Hello everyone,
I have two non-linear autonomous second order ODE with initial ,boundary and continuity conditions(4 conditions) to find four constants. Which method can be used to solvefor this complex non-linear case? Answers would be much appreciated. Thanks !
He wants analytic solution, not numerical solution, previous responder.
My suggestion would to represent each solution as an infinite power series, and then substitute of the solution and its derivatives into your system. The goal would be to find a recursive relationship between the coefficients in the power series - your initial/boundary conditions will start the recursive pattern. You can either leave your solution as an infinite series, or cut off higher order terms in the power series if it has asymptotic properties - based on what your goal is.
Since they are autonomous, you could try using the quadrature method developed by Laetsch. You can find more information on the quadrature method in Laetsch's original paper: T. Laetsch. The number of solutions of a nonlinear two point boundary value problems, Indiana University mathematics Journal, Vol.20, No.1, (1970), 1-13.
My suggestion is to contact or see the work of L.N.T in spectral methods.
Hope you will find some idea to solve your problem.
Prof Rath
Dear,
You could convert into the system of first order equations then solve by method of first order system ODEs.
Dear,
Vivekananthan Balakrishnan see the attachment. Also see the Sachin Bhalekar and Varsha Daftardar-Gejji work.
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