Time fractional PDE

General Discussion
1

time-fractional pde
time-fractional pde1063Ă—249 37.9 KB

Hello,

I want to solve this problem by FreeFEM for alpha = 0.6 in a unit square domain.
Kindly share your suggestions.
Thanks

2

I sorry, no real idea but a problem is due to the no local in time of the time fractional term, so I think the simple way is to use Fourier method.

1 Like
3

Dear Sir,

I somehow tried to solve the problem, but I don’t know where I am going wrong as answer is not what is expected.

border AB(t= 0, 1){x=t; y=0; label = 1;};
border BC(t= 0, 1){x=1; y=t; label = 2;};
border CD(t= 1, 0){x=t; y=1; label = 3;};
border DA(t= 1, 0){x=0; y=t; label = 4;};
int n=20;
mesh Th = buildmesh(AB(n)+BC(n)+CD(n)+DA(n));
plot(Th, wait=1);

func u0 = sin(pi*x)sin(piy);
real T= 1;
real dt= 0.1;
real alpha = 0.7;
real gm = 1/tgamma(2- alpha);

fespace Vh(Th, P1);
Vh u, v;

Vh[int] uu(15);
Vh[int] sum11(15);
// first iteration

//for(real t = 0; t < T; t += ){
for(int k=0; k < 10 ; k++){
uu[0]=u0;
u= uu[k+1];

sum11[0] =0;
sum11[1] =0;
for(int j=0; j<= k; j++){
sum11[3] = sum11[0] + uu[k+1-j]((j+1)^(1-alpha))/(dt)^alpha - uu[k+1-j]((j)^(1-alpha))/(dt)^alpha ;
sum11[2] = sum11[1] + uu[k-j]((j+1)^(1-alpha))/(dt)^alpha - uu[k-j]((j)^(1-alpha))/(dt)^alpha ;
sum11[0] = sum11[3];
sum11[1] = sum11[2];
}

solve thermic(u, v)
= int2d(Th)( sum11[0]vgm)
- int2d(Th)( sum11[1]vgm)
+ int2d(Th)(dx(u) * dx(v)+ dy(u) * dy(v))
+ on(1,2,3,4, u=0);

plot(u, cmm=“time=”+k, fill=1, wait=0, value=1);

uu[k+1] = u;
//dt = 0.1 + dt;
}