# Question:The solution of a PDE

## Question:The solution of a PDE

Maple 2020

Dear all,

I have a time-fractional PDE as follows.  ( denotes Caputo fractional derivative with respect to t)

for alpha=1, this is a classical PDE and the exact solution is given as follows (in a book)

Question:

1) for alpha=1, I want to find the L2 errors and L∞ errors in a table.

2) for alpha=0.5, Can Maple find a solution (numeric or exact)?

MY TRY: (MAPLE 2020.2)

```restart:
with(plots):
PDE:=diff(y(x,t),t)=y(x,t)*diff(y(x,t),x\$3)+y(x,t)*diff(y(x,t),x)+3*diff(y(x,t),x)*diff(y(x,t),x\$2) ;

#c is an arbitratry constant
c:=4:
exact_sol:=(x,t)->-8*c/3*(cos ((x-c*t)/4))^2;

# I selected initial and boundary conditions as follows
IBC := { y(x,0)=exact_sol(x,0),y(0,t)=exact_sol(0,t),D[1](y)(0,t)=D[1](exact_sol)(0,t),y(1,t)=exact_sol(1,t)};

numeric_sol := pdsolve(PDE,IBC,numeric);

num3d:=numeric_sol:-plot3d(t=0..1,x=0..1,axes=boxed,
color=[0,0,y]);
exact3d:=plot3d(exact_sol(x,t),t=0..1,x=0..1,axes=boxed);
display(exact3d,num3d);

pdetest(y(x,t)=exact_sol(x,t),PDE,IBC);

```
﻿