The specific method will depend upon the equations and kinds of terms that comprise them. You haven't given any real specifics, except to say they are non-linear. Are they "standard" equations, e.g. from mechanics, or is this a novel set of equations for which numerical solutions have not been attempted before?

Dear Dr Ayesha,

                      You have to study my paper in which I discuss a short topic about this issue.The result is not enough strong but I think it will make a sense to you. 

Thanks a lot for the answers and the input. yes it is a novel set of eqs Michael. I have solved the problem using iterative procedure of a variational method. 

If the set of equations is in stationary state, then a numerical approach you can use is the finite difference of elliptic equations.

if the set of eqs. varies in time, finite difference of parabolic approaches is a feasible numerical method.

Or well, if boundary conditions are difficult to establish, the finite element approach is a suitable numerical alternative.

Good luck!

Have you considered an analog computer?

you can use 

FEM

ADM

HPM

HAM

FDM

OHAM

VIM

I fully agree with the above answer, the collocation method based on Gauss-Lobatto quadrature nodes is more accurate numerical method. If the kernel is singular it is better to Gauss quadrature nodes as collocation points.

Best wishes

I think that spectral methods such as collocation or Galerkin methods are the best solvers to approximate these equations. But for weakly singular kernels a regularization proicess or suitable basis functions must be used to obtain a satisfactory rate of convergence.

You can see my papers about this topic.

Many of the suggestions here are potential approaches. I will add one more variant with which I found success - a "Method of Lines" approach coupled with a Gauss or Gauss-Lobatto quadrature. This method is described in my dissertation, and also in the following paper:

V.A. Jairazbhoy & L.L. Tavlarides, “A Numerical Technique for the Solution of Integrodifferential Equations Arising from Balances over Populations of Drops in Turbulent Flows”, Computers and

Chemical Engineering, Vol. 23, pp. 1725-1735 (2000)