Question

Example 9.2. One-Dimensional Parabolic PDE: Heat Flow Equation Consider the parabolic PDE 5 1.0515 for 0s1,0S10.1(E92.1) with the initial condition and the boundary conditions E9 2.2 Solve the paraholic PDE ing the Expict Forward Ealer and Crank-Nicholson methods both asalyically and aunericllyMATLAB code) Plol 2-D and 3D gnpha.

Answer 1

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MATLAB Code for explicit forward Eular method:

clc; clear;
delta_t = 0.001; %%% grid size in time
delta_x=0.1; %%% grid size in space
m = 10; %%% number of grid points
N = 100; %%% levels in time
lemda =delta_t/delta_x^2;
x(1:m+1)= delta_x * ((1:m+1)-1);
%%%%%%%%%%% initial condition
for i = 1:m+1
u(1,i) = sin(pi * x(i));
end
%%%%% boundary conditions
u(1,1)= 0;
u(1,m+1)= 0;
%%%%% solution of the equation
for n = 1:N+1
u(n+1,1)= 0;
u(n+1,m+1)= 0;
for i = 2 : m
u(n+1,i) =lemda*(u(n,i+1)+u(n,i-1))+(1-2*lemda)* u(n,i)
end
end

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MATLAB Code for Crank Nicholson method:

clc; clear;
delta_t = 0.001; %%% grid size in time
delta_x=0.1; %%% grid size in space
m = 10; %%% number of grid points
N = 100; %%% levels in time
lemda =delta_t/delta_x^2;
x(1:m+1)= delta_x * ((1:m+1)-1);
a1(1:m-1)=1+lemda;
a2(1:m-2)= -lemda/2;
a3(1:m-2)= -lemda/2;
A=diag(a1)+ diag(a2,1)+ diag(a3,-1);
b1(1:m-1)=1-lemda;
b2(1:m-2)= lemda/2;
b3(1:m-2)= lemda/2;
B=diag(b1)+ diag(b2,1)+ diag(b3,-1);
for i = 1:m+1
u(1,i) = sin(pi * x(i));
end
%%%%% boundary conditions
u(1,1)= 0;
u(1,m+1)= 0;
%%%%% solution of the equation
for n = 1:N+1
C = B* u(n,2:m)';
u(n+1,2:m) = inv(A)* C
end

Comments:

the 2D and 3D plots can be drawn by using plot (x,y) and plot3 (x,y,z) command in MATLAB. I will upload them soon.