How can we find exact, analytic and numerical solution for any Second order PDEs. For Example Elliptic Equations
If your equations only involve slow-variant parameters, you may use WKB method to get the approximate analytical solution. Or you may try to assume that the solution takes the form of power series and get the semi-analytical solution. For the numerical solution, there are a variety of methods, such as finite difference method, finite element method. But for the further discussion in more details, it would be the best that you could post the equations here.
Your question is too general. Different methods will often apply depending on whether your PDEs are elliptic, parabolic or hyperbolic. Some are better solved with finite elements, some with spectral methods, while others may be solved with a variety of methods. As Dr Zhang says, semi-analytical methods could be used if there are two very different timescales in the solution. Then there is the question of how accurate should the method be: 2nd order, 4th order or even better than that? Explicit/implicit schemes? Is your equation of second order (with a Laplacian), or fourth order (with the biharmonic operator), or are you solving a coupled set of PDEs?
Check out the following links:
http://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01379-4/S0273-0979-2012-01379-4.pdf
http://aero-comlab.stanford.edu/Papers/pde-mixed-types.pdf
1. Dautray, R. and Lions, J.-L. (2000), Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4., Springer
2. Samarskii, A. A. (1971), Introduction to the Theory of Difference Schemes, Nauka, Moscow (Russian).
Perhaps we should wait for Dr Abdallah to post the equations he has in mind, as suggested by Dr Zhang.
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