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2016, 6(2): 175-181. doi: 10.3934/naco.2016007

On general form of the Tanh method and its application to nonlinear partial differential equations

1. 

Department of Mathematics, Middle East Technical University, Ankara, Turkey

Received  February 2016 Revised  April 2016 Published  June 2016

The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.
Citation: Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
References:
[1]

M. J. Ablowitz and H. Segur, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511623998. Google Scholar

[2]

M. Coffey, On series expansions giving closed-form solutions of Korteweg-de Vries-Like equations,, SIAM J. Appl. Math., 50-6 (1990), 50. doi: 10.1137/0150093. Google Scholar

[3]

W. Hereman and M. Takaoka, Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA,, J. Phys. A: Math. Gen., 23 (1990), 4805. Google Scholar

[4]

W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. V. Immerzeele and A. Meerpoel, Exact solitary wave solutions of non-linear evolution and wave equations,, J. Phys. A Math. Gen., 19 (1986), 607. Google Scholar

[5]

W. Hereman and W. Malfliet, The tanh method: a tool to solve nonlinear partial differential equations with symbolic software,, 9th World Multi-Conference on Systemics, 7 (2005), 165. Google Scholar

[6]

S. A. Khuri, Exact solutions for a class of nonlinear evolution equations: A unified ansatze approach,, Chaos, 36 (2008), 1181. doi: 10.1016/j.chaos.2006.09.066. Google Scholar

[7]

W. Malfliet and W. Hereman, The tanh method: II. Perturbation technique for conservative systems,, Phys. Scr., 54 (1996), 569. doi: 10.1088/0031-8949/54/6/004. Google Scholar

[8]

W. Malfliet, The tanh method, a tool for solving certain classes of nonlinear PDEs,, Mathematical Methods in the Applied Sciences, 28-17 (2005), 28. doi: 10.1002/mma.650. Google Scholar

[9]

W. Malfliet and W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations,, Phys. Scr., 54 (1996), 563. doi: 10.1088/0031-8949/54/6/003. Google Scholar

[10]

W. Malfliet, The tanh method, a tool for solving certain classes of nonlinear evolution and wave equations,, J. Comput. Appl. Math., 164 (2004), 529. doi: 10.1016/S0377-0427(03)00645-9. Google Scholar

[11]

J. Murray, Mathematical Biology,, Springer Verlag, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar

[12]

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations,, 2nd edition, (2012). Google Scholar

[13]

N. Taghizadeh, M. Mirzazadeh and A. S. Paghaleh, The first integral method to nonlinear partial differential equations,, Appl. Appl. Math., 7-1 (2012), 7. Google Scholar

[14]

G. Whitham, Linear and Nonlinear Waves,, Wiley, (1974). Google Scholar

show all references

References:
[1]

M. J. Ablowitz and H. Segur, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511623998. Google Scholar

[2]

M. Coffey, On series expansions giving closed-form solutions of Korteweg-de Vries-Like equations,, SIAM J. Appl. Math., 50-6 (1990), 50. doi: 10.1137/0150093. Google Scholar

[3]

W. Hereman and M. Takaoka, Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA,, J. Phys. A: Math. Gen., 23 (1990), 4805. Google Scholar

[4]

W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. V. Immerzeele and A. Meerpoel, Exact solitary wave solutions of non-linear evolution and wave equations,, J. Phys. A Math. Gen., 19 (1986), 607. Google Scholar

[5]

W. Hereman and W. Malfliet, The tanh method: a tool to solve nonlinear partial differential equations with symbolic software,, 9th World Multi-Conference on Systemics, 7 (2005), 165. Google Scholar

[6]

S. A. Khuri, Exact solutions for a class of nonlinear evolution equations: A unified ansatze approach,, Chaos, 36 (2008), 1181. doi: 10.1016/j.chaos.2006.09.066. Google Scholar

[7]

W. Malfliet and W. Hereman, The tanh method: II. Perturbation technique for conservative systems,, Phys. Scr., 54 (1996), 569. doi: 10.1088/0031-8949/54/6/004. Google Scholar

[8]

W. Malfliet, The tanh method, a tool for solving certain classes of nonlinear PDEs,, Mathematical Methods in the Applied Sciences, 28-17 (2005), 28. doi: 10.1002/mma.650. Google Scholar

[9]

W. Malfliet and W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations,, Phys. Scr., 54 (1996), 563. doi: 10.1088/0031-8949/54/6/003. Google Scholar

[10]

W. Malfliet, The tanh method, a tool for solving certain classes of nonlinear evolution and wave equations,, J. Comput. Appl. Math., 164 (2004), 529. doi: 10.1016/S0377-0427(03)00645-9. Google Scholar

[11]

J. Murray, Mathematical Biology,, Springer Verlag, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar

[12]

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations,, 2nd edition, (2012). Google Scholar

[13]

N. Taghizadeh, M. Mirzazadeh and A. S. Paghaleh, The first integral method to nonlinear partial differential equations,, Appl. Appl. Math., 7-1 (2012), 7. Google Scholar

[14]

G. Whitham, Linear and Nonlinear Waves,, Wiley, (1974). Google Scholar

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