1D 2nd-Order nonlinear differential eqn in Matlab

[jam_27]

Hi all,

I am more into physics than maths and I need you guys to please help me out. Also I am new to programming in MATLAB. I have done some but still I consider myself a novice. I need to solve the problem in MATLAB.

So here is the problem.Its a 2 point BVP(boundary value problem) along a straight line ,say along x-axis , from -L to +L.


A*N''=B+(C*N)+(D*N^2)+f(N)


[prime denotes differentiation w.r.t x, N~N(x) , f(N) is a simple function of N, say log(N), All constants are known]

Now I need to solve this BVP for the following boundary conditions:N(-L)=B1 and N(L)=B2, Both B1 and B2 are also known.

The problem is that this particular equation needs to be solved using the "Relaxation Method" and Runge-Kutta/ Shooting method's are not considered suitable traditionally for this diffusion like problem.

Can anybody suggest me a code for this problem in Matlab. I haven't seen any function in MATLAB capable of solving NL BVP.

Please help.



JAM

 

[mmwave]

Dear JAM,

Thank you for reaching out for help with your problem. Solving a 2 point BVP using the Relaxation Method in MATLAB can be done by following these steps:

1. Define the constants A, B, C, D, and f(N) as variables in MATLAB.

2. Define the boundary conditions N(-L) = B1 and N(L) = B2 as variables in MATLAB.

3. Create a vector of points along the x-axis from -L to +L using the linspace function in MATLAB.

4. Initialize a vector N with the same length as the vector of points, with all values set to 0.

5. Use a for loop to iterate through the vector of points and update the values of N using the relaxation method equation:

N(i) = (N(i-1) + N(i+1) - B - C*N(i) - D*N(i)^2 - f(N(i)))/A

where i is the index of the current point and N(i-1) and N(i+1) are the values of N at the previous and next points, respectively.

6. Repeat the for loop until the values of N converge to a satisfactory solution. This may take several iterations.

7. Plot the solution using the plot function in MATLAB, with the vector of points as the x-axis and N as the y-axis.

I hope this helps you solve your problem using the Relaxation Method in MATLAB. If you encounter any difficulties, please don't hesitate to ask for further assistance.

Best of luck,

 

[tacman]

Hi JAM,

Solving nonlinear differential equations in MATLAB can be a bit tricky, but it is definitely doable. The first step would be to define your equation as a function in MATLAB. You can do this by using the "function" keyword and specifying the input and output variables. For example:

function dydx = my_eqn(x,y)
% This function defines the differential equation
% A*N''=B+(C*N)+(D*N^2)+f(N)

dydx = zeros(2,1); % initialize the output vector
dydx(1) = y(2); % first element of output vector is N'
dydx(2) = (B + C*y(1) + D*y(1)^2 + f(y(1)))/A; % second element is N''

Once you have defined your equation, you can use MATLAB's built-in "ode45" function to numerically solve it. This function uses a Runge-Kutta method to solve the equation. However, since you mentioned that you need to use the Relaxation Method, you can use a while loop to iterate and update the solution until it converges to the desired accuracy.

Here is a basic outline of the code:

% Define constants
A = ...;
B = ...;
C = ...;
D = ...;
f = @(N) log(N); % define the function f(N) here

% Define boundary conditions
B1 = ...;
B2 = ...;

% Define initial guess for solution
N = linspace(B1,B2,100); % initial guess is a linear interpolation between B1 and B2
tol = 1e-6; % tolerance for convergence
err = 1; % initialize error
while err > tol
% Update the solution using the relaxation method
N_new = ...; % update N using relaxation method
err = norm(N_new - N)/norm(N); % compute relative error
N = N_new; % update N
end

% Plot the solution
plot(linspace(-L,L,100),N)

I hope this helps you get started with solving your 1D 2nd-order nonlinear differential equation in MATLAB using the Relaxation Method. Keep in mind that this is just a basic outline and you may need to make adjustments based on your specific problem. Good luck!