Can someone help me to correct the code ​​for this problem using Crank-Nicolson finite difference implicit method?

Question is:

Governing equations:

Boundary conditions:

The Crank-Nicolson finite difference implicit method is used to solve the above equations:

As tried here, I want to plot graphs for velocity profile using parameters like Pe, beta, and N.

This trial code generates plain plot without errors.

Can anyone help me solve this problem?

Trial code

% Trialcodecranknicolson analytical code

function vj_crank

clear all;

clc;

mr=41; mz=41; mt=1000; Pe=10; Kr=1; Sc=1; lambda=1; gamma=pi/8;

tp=600; ttp=500; tttp=100; Gr=1; Gc=1; S=1; Re=1; M=0.5; t=0.1; w=pi; N=1; Da=0.5; beta=1;

dely=0.05;

g=exp(1.i*w*t);

Peva=[1 2 3 4];

for iiii=1:4

Pe=Peva(iiii);

X=linspace(-1,1,mr+1);

V=zeros(mr,mt);

T=zeros(mr,mt);

C=zeros(mr,mt);

%Initial Conditions

for i=1:mr+1

for j=1:mt+1

for k=1:mz+1

V(i,1)=0;

T(i,1)=1;

C(i,1)=1;

end

end

end

for i=1:mr+1

for j=1:mt+1

for k=1:mz+1

V(1,j)=0;

T(1,j)=0;

C(1,j)=0;

end

end

end

% Boundary Conditions

for i=1:mr+1

for j=1:mt+1

for k=1:mz+1

V(mr+1,j)=0;

T(mr+1,j)=0;

C(mr+1,j)=0;

end

end

end

for j=1:mt

for k=2:mz

for i=2:mr

V(i,j+1)=(1/1+beta*1.i*w)*((V(i-1,j)-(2*(V(i,j)))+V(i+1,j)+V(i-1,j+1)-(2*(V(i,j+1)))+V(i+1,j+1))/(2*(dely)^2))+(Re*S)*((V(i+1,j)-V(i,j))/(dely))-(lambda)+(Gr*sin(gamma))*((T(i,j+1)+T(i,j))/(2))+(Gc*sin(gamma))*((C(i,j+1)+C(i,j))/(2))-((Da+M-(Re*1.i*w))*((T(i,j+1)+T(i,j))/(2)));

T(i,j+1)=((T(i-1,j)-(2*(T(i,j)))+T(i+1,j)+T(i-1,j+1)-(2*(T(i,j+1)))+T(i+1,j+1))/(2*(dely)^2))+S*((T(i+1,j)-T(i,j))/(dely))+((N-(Pe*1.i*w))*((T(i,j+1)-T(i,j))/2));

C(i,j+1)=(Sc)*((C(i-1,j)-(2*(C(i,j)))+C(i+1,j)+C(i-1,j+1)-(2*(C(i,j+1)))+C(i+1,j+1))/(2*(dely)^2))+S*((C(i+1,j)-C(i,j))/(dely))-((Kr+1.i*w)*((C(i,j+1)-C(i,j))/2));

end

end

end

figure (1)

hold on

grid off

plot(X,V(:,tp)*g,'-','linewidth',2);

hold off

end

[Torsten]

Crank-Nicolson scheme for a stationary boundary value problem ? I cannot imagine how you want to apply it in this case.

Use bvp4c or bvp5c instead.

[VAIDEHI]

Okay, Thank you sir.

[VAIDEHI]

Dear sir,

Can I use Crank Nicholson method to solve the below equations?

[Torsten]

Can I use Crank Nicholson method to solve the below equations?

Maybe, but I have no idea how to discretize and solve the equation for u together with the third-order term.

You have a fourth equation for p ?

[VAIDEHI]

Yes sir