Fourier method (old but renewed in the method of pseudo differential operators and Fourier integral operators), spectral decomposition, small parameter method , perturbation method.

Dear Gro... In recent years, quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have been proposed. A variety of powerful methods, such as, tanh-sech method , extended tanh method], hyperbolic function method , Jacobi elliptic function expansion method, F-expansion method, First Integral method , The sine-cosine method, has been used to solve different types of nonlinear systems of PDEs. Some of these methods present exact solutions, the question is which of these methods presents the best exact solution for the same boundary and initial conditions of the problem?

Dear Anwar

Will it be possible for you to share literature for these methods.

Dear Mazhar

Kindly you can see the applications of these methods on my contributions on the link:

https://www.researchgate.net/profile/Anwar_Jawad/contributions/?ev=prf_act

Regards

Anwar

Contributions are helpful to many researchers

Yes and all contributions can be free download to help researchers

Shouldn't we limit the discussion of PDE type first? I don't think there is one method good for all three types of PDE.

Ok.

I think the solution calculated from any of these methods that verify the nonlinear PDE will be the ( exact solution ) with no error, Other solutions (Numerical or analytical ) calculated by the other methods will be compared with the Exact solution and decide its reliability and effectiveness.

Fritz Schwarz, Fraunhofer Institut, Sankt Augustin

For linear equations - ordinary and partial - the best method is determining the

Loewy Decomposition of the given equation. There is a recent book on the subject

entitled "Loewy Decomposition of Linear Differential Equations" (Springer Publisher); a shorter review is available here:

http://link.springer.com/article/10.1007%2Fs13373-012-0026-7

The methods listed in the question are in fact closely related to the invariant solution method and, in a sense, provide some explicit recipes on how to find solutions for the equations (usually ODEs) describing invariant solutions. In turn, the invariant solution method is a part of the large set of symmetry-based methods which are pretty much the most powerful and universal methods for finding analytic, closed-form solutions for *nonlinear* ODEs and PDEs. The symmetry approach has several main flavors: essentially you can employ symmetries to either lower the order of your system, find a transformation which brings your system to a simpler form (e.g. linearizes it), or find solutions invariant under a given symmetry. There are many excellent books on symmetries, see e.g. P.J. Olver, Applications of Lie Groups to Differential Equations (http://www.springer.com/mathematics/analysis/book/978-0-387-95000-6) and H. Stephani, Differential Equations: Their Solution Using Symmetries (http://books.google.com/books?isbn=0521366895).

On the oher hand, the invariant solutions method can be viewed as a special case of the method of differential constraints. For an up-to-date introduction to the latter, see e.g. the book by S.V. Meleshko (https://www.researchgate.net/profile/Sergey_Meleshko/), Methods for Constructing Exact Solutions of Partial Differential Equations (http://books.google.com/books?id=nc4RrvrHATkC).

For certain exceptional nonlinear ODEs and PDEs which are *completely integrable* there exist other solution methods based on existence of the so-called Lax representation (the inverse scattering transform and many other related techniques) enabling one to construct e.g. the famous multisoliton and finite-gap solutions of the Korteweg--de Vries and other equations and (anti)instanton and monopole solutions of the Yang--Mills equations. Again, there are many excellent books on completely integrable systems, see e.g. a recent book by M. Dunajski (https://www.researchgate.net/profile/Maciej_Dunajski) Solitons, Instantons, and Twistors (http://books.google.com/books?isbn=0198570627) and references therein.

For completely integrable systems it is also often possible to generate new solutions from the known ones using Bäcklund or Darboux transformations, see e.g. the books C. Rogers, W.K. Schief, Bäcklund and Darboux transformations, 2002 (https://www.researchgate.net/publication/252949404_Bcklund_and_Darboux_Transformations) and C. Rogers, W.F. Shadwick, Bäcklund transformations and their applications, Academic Press, 1982 (http://books.google.com/books?isbn=008095667X) and references therein.

Article Bäcklund and Darboux Transformations