Can Matlab ODE solver solve this problem

Can Matlab ODE solver solve this problem

[Torsten]

Why are there two independent variables (eta and lambda) ?

Best wishes

Torsten.

[WEI JIN]

Yes, eta and lambda are two different independent variables. The function to be solved q(eta) only depends on eta, and the function p(lambda) only depends on lambda. Can ordinary differential equation solver solve this problem? Or still consider the system as a partial differential equation with two independent variables, the two functions to be solved depends on both independent variables, i.e., q(eta,lambda), p(eta,lambda), with the condition that q^{\prime}(\lambda) =0 and p^{\prime}(\eta)=0.

[Jan]

It depends on what eta, lambda, q, p, q' and p', f and g are. Please provide any details. If the problem can be solved at all (numerically or symbolically?), then it is possible with Matlab also - most likely.

[WEI JIN]

These are two implicit ordinary differential equations, so the function form f and g are known. The unknown functions to be solved are q and p. The question is can we use ODE solver to numerically solve the function q=q(eta) and p=p(lambda) that satisfy the two equations.

Answer 1

[Torsten]

Given p(lambda_i) and q(eta_i), you can determine p(lambda_i+delta_lambda) and q(eta_i+delta_eta) by solving the system of equations

f(eta_i+delta_eta,lambda_i+delta_lambda,q(eta_i+delta_eta),p(lambda_i+delta_lambda),(q(eta_i+delta_eta)-q(eta_i))/delta_eta,(p(lambda_i+delta_lambda)-p(lambda_i))/delta_lambda)=0

g(eta_i+delta_eta,lambda_i+delta_lambda,q(eta_i+delta_eta),p(lambda_i+delta_lambda),(q(eta_i+delta_eta)-q(eta_i))/delta_eta,(p(lambda_i+delta_lambda)-p(lambda_i))/delta_lambda)=0

using "fsolve", e.g.

So starting from p(lambda_start) and q(eta_start) (initial values), you can integrate your system up to specified values for lambda_end and eta_end.

Look up "method of characteristics" for more details.

Best wishes

Torsten.