What are the standard references for closed form solutions of nonlinear partial difference equations?
I do not think that there is one reference. Such solution is a piece of art. For example, Korteweg-de-Vries equation du/dt = u du/dx+d3u/dx3 has solutions of periodic and solitary type. While the fist solutions were obtained by seeking them in a form of wave z=x-vt, it took much time and rather sophisticated theory to find multi-soliton solutions. The similar situation is with other PDE.
Dear Prof Yuri Yegorov
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Dr/Tarek Ibrahim ( T. F. Ibrahim )
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¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia
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Plethora of such solutions are available for integrable partial difference equations. For a brief introduction to the subject and some further references see e.g. the links below.
http://www.icmat.es/research/international-grants/DMGILBS/NDDMGILBS/abstracts/SURIS.pdf
https://www.newton.ac.uk/files/seminar/20090212110012001-152100.pdf
Another class of partial difference equations admitting closed form solutions are those possessing symmetries, see e.g. the survey linked below
Article TOPICAL REVIEW: Continuous symmetries of difference equations
Handbook of Nonlinear Partial Differential Equations, Polyanin & Zaitsev. Chapman & Hall/CRC 2004
Dear Prof Claude Pierre Massé
Thank you very much for your interesting and your answer .
Best Regards ,
Dr/Tarek Ibrahim ( T. F. Ibrahim )
Associate professor
¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia
²Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
E-mails: [email protected]
Personal Websites :
http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=A/
http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=E
https://www.researchgate.net/profile/Tarek_Ibrahim2/?ev=hdr_xprf
http://scholar.google.com/citations?hl=en&user=K0uEihsAAAAJ&sortby=pubdate&view_op=list_works&pagesize=100
What are the standard references for closed form solutions of nonlinear partial difference equations? - ResearchGate. Available from: https://www.researchgate.net/post/What_are_the_standard_references_for_closed_form_solutions_of_nonlinear_partial_difference_equations#56f478e340485455ce7e5491 [accessed Mar 25, 2016].
Solving non-linear partial differential equations is a challenging task. First they do not admit separation of variables, however with special techniques generalized separation of variables method can be applied as Burger's equation is converted to heat equation and then heat equation is solved. The book "Handbook of Nonlinear Partial Differential Equations, Polyanin & Zaitsev. Chapman & Hall/CRC 2004" contains the generalized separation of variables method. Moreover , some semi analytic method such as Adomian Decomposition Method, He's method etc available in the literature can be used.
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