Brewing Company where three enthusiasts sat next to us were sharing the largest sampler flight I have ever seen! Basically it was a 4oz pour of every beer on tap… we on the other hand by this time of night were more restrained.
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yes Paul, the Mash Filter squeezes the water out of the grain using hydraulics with the resultant grain 'wafers' being almost bone dry and they drop out of the bottom when the press is released.


text:
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Despite the situation, my lips twitched. "What are you going to do? Wear a disguise?"
His nod was swift. "If it'll help, absolutely."
"I appreciate the suggestion, but maybe this was just a sign that it wouldn't work out anyway." I took a step back, towards the waiting taxi. "It was a fun couple of days, Ryan," I said, trying to inject some light into his sad expression.
The truth in those words took me by surprise. I had had fun with him, even if our interactions were brief. I'd miss his easy smile and the way he seemed to draw me to him like a magnet.
Wordlessly, he stepped around me and opened the taxi door. With a grateful smile, I slid in and buckled up.
Fun or not, I couldn't give in. At this point, it felt like I needed to prove that I could be happy without a man messing with my head. Breaking a promise to myself for a guy who had women tracking his every movement would be foolish.
I was giving the driver my address when Ryan knocked on my window. He gestured for me to roll it down and I hit the button, frowning.
Ryan leaned forward until his head was level with mine. Something had eased in him and he had that determination shining in his eyes again. My stomach dropped at the sight. In that second, my wishes to be a mind reader came true. I knew what he was going to say before he uttered the words.
"There's something important here and I'm not giving up yet."
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← BETWEEN TAKES IS LIVE Between Takes →


In receiving initiation into the Usui System of Reiki Healing, we are joining a long line of serious practitioners who have validated hands on healing for self-evolution and relieving the suffering of others.
The First Degree class reflects the simplicity of the practice.
"I'm never farther away from serenity and healing than my own hands."
It is most supportive for the student when first and second level classes are clear, simple, and do not include other practices.
"Reiki gives me a way to reach out and touch others in a loving context without fear of being misunderstood."