the cable equation is one of the most fundamental equations for modeling neuronal dynamics . due to its significant deviation from the dynamics of brownian motion ,
the anomalous diffusion in biological systems can not be adequately described by the traditional nernst 
very recently , a modified cable equation was introduced for modeling the anomalous diffusion in spiny neuronal dendrites .
the resulting governing equation , the so - called fractional cable equation , which is similar to the traditional cable equation except that the order of derivative with respect to the space and/or time is fractional . also , the proposed fractional cable equation model is better than the standard integer cable equation , since the fractional derivative can describe the history of the state in all intervals , for more details see   and the references cited therein .
the main aim of this work is to solve such this equation numerically by an efficient numerical method , fractional weighted average finite difference method ( fwa  fdm ) . in recent years
, considerable interest in fractional calculus has been stimulated by the applications that this calculus finds in numerical analysis and different areas of physics and engineering , possibly including fractal phenomena .
the applications range from control theory to transport problems in fractal structures , from relaxation phenomena in disordered media to anomalous reaction kinetics of subdiffusive reagents .
fractional differential equations ( fdes ) have been of considerable interest in the literatures , see for example [ 413 ] and the references cited therein , the topic has received a great deal of attention especially in the fields of viscoelastic materials , control theory , advection and dispersion of solutes in natural porous or fractured media , anomalous diffusion , signal processing and image denoising / filtering . in this section ,
liouville derivative of order  of the function y(x ) is defined by(1)dxy(x)=1(n-)dndxn0xy()(x-)-n+1d,x>0,where n is the smallest integer exceeding  and  ( . ) is the gamma function . if =nn , then ( 1 ) coincides with the classical n derivative y(x).definition  2the grnwald 
letnikov definition for the fractional derivatives of order   >  0 of the function y(x ) is defined by(2)dy(x)=limh01hk=0xhwk()y(x - hk),x0,where xh means the integer part of xh and wk( ) are the normalized grnwald weights which are defined by wk()=(-1)kk.the grnwald  letnikov definition is simply a generalization of the ordinary discretization formula for integer order derivatives . the riemann  liouville and the grnwald 
letnikov approaches coincide under relatively weak conditions ; if y(x ) is continuous and y(x ) is integrable in the interval [ 0 ,  x ] , then for every order 0  < 
  <  1 both the riemann  liouville and the grnwald  letnikov derivatives exist and coincide for any value inside the interval [ 0 ,  x ] . this fact of fractional calculus ensures the consistency of both definitions for most physical applications , where the functions are expected to be sufficiently smooth .the plan of the paper is as follows : in the second section , some fractional formulae and some discrete versions of the fractional derivative are given .
 numerical results  numerical solutions and exact analytical solutions of a typical fractional cable problem are compared .
the paper ends with some conclusions in section  conclusion and remarks.we consider the initial - boundary value problem of the fractional cable equation which is usually written in the following way(3)ut(x , t)=dt1-uxx(x , t)-dt1-u(x ,
t),a < x < b,0<tt , where 0  <   ,     1 ,  is a constant and dt1- is the fractional derivative defined by the riemann  liouville operator of order 1     , where   =   , . under the zero boundary conditions(4)u(a , t)=u(b , t)=0,and the following initial condition(5)u(x,0)=g(x).in the last few years , appeared many papers to study this model ( 3)(5 ) 
 [ 5,1922 ] , the most of these papers study the ordinary case of such system . in this paper
, we study the fractional case and use the fwa  fdm to solve this model . the riemann 
liouville derivative of order  of the function y(x ) is defined by(1)dxy(x)=1(n-)dndxn0xy()(x-)-n+1d,x>0,where n is the smallest integer exceeding  and  ( . )
is the gamma function . if =nn , then ( 1 ) coincides with the classical n derivative y(x ) .
letnikov definition for the fractional derivatives of order   >  0 of the function y(x ) is defined by(2)dy(x)=limh01hk=0xhwk()y(x - hk),x0,where xh means the integer part of xh and wk( ) are the normalized grnwald weights which are defined by wk()=(-1)kk .
letnikov definition is simply a generalization of the ordinary discretization formula for integer order derivatives . the riemann  liouville and the grnwald 
letnikov approaches coincide under relatively weak conditions ; if y(x ) is continuous and y(x ) is integrable in the interval [ 0 ,  x ] , then for every order 0  <    <  1 both the riemann  liouville and the grnwald  letnikov derivatives exist and coincide for any value inside the interval [ 0 ,  x ] .
this fact of fractional calculus ensures the consistency of both definitions for most physical applications , where the functions are expected to be sufficiently smooth .
the plan of the paper is as follows : in the second section , some fractional formulae and some discrete versions of the fractional derivative are given .
also , the fwa  fdm is developed . in the third section , we study the stability and the accuracy of the presented method . in section  numerical results  numerical solutions and exact analytical solutions of a typical fractional cable problem are compared .
 we consider the initial - boundary value problem of the fractional cable equation which is usually written in the following way(3)ut(x , t)=dt1-uxx(x , t)-dt1-u(x , t),a < x < b,0<tt , where 0  <   ,     1 ,  is a constant and dt1- is the fractional derivative defined by the riemann 
liouville operator of order 1     , where   =   , . under the zero boundary conditions(4)u(a , t)=u(b , t)=0,and the following initial condition(5)u(x,0)=g(x ) . in the last few years
, appeared many papers to study this model ( 3)(5 ) 
 [ 5,1922 ] , the most of these papers study the ordinary case of such system . in this paper
, we study the fractional case and use the fwa  fdm to solve this model .
in this section , we will use the fwa  fdm to obtain the discretization finite difference formula of the cable eq .
we use the notations t and x , at time - step length and space - step length , respectively .
the coordinates of the mesh points are xj  =  a  +  jx and tm  =  mt , and the values of the solution u(x , t ) on these grid points are u(xj , tm)ujmujm . for more details about discretization in fractional calculus see . in the first step ,
the ordinary differential operators are discretized as follows ( 6)utxj , tm+t2=tujm+12+o(t)ujm+1-ujmt+o(t),and(7)2ux2xj , tm=xxujm+o(x)2uj-1m-2ujm+uj+1m(x)2+o(x)2 . in the second step , the riemann 
liouville operator is discretized as follows(8)dt1-u(x , t)xj , tm=t1-ujm+o(t),where(9)t1-ujm1(t)1-k=0tmtwk(1-)u(xj , tm - kt)=1(t)1-k=0mwk(1-)ujm - k , where tmt means the integer part of tmt and for simplicity , we choose h  =  t .
there are many choices of the weights wk( ) 
 , so the above formula is not unique .
let us denote the generating function of the weights wk( ) by w(z ,   ) , i.e.,(10)w(z,)=k=0wk()zk.if(11)w(z,)=(1-z),then ( 9 ) gives the backward difference formula of the first order , which is called the grnwald 
=  1 the operator dt1- becomes the identity operator so that , the consistency of eqs .
( 8) and ( 9 ) requires w0(0)=1 , and wk(0)=0 for k    1 , which in turn means that w(z,0 )  =  1 .
now , we are going to obtain the finite difference scheme of the cable eq .
, we evaluate this equation at the intermediate point of the grid xj , tm+t2(13)ut(x , t)-dt1-uxx(x , t)xj , tm+t2+dt1-u(xj , tm)=0 .
then , we replace the first order time - derivative by the forward difference formula ( 6 ) and replace the second order space - derivative by the weighted average of the three - point centered formula ( 7 ) at the times tm and tm+1(14)tujm+12-t1-xxujm+(1-)t1-xxujm+1+t1-ujm = tejm+12,with  is being the weight factor and tejm+12 is the resulting truncation error .
now , by substituting from the difference operators given by ( 6 ) , ( 7 ) and ( 9 ) , we get(16)ujm+1-ujmt-1(t)1-r=0mwr(1-)uj-1m - r-2ujm - r+uj+1m - r(x)2-(1-)1(t)1-r=0mwr(1-)uj-1m+1-r-2ujm+1-r+uj+1m+1-r(x)2+1(t)1-r=0mwr(1-)ujm - r=0.put n=(t)(x)2,n=(t),=(1-)n , and under some simplifications we can obtain the following form(17)-uj-1m+1+(1 + 2)ujm+1-uj+1m+1=r , where(18)r = ujm+nr=0mwr(1-)+(1-)wr+1(1-)uj-1m - r-2ujm - r+uj+1m - r-nr=0mwr(1-)ujm - r .
fortunately , eq . ( 17 ) is tridiagonal system that can be solved using conjugate gradian method . in the case of  
=  1 and =12 , we have the backward euler fractional quadrature method and the crank  nicholson fractional quadrature methods , respectively , which have been studied , e.g. , in , but at   =  0 the scheme is called fully implicit .
in this section , we use the john von neumann method to study the stability analysis of the weighted average scheme ( 17).theorem  1the fractional weighted average finite difference scheme ( wads ) derived in 
 ( 17 ) 
 is stable at 
 012 
 under the following stability criterion(19)nn(2-1)2-1-2-.proofby using ( 18 ) , we can write ( 17 ) in the following form(20)-uj-1m+1+(1 + 2)ujm+1-uj+1m+1-ujm=-nr=0mwr(1-)ujm - r+nr=0mwr(1-)+(1-)wr+1(1-)uj-1m - r-2ujm - r+uj+1m - r.in the fractional john von neumann stability procedure , the stability of the fractional wads is decided by putting ujm=meiqjx . inserting this expression into the weighted average difference scheme ( 20 )
we obtain(21)-m+1eiq(j-1)x+(1 + 2)m+1eiqjx-m+1eiq(j+1)x-meiqjx = nr=0mwr(1-)+(1-)wr+1(1-)[eiq(j-1)x-2eiqjx+eiq(j+1)x]m - r-nr=0mwr(1-)m - reiqjx , substitute by   =  ( 1    )n and divide ( 21 ) by e we get(22)-(1-)nm+1e - iqx+(1 + 2(1-)n)m+1-(1-)nm+1eiqx-m - nr=0mwr(1-)+(1-)wr+1(1-)[e - iqx-2+eiqx]m - r+nr=0mwr(1-)m - r=0.using the known euler s formula ei=cos+isin we have(23)[1 + 2(1-)n-2(1-)ncos(qx)]m+1-m - nr=0mwr(1-)+(1-)wr+1(1-)[-2 + 2cos(qx)]m - r+nr=0mwr(1-)m - r=0.under some simplifications , we can write the above equation in the following form(24)1 + 4(1-)nsin2qx2m+1+nr=0mwr(1-)m - r-m+4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)m - r=0.the stability of the scheme is determined by the behavior of m . in the john von neumann method , the stability analysis is carried out using the amplification factor  defined by(25)m+1=m.of course ,  depends on m. but , let us assume that , as in ,  is independent of time . then , inserting this expression into eq .
( 24 ) , one gets(26)1 + 4(1-)nsin2qx2m+nr=0mwr(1-)-rm-m+4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)-rm=0,divide by m to obtain the following formula of (27)=1 - 4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)-r-nr=0mwr(1-)-r1 + 4(1-)nsin2qx2.the scheme will be stable as long as     1 , i.e.,(28)-11 - 4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)-r-nr=0mwr(1-)-r1 +
4(1-)nsin2qx21considering the time - independent limit value   =  1 and since 1 + 4(1-)nsin2qx2>0 , then-1 - 4(1-)nsin2qx21 - 4nsin2qx2r=0m(-1)-rwr(1-)+(1-)wr+1(1-)-nr=0mwr(1-)(-1)-r.from the above inequality , we can obtain-2 - 4(1-)nsin2qx2+nr=0mwr(1-)(-1)-r+4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)(-1)-r0.put =nsin2qx2 , we find(29)-2 - 4(1-)+nr=0mwr(1-)(-1)-r+4r=0mwr(1-)+(1-)wr+1(1-)(-1)-r0,which can be written in the form-2 - 4(1-)+nr=0mwr(1-)(-1)-r+4(1 - 2)r=1m(-1)r-1wr(1-)++(-1)m(1-)wm+1(1-)0.put(30)1mm=4(2-1)1-r=1m(-1)r-1wr(1-)+(-1)m(1-)wm+1(1-)2-nr=0mwr(1-)(-1)-r , one finds that the mode is stable when(31)11mm.although , mm depends on m , it turns out that mm tends toward its limit value(32)1m = limm1mm.in this limit the stability condition is(33)11m=4(2-1)1-r=1(-1)r-1wr(1-)+limm(-1)m(1-)wm+1(1-)2-nr=0wr(1-)(-1)-r , but from eqs .
( 10 ) and ( 11 ) with z  =  1 one sees that r=0(-1)rwr(1-)=21- , so that(34)1m=4(2-1)21-+limm(-1)m(1-)wm+1(1-)2-n21-,since =nsin2qx2 , replacing sin2qx2 by its highest value and since limm(-1)m(1-)wm+1(1-)=0 , therefore we find that the sufficient condition for the present method to be stable and this completes the proof of the theorem . 
remark  1for 12<1 , the stability condition ( 19 ) can be satisfied under specific values of n=(t)(x)2 .
the fractional weighted average finite difference scheme ( wads ) derived in 
 ( 17 ) 
 is stable at 
 012 
 under the following stability criterion(19)nn(2-1)2-1-2-. by using ( 18 ) , we can write ( 17 ) in the following form(20)-uj-1m+1+(1 + 2)ujm+1-uj+1m+1-ujm=-nr=0mwr(1-)ujm - r+nr=0mwr(1-)+(1-)wr+1(1-)uj-1m - r-2ujm - r+uj+1m - r . in the fractional john von neumann stability procedure
, the stability of the fractional wads is decided by putting ujm=meiqjx . inserting this expression into the weighted average difference scheme ( 20 ) we obtain(21)-m+1eiq(j-1)x+(1 + 2)m+1eiqjx-m+1eiq(j+1)x-meiqjx = nr=0mwr(1-)+(1-)wr+1(1-)[eiq(j-1)x-2eiqjx+eiq(j+1)x]m - r-nr=0mwr(1-)m - reiqjx , substitute by   =  ( 1    )n and divide ( 21 ) by e we
get(22)-(1-)nm+1e - iqx+(1 + 2(1-)n)m+1-(1-)nm+1eiqx-m - nr=0mwr(1-)+(1-)wr+1(1-)[e - iqx-2+eiqx]m - r+nr=0mwr(1-)m - r=0 . using the known euler s formula ei=cos+isin we have(23)[1 + 2(1-)n-2(1-)ncos(qx)]m+1-m - nr=0mwr(1-)+(1-)wr+1(1-)[-2 + 2cos(qx)]m - r+nr=0mwr(1-)m - r=0 . under some simplifications ,
we can write the above equation in the following form(24)1 + 4(1-)nsin2qx2m+1+nr=0mwr(1-)m - r-m+4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)m - r=0 . the stability of the scheme is determined by the behavior of m . in the john
von neumann method , the stability analysis is carried out using the amplification factor  defined by(25)m+1=m . of course ,  depends on m. but , let us assume that , as in ,  is independent of time . then , inserting this expression into eq .
( 24 ) , one gets(26)1 + 4(1-)nsin2qx2m+nr=0mwr(1-)-rm-m+4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)-rm=0,divide by m to obtain the following formula of (27)=1 - 4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)-r-nr=0mwr(1-)-r1 + 4(1-)nsin2qx2 .
  1 , i.e.,(28)-11 - 4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)-r-nr=0mwr(1-)-r1 + 4(1-)nsin2qx21considering the time - independent limit value   =  1 and since 1 + 4(1-)nsin2qx2>0 , then-1 - 4(1-)nsin2qx21 - 4nsin2qx2r=0m(-1)-rwr(1-)+(1-)wr+1(1-)-nr=0mwr(1-)(-1)-r.from the above inequality , we can obtain-2 - 4(1-)nsin2qx2+nr=0mwr(1-)(-1)-r+4nsin2qx2r=0mwr(1-)+(1-)wr+1(1-)(-1)-r0.put =nsin2qx2 , we find(29)-2 - 4(1-)+nr=0mwr(1-)(-1)-r+4r=0mwr(1-)+(1-)wr+1(1-)(-1)-r0,which can be written in the form-2 - 4(1-)+nr=0mwr(1-)(-1)-r+4(1 - 2)r=1m(-1)r-1wr(1-)++(-1)m(1-)wm+1(1-)0.put(30)1mm=4(2-1)1-r=1m(-1)r-1wr(1-)+(-1)m(1-)wm+1(1-)2-nr=0mwr(1-)(-1)-r , one finds that the mode is stable when(31)11 mm .
although , mm depends on m , it turns out that mm tends toward its limit value(32)1m = limm1 mm .
( 10 ) and ( 11 ) with z  =  1 one sees that r=0(-1)rwr(1-)=21- , so that(34)1m=4(2-1)21-+limm(-1)m(1-)wm+1(1-)2-n21-,since =nsin2qx2 , replacing sin2qx2 by its highest value and since limm(-1)m(1-)wm+1(1-)=0 , therefore we find that the sufficient condition for the present method to be stable and this completes the proof of the theorem . 
 for 12<1 , the stability condition ( 19 ) can be satisfied under specific values of n=(t)(x)2 .
in this section , we present two numerical examples to illustrate the efficiency and the validation of the proposed numerical method when applied to solve numerically the fractional cable equation.example  1consider the following initial - boundary problem of the fractional cable equation(35)ut(x , t)=dt1-uxx(x , t)-dt1-u(x , t)+f(x , t),on a finite domain 0  <  x  <  1 , with 0    t    t , 0  <   ,   <  1 and the following source term(36)f(x , t)=2t+2t+1(2+)+t+1(2+)sin(x),with the boundary conditions u(0 ,  t )  =  u(1,t )  =  0 , and the initial condition u(x ,  0 )  =  0.the exact solution of eq .
( 35 ) is u(x ,  t )  =  tsin(x).the behavior of the exact solution and the numerical solution of the proposed fractional cable eq .
( 35 ) by means of the fwa  fdm with different values of  ,  ,  , t , x and the final time t are presented in figs .
15.in table 1 , we presented the behavior of the absolute error between the exact solution and the numerical solution of eq .
( 35 ) at =1,=0.9,=0.9,x=110,t=13000 and t  =  0.01.also , in table 2 , we presented the maximum error of the numerical solution for   =  0,  =  0.2 ,   =  0.7 , t  =  0.1 with different values of x and t.example  2consider the following initial - boundary problem of the fractional cable equation(37)ut(x , t)=dt1-uxx(x , t)-0.5dt1-u(x , t),0<x<10,0<tt , with u(0 ,  t )  =  u(10 ,  t )  =  0 and u(x ,  0 )  =  10(x    5 ) , where (x ) is the dirac delta function.the numerical solutions of this example are presented in figs .
 ,  ,  , x , t and the final time t. consider the following initial - boundary problem of the fractional cable equation(35)ut(x , t)=dt1-uxx(x , t)-dt1-u(x , t)+f(x , t),on a finite domain 0  <  x  <  1 , with 0    t    t , 0  <   ,   <  1 and the following source term(36)f(x , t)=2t+2t+1(2+)+t+1(2+)sin(x),with the boundary conditions u(0 ,  t )  =  u(1,t )  =  0 , and the initial condition u(x ,  0 )  =  0
the behavior of the exact solution and the numerical solution of the proposed fractional cable eq .
( 35 ) by means of the fwa  fdm with different values of  ,  ,  , t , x and the final time t are presented in figs .
in table 1 , we presented the behavior of the absolute error between the exact solution and the numerical solution of eq .
( 35 ) at =1,=0.9,=0.9,x=110,t=13000 and t  =  0.01 . also , in table 2 , we presented the maximum error of the numerical solution for   =  0,  =  0.2 ,   =  0.7 , t  =  0.1 with different values of x and t .
consider the following initial - boundary problem of the fractional cable equation(37)ut(x , t)=dt1-uxx(x , t)-0.5dt1-u(x , t),0<x<10,0<tt , with u(0 ,  t )  =  u(10 ,  t )  =  0 and u(x ,  0 )  =  10(x    5 ) , where (x ) is the dirac delta function .
to execute this aim , we have resorted to the kind of fractional john von neumann stability analysis . from the theoretical study
, we can conclude that this procedure is suitable and leads to very good predictions for the stability bounds .
the presented stability of the fractional weighed average finite difference scheme depends strongly on the value of the weighting parameter . numerical solutions and exact solutions of the proposed problem are compared and the derived stability condition is checked numerically . from this comparison , we can conclude that the numerical solutions are in excellent agreement with the exact solutions .
 in this article , numerical study for the fractional cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods . 
 the stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard john von neumann stability analysis . 
 a simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically . 
 numerical results , figures , and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method . 
Introduction
Finite difference scheme for the fractional Cable equation
Stability analysis
Numerical results
Conclusion and remarks
Conflict of interest