How should I solve coupled differential non linear equations? Given below.

I want to solve equations A and B to find beta. Please help. lambda = 1.550; zmax = 2290; zmin = 300; N = 50; z = zeros(0,N); phi = zeros(0,N); r2 = 5; phimax = 50; phimin = 1; mu0 = 4*pi*10^-1; epsilon0 = 8.85*10^-6; k = 2*pi/lambda; for n = 1:N z(n)= ((n-1)*(zmax-zmin))/(N-1)+zmin; phi(n)= ((n-1)*(phimax-phimin))/(N-1)+phimin;

end

n1 = z.*(1-0.015.*exp(-r2)/1.795); h(phi) = ((sqrt(epsilon0/mu0).*(1/k).*(beta(z).*e(z)+W)));

e(phi) = -((sqrt(mu0/epsilon0).*(1./k.*n(z).^2).*(beta(z).*h(z)+V))); e(z)=-1/1i.*((sqrt(mu0/epsilon0).*(1./k.*n(z).^2).*(T-Q))); h(z)= 1/1i.*((sqrt(epsilon0/mu0).*(1/k.*z).*(U-S)));

syms z f(z)=1i.*h(z); V=diff(f,z); syms z f(z)=1i.*e(z); W=diff(f,z);

syms z f(z)=z.*h(phi); T=diff(f,z); syms z f(z)=z.*e(phi); U=diff(f,z); syms z f(z) = ln(n1(z).^2); df = diff(f,z); syms z f(z)=e(z); P=diff(f,z); syms z f(z)=h(z); Q=diff(f,phi); syms z f(z)=h(z); R=diff(f,phi); syms z f(z)=e(z); S=diff(f,phi);

A= del2(e(z))+(n1(z)^2.*k^2-beta(z)).*e(z)-(df.*(beta(z)./((n1(z)^2.*k^2-beta(z)))).*(beta(z).*P+sqrt(mu0/epsilon0).*(k./z).*R)); B = del2(h(z))+(n1(z)^2.*k^2-beta(z)).*h(z)-(df.*(beta(z)./((n1(z)^2.*k^2-beta(z)))).*(Q-(beta(z)./k.*z).*sqrt(epsilon0/mu0).*S));

[Star Strider]

If they are nonlinear, it is best to not use the Symbolic Math Toolbox, since very few nonlinear equations have analytic solutions.

Integrate them numerically with ode45 (or ode15s or whatever ODE solver is most appropriate to your functions).

[chetna sharma]

Thanks Star. I was trying to use ode45- couplode = @(t,z) [ del2(e(z))+(n1(z)^2.*k^2-beta(z)).*e(z)-(df.*(beta(z)./((n1(z)^2.*k^2-beta(z)))).*(beta(z).*P+sqrt(mu0/epsilon0).*(k./z).*R));del2(h(z))+(n1(z)^2.*k^2-beta(z)).*h(z)-(df.*(beta(z)./((n1(z)^2.*k^2-beta(z)))).*(Q-(beta(z)./k.*z).*sqrt(epsilon0/mu0).*S))]; [t,y] = ode45(couplode, [0 0.49*pi], [1;1]*1E-8); figure(1) plot(t, y) grid str = {'$$ \dot{y} $$', '$$ y $$', '$$ \dot{x} $$', '$$ x $$'}; legend(str, 'Interpreter','latex', 'Location','NW') after taking help from one of the answers here posted by someone.

But its not working. I don't know how to use it correctly as I have never used ode before. Can you please help ?

Thanks Chetna.