Do You Have to Make Up Missed Fasts in Order?
Question

Is it essential to make up missed Ramadans in order, such as making up for 1440 first, then 1439? Is the fasting invalidated if one Ramadan is done before another?
Praise be to Allah.The jurists differed as to whether it is essential to make up missed Ramadans in order or not. There are two views:

The first view is that they must be made up in order. This is the view of the Hanbalis.

It says in Kashshaf Al-Qina` (2/308): “If someone fasts Sha`ban for three consecutive years, and his fasting in Sha`ban for these three years was a total of three months with the intention of making up what he had missed of Ramadans , one after another, in order, like making up missed prayers, then his fasting is valid. As making up prayers in the right order is obligatory, the same applies when making up missed Ramadans.”

The second view is that it is not obligatory to do them in order . This is the view favoured by the Hanafis and is the view of the Malikis.

It says in Tabyin Al-Haqa’iq (6/220): 
“If he has to make up two days from one Ramadan, and he fasted but did not specify which day he was making up, that is permissible. The same applies if they were two days from two different Ramadans, according to the correct view, and even if he intended to make up what he missed without being more specific, it is permissible." (Fath Al-Bari)
It says in Minah Al-Jalil (2/124): “One who has to make up two missed Ramadans must start with the first of them, but if he does it the other way round, it is permissible.” 
The Standing Committee issued a Fatwa stating that it is obligatory to do them in order. They were asked the following question:
My mother has been suffering from kidney failure for nine years, and when Ramadan comes she is doing dialysis. She goes to the hospital three times a week, on Saturdays, Mondays and Wednesdays. The way in which it is done is with a catheter inserted into the veins; this is what breaks the fast if the patient is fasting. Nine Ramadans have gone by in the last nine years during which she did not fast on the days on which she did dialysis. When Ramadan ends, she gives Fidyah for the days on which she did not fast, without making up the fast, because she goes to the hospital three times every week. What should she do, even though nine Ramadans have passed? Is there any sin on her, because she paid Fidyah for every day and she did not make up the fast? Please advise us, may Allah reward you with good.
Answer:
“If the matter is as mentioned, she must make up the days on which she did not fast during the previous Ramadans, and making up the fasts should be done in order. So she should fast the days when she did not fast from the first Ramadan, then the second, and so on, and she should feed one poor person for each day, as an expiation for the delay in making up the fasts. The amount to be given to each is one and a half kilograms of wheat, rice, dates or other local staple food, because of the general meaning of the verse in which Allah, may He be exalted, says (interpretation of the meaning):
{and whoever is ill or on a journey - then an equal number of other days.} [Al-Baqarah 2:185]
And Allah is the source of strength. May Allah send blessings and peace upon our Prophet Muhammad and his family and companions.” (Bakr Abu Zayd, Salih Al-Fawzan, `Abdullah ibn Ghadyan, `Abd Al-`Aziz ibn `Abdullah Al Ash-Shaykh (Fatawa Al-Lajnah Ad-Da’imah, vol. 2   9/105)
Making up the missed fasts in order is more prudent, so make up what you missed of Ramadan 1439, then what you missed of Ramadan 1440, then what you missed of Ramadan 1441.
But for one who does not know the order, or has forgotten it, and has made up what he owed of fasts from previous years, there is no blame on him and his making up of the fasts is valid.
The one who delayed making up what he owed of Ramadan fasts until the following Ramadan began, without an excuse, must – in addition to making up the fasts – offer the Fidyah , according to the majority of scholars. What is required is to feed one poor person for each day. This Fidyah is to be given once for each fast that was missed and the making up thereof was delayed, regardless of the number of years the person delayed making up the fast .
And Allah knows best.