How can I program the code for solving PDE equation using finite difference method?

Hi, I am trying to solve a PDE governing equation using finite differenc method and I am having trouble to set up the programming code for this equation together with the initial and boundary condition. I tried to do the programming code, but it always show 'Unable to perform assignment because the size of the left side is 1-by-1 and the size of the right side is 1-by-300'. The equation is dudt= A0+A1*cos(omega*t) +Beta1* [d2udr2+1/r(dudr)], initial condition is u(r,0) = 0, and boundary condition is u(1,t) = 0, t>0.

[Voss]

Please show MATLAB code.

[Nur Nadhirah Syed Malik]

%Parameters to define the governing casson fluid equation and the

%parameters value range

L = 1; % Length of the artery

maxk= 2500; % Number of time steps

beta1 = 0.025; % Beta parameter

delta_t = 0.0001; % Time step

delta_r = 0.025; % Radial direction

A0 =0.2; % Amplitude of systolic pressure gradient

A1 =0.4; % Amplitude of diastolic pressure gradient

n = 50; % Number of space steps

du = L/n;

r = 1; %Radius of artery

omega = pi/4;

%Initial conditions of velocity

for i = 1:n+1

u(i) =(i-1)*du;

u(r,1)= 0; % 0<=r<=1

end

% Boundary conditions

for k=1:maxk+1

u(1,k) = 1.;

u(n+1,k)=0.;

end

% Implementation of the explicit

for k=1:maxk % Time Loop

for i=2:n % Space Loop

u(i,k+1) = u(i,k)+ delta_t*(A0 +A1*cos(omega*t)+beta1*((u(i+1,k)-2*u(i,k) + u(i-1,k))/(delta_r)^2 + 1/r *(u(i+1,k)-u(i-1,k))/(2*delta_r)));

end

end

% Graphical representation of the velocity at different selected times

figure(1)

plot(u(:,1),y,'-',u(:,100),y,'-',u(:,300),y,'-',u(:,600),y,'-')

tittle('velocity within explicit method')

xlabel('r')

ylabel('t')

legend('dt=1','dt=100','dt=300','dt=600')

Answer 1

[Rishabh Singh]

Hi,

In the line below,

u(i,k+1) = u(i,k)+ delta_t*(A0 +A1*cos(omega*t)+beta1*((u(i+1,k)-2*u(i,k) + u(i-1,k))/(delta_r)^2 + 1/r *(u(i+1,k)-u(i-1,k))/(2*delta_r)));

make sure to replace 't' with 'k'. Also the code which you have provided does not produce the error as mentioned by you. Also can you provide more information regarding the variable which you are trying to plot. Is it velocity vs time or something else?