How to solve differential equations while variables are coupled?

(p-1)*U1r-d(U1i)/dx+L*U2r-n*d^2(U1r)/dx^2=0;

(p+1)*U1i+d(U1r)/dx+L*U2i+n*d^2(U1i)/dx^2=0;

(p-1)*U2r-d(U2i)/dx+L*U1r=0;

(p+1)*U2i+d(U2r)/dx+L*U1i=0.

How to solve U1r, U1i, U2r, and U2i by using MATLAB symlbolic tool ?

Thanks in advance.

[John D'Errico]

1. Please don't post an answer just to thank the person who answered your question.
2. DO learn to use comments.
3. And don't accept your own non-answer.

I've unaccepted your non-answer, and accepted Ameer's answer for you.

Answer 1

[Ameer Hamza]

Try this code.

syms U1r(t) U1i(t) U2r(t) U2i(t) p n L

eq1 = (p-1)*U1r - diff(U1i) + L*U2r - n*diff(U1r,2) == 0; 
eq2 = (p+1)*U1i + diff(U1r) + L*U2i + n*diff(U1i,2) == 0; 
eq3 = (p-1)*U2r - diff(U2i) + L*U1r == 0;
eq4 = (p+1)*U2i + diff(U2r) + L*U1i == 0;

eq = [eq1; eq2; eq3; eq4];
cond = [U1r(0)==0 U1i(0)==0 U2r(0)==0 U2i(0)==0];
sol = dsolve(eq, 'IgnoreAnalyticConstraints', true, 'MaxDegree', 4)

However, the analytical solution is a "bit" messy. Following is output of Live script.

disp(sol.U1r)

disp(sol.U1i)

disp(sol.U2r)

disp(sol.U2i)

I recommend using a numerical method.

[Md Bellal Hossain]

Thanks for your answer.

But How long it takes to get the solutions? I have run the code in the morning, but I still don't have the solutions.

[Ameer Hamza]

Which MATLAB release are you using? On R2020a, I got a solution within a few seconds. (MATLAB have made improvements in the symbolic engine in R2020a). I am using a typical i7 laptop, so computation speed is not a factor.

[Md Bellal Hossain]

Today I installed R 2019b and I got the solution. Actually, I need to compare the analytical solution with the numerical solution and I already have the numerical solution. Is there any other ways to solve this equations analytically? I want to have the solution a little bit simple form so that I can compare it with the numerical solution. Can you do that for me?

[Md Bellal Hossain]

How can I get the full solution like you?

When I run then after some result I found ... Output truncated. Text exceeds maximum line length for Command Window display.

[Ameer Hamza]

Run this code in Live script. I guess live script was introduced somewhere near R2017.

[Md Bellal Hossain]

Thank you very much.