I'm trying to redo the Poisson equation so that ∂²Φ /∂x² = 2x² - 0.5x + exp(x). Every time I try and input the right-hand side of the equation it comes up with a syntax error, any help would very much be appreciated.
% Solving of 1D Poisson equation
% using finite differences
% Clearing memory and figures
clear all
clf
% Define 1D numerical model
xsize=100; % Model size in horizontal direction, m
Nx=101; % Number of grid points
dx=xsize/(Nx-1); % Gridstep, m
x=0:dx:xsize; % Coordinates of grid points, m
% Defining global matrixes
L=zeros(Nx,Nx); % Koefficients in the left part
R=zeros(Nx,1); % Right parts of equations
% Composing global matrixes
% by going through all grid points
for j=1:1:Nx
% Discriminating between BC-points and internal points
if(j==1 || j==Nx)
% BC-points
L(j,j)=1; % Left part
R(j)=0; % Right part
else
% Composing Poisson eq. d2T/dx2=1
%
% ---T(j-1)------T(j)------T(j+1)---- STENCIL
%
% 1/dx2*T(j-1)+(-2/dx2)*T(j)+1/dx2*T(j+1)=1
%
% Left part of j-equation
L(j,j-1)=1/dx^2; % T(j-1)
L(j,j)=-2/dx^2; % T(j)
L(j,j+1)=1/dx^2; % T(j+1)
% Right part
R(j)=2x;
end
end
% Solve the matrix
S=L\R;
% Reload solutions
T=zeros(1,Nx); % Create array for temperature, K
for j=1:1:Nx
T(j)=S(j);
end
% Visualize results
figure(1); clf
plot(x,T,'o r') % Plotting results
Again, I appreciate any help in this matter
The problem is clear: "input the right-hand side of the equation it comes up with a syntax error", to correct the error you should properly define the right-hand side of the equation:
R(j)=2*x(j)^2-0.5*x(j) + exp(x(j));
Since it depends on x
, you should point to each element j
when calculating the coefficients in the for loop on the left and right hand sides of the equation.
Analytical solution of the PDE is:
Changing this line of code and plotting the discretized and analytical solutions produces
% Visualize results
figure(1); clf
plot(x,T,'o r');
hold on
plot(x , 0.166667*x.^4-0.0833333*x.^3-268811714181613573321934397213431784538112*x+exp(x)-1)
which seems correct. Differences are due to discretization itself.