Finite difference method solving boundary condition

Oluwaseun Adisa

2021 5 22

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Hi try to solve the nonlinear ordinary difference equation by finite difference method with boundary conditions: U(-1) = 0, U(1)=0. The code below.

Function [r, sol] =res(U)

n = 10;

r = ((1:n)-1)/(n-1);

r0=zeros(n+1,1);

color={'r','m','c','b');

For delta = 2:5

Sol = solve(@(r)fun(r, n), r0);

norm(fun(sol, n));

Plot(r, sol(1:n), color{delta-1});hold on;

end

legend('delta=2', 'delta=3','delta=4','delta=5');

Function res=fun(z, n)

lambda =0.1;

beta = 0.1;

eps = 0.1;

m=0;

r = ((1:n)-1)/(n-1);

h=1/(n-1);

U = z(1:n);

res_U = zeros(n, 1);

res_U(1)=0;

For i=2:n-1

res_U(i)=((1+beta*U(i))/h^2)*(U(i+1)-2*U(i)+U(i-1))+((1+beta*U(i))/(r*h))*(U(i+1)-U(i-1))+beta*((U(i+1)-U(i-1))/h)^2+lambda*((1+eps*U(i))^m)*exp(U(i)/(1+eps*U(i)))-delta*U(i);

end

res_U(n)=0;

res = res_U;

end

Outcome

>> res

Warning: Trust-region-dogleg algorithm of Fsolve can not handle non-square systems; using levenberg-marquardt algorithm instead.

Equation solved. Fsolve completed because the vector of function values is near zero as measured by the default value of the function tolerance, and the problem appears regular as measured by the gradient.

'Please help me correct this errors'

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Torsten 2021-5-28

Is it necessary to write your own code ? If not, you should use sophisticated matlab code especially designed for this problem class, i.e. bvp4c in your case.