The Fornberg-Whitham Equation Solved by the Differential Transform Method

Main Article Content

Helena Nayar
Patrick Azere Phiri


The Differential Transform Method is a powerful analytical method that can solve nonlinear partial differential equations. Yet, the method cannot be used to solve time-dependent partial differential equations that involve more than one partial derivative with respect to the temporal variable t when they are of the same order, as in the case of the Fornberg-Whitham type equations. In this paper, a new theorem is devised to overcome the aforementioned problem of
the method, and it has been successfully applied to solve the Fornberg-Whitham equation. The other equations belonging to this group of equations, such as the Camassa-Holm equation and the Degasperi-Procesi equation, may also be solved by this approach.

Differential Transform method, Fornberg-Whitham equation.

Article Details

How to Cite
Nayar, H., & Phiri, P. A. (2020). The Fornberg-Whitham Equation Solved by the Differential Transform Method. Journal of Advances in Mathematics and Computer Science, 35(7), 85-95.
Original Research Article


Sutkar PS. Solution of some differential equations by using transform method. International Journal of Scientific and Innovative Mathematical Research. 2017;5(5):17-20.

Jang M, Chen C, Liu Y. Two-dimensional differential transform for partial differential equations. Applied Mathematics and Computations. 2001;121:261-270.

Biazar J. Application of differential transform method to the Sine-Gordon equation. International Journal of Nonlinear Sciences. 2010;9(4):444-447.

Soltanalizadeh B. Differential transformation method for solving one-dimensional telegraph equation. Comput.Appl.Math. 2011;3(3):639-653.

Sotanalizadeh B, Babayar B. Application of differential transformation method to the Fisher equation. Mathematics Scientific Journal. 2012;8(1):85-95.

Ahmad J, Bajwa S, Siddique I. Solving the Klein-Gordon equations via differential transform method. Journal of Science and Arts. 2015;1(30):33-38.

Chamekh M, Elzaki TM, Brik N. Semi-analytical solution for some proportional delay differential equations. SN Applied Sciences. 2019;1(2):148.

Moghimi MB, Borhanifar A. Solving a class of nonlinear delay integro-differential equations by using differential transform method. Applied and Computational Mathematics. 2016;5(3):142-149.

Mirzaee F, Yari MK. A novel computing three-dimensional differential transform method for solving fuzzy partial differential equations. Ain Shams Engineering Journal. 2016;7:695-708.

Ravi Kanth ASV, Aruna K. Solution of fractional third order dispersive partial differential equations. Egyptian Journal of Basic and Applied Sciences. 2015;2:190-199.

Mehne HH, Esmaeili M. Analytical solution to the boundary layer slip flow and heat transfer over a flate plate using the switching differential transform method. Journal of Applied Fluid Mechanics. 2019;12(2):433-444.

Kumar M. Study of differential transform technique for transient hydromagnetic Jeffrey fluid flow from a stretching sheet. Nonlinear Engineering. 2020;9:145-155.

Abidi F, Omrani K. The homotopy analysis method for solving the Fornberg Whitham equation and comparison with the Adomian Decomposition method. Computers and Mathematics with Applications. 2010;59:2743-2750.

Lu J. An analytical approach to the Fornberg Whitham type equations by using the variational iteration method. Computers and Mathematics with Applications. 2011;61:2010-2013.

Hesam S, Nazemi A, Haghbin A. Reduced differential transform method for solving the fornberg-whitham type equation. International Journal of Nonlinear Science. 2012;13(2):158-162.

Karbalaie A, Muhammed HH, Shabani M, Montazer MM. Exact solution of partial differential equations using homo-separation of variables. International Journal of Nonlinear Science. 2014;17(1):84-90.

Fornberg B, Whitham GB. A numerical and theoretical study of certain nonlinear wave phenomena. Philosophical Transactions of The Royal Society of London. 1978;289(1361):373-404.

Whitham GB. Variational methods and applications to water waves. Proceedings of The Royal Society A. 1967;299(1456). Available:

Merdan M, Gokdogan A, Yildirim A, Mohyud-Din ST. Numerical simulation of fractional Fornberg-Whitham equation by differential transform method. Abstract and Applied Analysis; 2012. Hindawi

Ayaz F. Solutions of systems of differential equations by differential transform method. Applied Mathematics and Computation. 2004;147:547-567.

Wu Y, Zhao P. A note on the generalized Camassa-Holm equation. Journal of Function Spaces; 2014. Available:

Chen W. On solutions to the Degasperis-Procesi equation. Journal of Mathematical Analysis. 2011;379:351-359.