A new technique of Laplace Padé reduced differential transform method for (1+3) dimensional wave equations

Abstract

The aim of this paper is to give a good strategy for solving some linear and non-linear partial differential equations in mechanics, physics, engineering and various other technical fields by Modified Reduced Differential Transform Method. In this article we use the method named with Laplace-Padé Reduced Differential Transform Method. This method is obtained by combining Laplace-Padé resummation method, which is a useful technique to find exact solutions, and the Reduced Differential Transform Method. We apply the method to the wave equations and give some examples to see its effectiveness and usefulness. The results and the findings showed that this method leads us to exact solutions with a few iterations or the approximate solutions with small errors.

Keywords

• Laplace-Padé Reduced Differential Transform Method (LPRDTM),Modified Reduced Differential Transform Method (MRDTM),Reduced Differential Transform Method (RDTM),Partial differential equations (PDEs),wave equation

Kaynakça

1. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Springer Science & Business Media, 2011. https://books.google.com/books?id=Ir4yXgBesAsC&pgis=1 (accessed December 10, 2014).
2. A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education press and Springer Vergal, 2009.
3. R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, 2000. https://books.google.com/books?hl=tr&lr=&id=xMaAqOWMgXkC&pgis=1 (accessed June 18, 2015).
4. V.K. Srivastava, M.K. Awasthi, (1+n)-Dimensional Burgers’ equation and its analytical solution: A comparative study of HPM, ADM and DTM, Ain Shams Eng. J. 5 (2014) 533–541.
5. F. Mirzaee, M. Komak Yari, A novel computing three-dimensional differential transform method for solving fuzzy partial differential equations, Ain Shams Eng. J. (2015).
6. M. Jafaryar, S.I. Pourmousavi, M. Hosseini, E. Mohammadian, Application of DTM for 2D viscous flow through expanding or contracting gaps with permeable walls, New Trends Math. Sci. 2 (2014) 145–158.
7. M. Kurulay, M. Bayram, Approximate analytical solution for the fractional modified KdV by differential transform method, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1777–1782.
8. Y. Keskin, G. Oturanç, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 741–749.

9. O. Acan, Y. Keskin, Approximate solution of Kuramoto–Sivashinsky equation using reduced differential transform method, in: Proc. Int. Conf. Numer. Anal. Appl. Math. 2014, AIP Publishing, 2015: p. 470003. doi:10.1063/1.4912680.
10. O. Acan, Y. Keskin, Reduced differential transform method for (2+1) dimensional type of the Zakharov–Kuznetsov ZK(n,n) equations, in: Proc. Int. Conf. Numer. Anal. Appl. Math. 2014, AIP Publishing, 2015: p. 370015. doi:10.1063/1.4912604.
11. J. Yu, J. Jing, Y. Sun, S. Wu, On Finding Exact and Approximate Solutions to Some PDEs Using the Reduced Differential Transform Method, Appl. Math. Comput. 273 (2016) 697–705.
12. A. Babaei, A. Mohammadpour, Solving an inverse heat conduction problem by reduced differential transform method Solving an inverse heat conduction problem by reduced differential transform method, New Trends Math. Sci. 3 (2016) 65–70.
13. K. Yildirim, B. İbiş, M. Bayram, New solutions of the nonlinear Fisher type equations by the reduced differential transform, Nonlinear Sci. Lett. A. 3 (2012) 29–36.
14. B. İbiş, M. Bayram, Approximate Solutions for Some Nonlinear Evolutions Equations By Using The Reduced Differential Transform Method, Intenational J. Appl. Math. Res. 1 (2012) 288–302.
15. O. Acan, Y. Keskin, A Comparative Study of Numerical Methods for Solving (n+1) Dimensional and Third-Order Partial Differential Equations, J. Comput. Theor. Nanosci. (2016) (In press).
16. Y. Zhang, Formulation and solution to time-fractional generalized Korteweg-de Vries equation via variational methods, Adv. Differ. Equations. 2014 (2014) 65.
17. M.G. Sakar, F. Erdogan, A. Yldrm, Variational iteration method for the time-fractional Fornberg-Whitham equation, Comput. Math. with Appl. 63 (2012) 1382–1388.
18. Q. Zhao, H. Xu, T. Fan, Analysis of three-dimensional boundary-layer nanofluid flow and heat transfer over a stretching surface by means of the homotopy analysis method, Bound. Value Probl. 2015 (2015) 64. doi:10.1186/s13661-015-0327-3.
19. S. Sattari, M. Jahani, M. Gorji-bandpy, Application of different analytical methods to equation system of Bodewadt ’ s fixed disc and rotating stream, New Trends Math. Sci. 163 (2015) 149–163.
20. M.G. Sakar, F. Erdogan, The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian’s decomposition method, Appl. Math. Model. 37 (2013) 8876–8885.
21. G. Akram, I.A. Aslam, Solution of fourth order three-point boundary value problem using ADM and RKM, J. Assoc. Arab Univ. Basic Appl. Sci. (2014). doi:10.1016/j.jaubas.2014.08.001.
22. G.A. Baker Jr, Essentials of Padé approximants, Academic Press, UK, 1975.
23. A.S.-R. Brahim Benhammouda, Hector Vazquez-Leal, Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations, 2014 (2014). doi:http://dx.doi.org/10.1155/2014/279481.
24. S. Momani, G.H. Erjaee, M.H. Alnasr, The modified homotopy perturbation method for solving strongly nonlinear oscillators, Comput. Math. with Appl. 58 (2009) 2209–2220.
25. P.-Y. Tsai, C.-K. Chen, An approximate analytic solution of the nonlinear Riccati differential equation, J. Franklin Inst. 347 (2010) 1850–1862.
26. A.E. Ebaid, A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 528–536.