A NEW MODIFICATION OF THE DIFFERENTIAL TRANSFORM METHOD

Main Article Content

HELENA NAYAR
PATRICK AZERE PHIRI

Abstract

In this article, a new modification of the Differential Transform Method, named the ModDTM, is being introduced as a method suitable for solving (1 + 1) partial differential equations, with initial conditions that are specified at the initial value of the spatial variable. Definitions and properties of the method are developed and then applied for solving two nonlinear partial differential equations, namely, the Rosenau-Hyman equation and the Newell-Whitehead-Segel equation, to examine the effectiveness of the method. The solutions obtained are compared with the exact solutions and the solutions by the Reduced Differential Transform Method. It is assessed that the ModDTM is an effective method for solving (1+1) partial differential equations when the attached initial conditions pertain to the spatial variable.

Keywords:
ModDTM, differential transform method, reduced differential transform method, Rosenau-Hyman equation, Newell-Whitehead-Segel equation.

Article Details

How to Cite
NAYAR, H., & PHIRI, P. A. (2020). A NEW MODIFICATION OF THE DIFFERENTIAL TRANSFORM METHOD. Asian Journal of Mathematics and Computer Research, 27(3), 38-51. Retrieved from https://ikprress.org/index.php/AJOMCOR/article/view/5475
Section
Original Research Article

References

Gundogdu H, Gozukizil OF. Solving Nonlinear Partial differential equations by using adomian decomposition method, modified decomposition method and laplace decomposition method. MANAS Journal of Engineering. 2017;5(1):1-13.

Hussain S, Shah A, Ayub S, Ullah A. An approximate analytical solution of the AllenCahn equation using homotopy perturbation method and homotopy analysis method. Heliyon. 2019;5(12):e03060.

Sheth SS, Singh TR. Analytical approximate solution of Nonlinear Partial differential equations using VIM, VIADM and New modified KVIADM. Journal of Physics. Conference Series. 2020;1473.

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

Mehne HH, Esmaeili M. Analytical solution to the boundary layer slip flow and heat transfer over a flat 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.

Eslami M, Taleghani SA. Differential transform method for conformable fractional partial differential equations. Iranian Journal of Numerical Analysis and Optimization. 2019;9(2):17- 29.

Liu B, Zhou X, Du Q. Differential transform method for some delay differential equations. Applied Mathematics. 2015;6:585-593.

Rizkalla RR, Tantawy SSH, Taha MH. Applications of differential transform method for solving singularly perturbed Volterra integral equations and integro-differential equations. International Journal of Mathematical Trends and Technology. 2015;23(1):41-55.

Kadkhoda N, Roushan SS, Jafari H. Differential transform method. A tool for solving fuzzy differential equations. Int. J. Appl. Comput. Math. 2018;4:33. Available:https://doi.org/10.1007/s40819-017-0471-9

Turgut AK, Sharanjeet D. A practical and powerful approach to potential KdV and Benjamin equations. Beni-Suef University Journal of Basic and Applied Sciences. 2017;6:383-390.

Mohammed O. Al-Amr. New applications of reduced differential transform method. Alexandria Engineering Journal. 2014;53:243-247.

Ramesh Rao TR. Numerical solution of Sine Gordon equation through reduced differential transform method. Global Journal of Pure and Applied Mathematics. 2017;7(13):3879-3888.

Mohammed O Al-Amr. Solution of the coupled Boussinesq-Burgers equation by reduced differential transform method. The 15th International Conference for Informatics and Information Technology. 2018;CIIT.

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.

Arslan D. The comparison study of hybrid method with RDTM for solving Rosenau-Hyman equation. Applied Mathematics and Nonlinear Sciences. 2020;5(1):267-274.

Gupta S, Goyal M, Prakash A. Numerical treatment of Newell Whitehead Segel equation. J. App. Eng. Math. 2020;10(2):312-320.

Baskonus HM. Complex soliton solutions to the Gilson-Pickering model. Axioms. 2019;8(18). Available:https://doi.org/10.3390/axioms8010018

Karmina KA, Dutta H, Yilmazer R, Noeiaghdam S. On the new wave behaviours of the GilsonPickering equations. Frontiers in Physics. 2020;8. Available:https://doi.org/10.3389/fphy.2020.00054

Kumari P, Gupta RK, Kumar S. On exact solutions, conservation laws and invariant analysis of the generalised Rosenau-Hyman equation; 2020. arXix (pre-print) arVix 2006.10551v1

Rus F, Villatoro FR. A repository of equations with cosine/sine compactons. Applied Mathematics and Computations. 2009;215(5):1838-1851.

Nadeem M, Yao SW, Parveen N. Solution of Newell-Whitehead-Segel equation by variational iteration method with He’s polynomials. Journal of Mathematics and Computer Science. 2020;20:21-29.

Latif B, Selamat MS, Rosli AN, Yusoff AI, Hasan NM. The semi-analytics iterative method for solving Newell-Whitehead-Segel equation. Mathematics and Statistics. 2020;8(2):87-94.

Patade J, Bhalekar S. Approximate analytical solutions of the Newell-Whitehead-Segel equation using a new iterative method. World Journal of Modelling and Simulation. 2015;11(2):94-103.

Zellal M, Belghaba K. Applications of homotopy perturbation transform method for solving Newell-Whitehead-Segel equation. General Letters in Mathematics. 2017;3(1):35-46