Dan Kelley is right, your question is far too broad to answer in any meaningful way without knowing more detail. Have you established the overall nature of the system? This is one of the first steps you will need to take, as numerical methods vary depending upon whether, for example, the system is parabolic or hyperbolic, what sort of variable coupling exists amongst the equations, etc. You may want to post more details here. Also, I would start with a good general reference on numerical solution of PDEs, like Tannehill Anderson and Pletcher, for example. If nothing else, you'll get a lot of guidance on what the preliminary steps are to point you in the right direction. It sounds like your problem is not at all trivial.

Could one go for a discretization in time first to reduce the system to a system of ODEs in the spatial variable and then employ the standard ODE methods to solve the problem? 

You are asking a very broad question on a complicated topic.

If you're at a university, walk over to the library and look on the shelves with call number starting QA.  Flip through some books until you find one that addresses problems of interest in a method that suits you.

Dan Kelley is right, your question is far too broad to answer in any meaningful way without knowing more detail. Have you established the overall nature of the system? This is one of the first steps you will need to take, as numerical methods vary depending upon whether, for example, the system is parabolic or hyperbolic, what sort of variable coupling exists amongst the equations, etc. You may want to post more details here. Also, I would start with a good general reference on numerical solution of PDEs, like Tannehill Anderson and Pletcher, for example. If nothing else, you'll get a lot of guidance on what the preliminary steps are to point you in the right direction. It sounds like your problem is not at all trivial.

I realise that I have not provided the details of the problem. The equations arise in context of 1-d non newtonian fluid flow and involve conservation equations. The steady state equations do not have enough boundary conditions on one end so that a shooting method is used to compute the stationary solutions. This is done by guessing one boundary condition at one end and then matching it with another boundary condition prescribed at the other end. I am aware that for the time dependent case there are methods such as FEM that can used, but I  was wondering if there could be an easier method as I am not very comfortable with FEM. As I write this post I realise how little information I gave in my question! I apologise as this is my first interaction at this site and I thank everyone for taking time to respond to the question nevertheless.

Are you implementing the calculation method yourself, or are you trying to find a good established program to use? One problem with non-Newtonian simulation is stress singularities. I would recommend reading a paper from a while back by Gyanendra Sasmal (1995, J. Non-Newtonian Fluid Mech. 56(1), 15-47) who described a novel finite-volume (not FEM) method for resolving the singularity problem. He ran cases simulating converging flow for Deborah numbers up to around 6, if I remember correctly. This paper might help you along with your own problem.

Thanks Michael, this paper might be pretty helpful. As of now, even for the steady state case we are able to do simulations only for  Deborah numbers close to 1. Realistic values of Deborah number are much higher and the goal is to simulate the behaviour as physically close to reality as possible. 

One should be careful when quantizing PDEs.  For example, with non-linear hyperbolic conservation laws, quantizing one dimension (e.g., space) and leaving the other dimension continuous (obtaining a system of ODEs) can introduce 'numerical viscosity' (also sometimes called numerical diffusion), which (depending on application) produces possibly highly undesirable behavior in the solution, which should not be there in the first place.

I believe that  Laplace decomposition method be a good tool for solving this kind of PDEs

I don't understand why 1-D non-Newtonian problem needs to use shooting method for the BC. The the GE the ordinary conservative equations or the re-derivative PDE. If we dealing with the ordinary continuity, momentum and energy equation, there have been many approaches for this problem. Check with the book by  Tannehill, Anderson & Pletcher, or Patankar.

I would follow the advice of Paul D with highest chance to solve the problem.

Hello Renu Dhalwal

You say have a system of 6 nonlinear PDEs and that these equations involve a time derivative and one spatial derivative. That means that you have 6 dependent variables which is quite unusual. You say you are not comfortable with the FEM but that is exactly the simplest tool you need to solve your problem numerically. Would you refer to the text book by   I. M. Smith and D. V. Griffiths " Programming the finite element method". It is time for you to learn FEA. Look p this book online or in your library. This book is written by engineers and gives an elementary presentation of the FEM: viz how the stiffness or coefficient matrix is derived using the Galerkin weighted residual procedure. In addition it explains the FEM solution of the 2-D time independent (steady state)  Navier-Stokes PDES  involving 3 nonlinear equations is obtained. It is also explains how 1D time dependent PDEs are set up and marched forward in time. It is unfortunate, but FE books written by mathematicians are not readily understood. You did not provide the PDES hence one can't really say much. It would be desirable for you to provide these equations with some alterations (if you are afraid of compromising your copyright) so that the 'stiffness' matrix can be developed for you. You can then consult a colleague with expertise in FE programming to help you solve your problem.

Try EASY-FIT which can be downloaded from http://www.klaus-schittkowski.de//home.htm

Hi everyone. I am trying to solve a similar problem having details as follows.

I am checking different discretisation methods for solving a second degree PDE for space and first degree for time. More of like a diffusion equation. What best discretisation method can I use to retain stability? I tried pdepe solver in MATLAB but it doesn't give perfect results (which uses method of lines).

Hello  Sanket Jadhav

Please, ask this your question under your own name. This forum belongs to  Renu Dhadwal.