In this paper, differential transformation method is used to find exact solutions of nonlinear delay integro– differential equations. Many theorems are presented that required for applying differential transformation method for nonlinear delay integro–differential equation. The validity and efficiency of the proposed method are demonstrated through several tests.
| Published in | Applied and Computational Mathematics (Volume 5, Issue 3) |
| DOI | 10.11648/j.acm.20160503.18 |
| Page(s) | 142-149 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2016. Published by Science Publishing Group |
Delay Integro–differential Equation, Delay Differential Equation, Differential Transformation Method, Closed Form Solution
| [1] |
M. M. Kabir, A. Borhanifar and R. Abazari, Application of  –expansion method to Regularized Long Wave (RLW) equation, Computers and Mathematics with Applications 61 (8) (1991), 233-241.
|
| [2] |
A. Borhanifar, A. Zamiri, Application of the –expansion method for the Zhiber-Shabat equation and other related equations, Mathematical and Computer Modeling 54 (9-10) (2011), 2109-2116.
|
| [3] |
A. Borhanifar, R. Abazari, General Solution of Generalized (2+1)-Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –expansion method, American Journal of computational Mathematics 1 (2011), 219-225.
|
| [4] | A. Borhanifar, M. M. Kabir and L. Maryam Vahdat, New periodic and soliton wave solutions for the generalized Zakharov system and (2 + 1)-dimensional Nizhnik-Novikov-Veselov system, Chaos, Solitons and Fractals 42 (2009), 1646-1654. |
| [5] | A. Borhanifar, M. M. Kabir, New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations, Journal of Computational and Applied Mathematics 229 (2009), 158-167. |
| [6] | H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math. 1 (1994), 309-322. |
| [7] | H. Brunner, W. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal. 6 (1999), 525-533. |
| [8] | H. Brunner, On the discretization of differential and Volterra integral equations with variable delay, BIT 37 (1997), 1–12. |
| [9] | H. Brunner, The numerical analysis of functional integral and integro-differential equations of Volterra type, Acta Numerica. (2004), 55–145. |
| [10] | H. Brunner, Q.-Y. Hu, Superconvergence of iterated collocation solutions for Volterra integral equations with variable delays, SIAM J. Numer. Anal. 43 (2005), 1934–1949. |
| [11] | H. Brunner, Q.-Y. Hu, Optimal superconvergence results for delay integro-differential equations of pantograph type, SIAM J. Numer. Anal. 45 (2007), 986–1004. |
| [12] | H. Brunner, Q.-Y. Hu, Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal. 21 (2001), 783–798. |
| [13] | H. Brunner, R. Vermiglio, Stability of solutions of neutral functional integro-differential equations and their discretizations, Computing 71 (2003), 229–245. |
| [14] | H. Brunner, Recent advances in the numerical analysis of Volterra functional differential equations with variable delays, J. Comput. Appl. Math. 228 (2009), 524-537. |
| [15] | C. T. H. Baker, N. J. Ford, Asymptotic error expansions for linear multistep methods for a class of delay integro-differential equations, Bull. Soc. Math. Greece (N. S.) 31 (1990), 5–18. |
| [16] | C. T. H. Baker, N. J. Ford, Stability properties of a scheme for the approximate solution of a delay-integro-differential equation, Appl. Numer. Math. 9 (1992), 357–370. |
| [17] | T. Koto, Stability of Runge–Kutta methods for delay integro-differential equations, J. Comput. Appl. Math. 145 (2002), 483–492. |
| [18] |
T. Koto, Stability of  -methods for delay integro-differential equations, J. Comput. Appl. Math. 161 (2003), 393–404.
|
| [19] | W. H. Enright, M. Hu, Continous Runge-Kutta methods for neutral Volterra integro-differential equations with delay, Appl. Numer. Math. 24 (1997), 175-190. |
| [20] | A. Borhanifar, R. Abazari, Numerical study of nonlinear Schrodinger and coupled Schrodinger equations by differential transformation method, Optics Communications 283 (2010), 2026-2031. |
| [21] | R. Abazari, A. Borhanifar, Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method, Computers and Mathematics with Applications 59 (2010), 2711-2722. |
| [22] | J. K. Zhou, Differential Transformation and its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986. |
| [23] | Fatma Ayaz, Solutions of the system of differential equations by differential transform method, Appl. Math. Comput. 147 (2004), 547-567. |
| [24] | M. Mossa Al-Sawalha, M. S. M. Noorani, Application of the differential transformation method for the solution of the hyperchaotic Rossler system, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 1509-1514. |
| [25] | M. Mossa Al-Sawalha, M. S. M. Noorani, A numeric-analytic method for approximating the chaotic Chen system, Chaos, Solitons & Fractals, 42 (2009), 1784–1791. |
| [26] | F. Ayaz, Application of differential transform method to differential-algebraic equations, Appl. Math. Comput. 152 (2004), 649-657. |
| [27] | A. Arikoglu, I. Ozkol, Solution of difference equations by using differential transform method, Appl. Math. Comput. 174 (2006), 1216–1228. |
| [28] | A. Arikoglu, I. Ozkol, Solution of differential–difference equations by using differential transform method, Appl. Math. Comput. 181 (2006), 153–162. |
| [29] | F. Kangalgil, F. Ayaz, Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos, Solitons & Fractals 41 (2009), 464–472. |
| [30] | C. K. Chen, Solving partial differential equations by two dimensional differential transform, Appl. Math. Comput. 106 (1999), 171-179. |
| [31] | M. J. Jang, C. L. Chen, Y. C. Liy, Two-dimensional differential transform for Partial differential equations, Appl. Math. Comput. 121 (2001), 261-270. |
| [32] | A. Kurnaz, G. Oturanc, ME. Kiris, n-Dimensional differential transformation method for solving linear and nonlinear PDE’s, Int J Comput Math 82 (2005), 369–380. |
| [33] | M. O. Al-Amr, New applications of reduced differential transform method, Alexandria Engineering Journal 53 (2014), 243–247. |
| [34] | S. Momani, Z. Odibat, I. Hashim, Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods, Topological method in Nonlinear Analysis 31 (2008), 211-226. |
| [35] | A. Arikoglu, I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Soliton. Fract. 34 (2007), 1473–1481. |
| [36] | A. letinkaya, O. Kıymaz, The solution of the time-fractional diffusion equation by the generalized differential transform method, Mathematical and Computer Modelling 57 (2013), 2349–2354. |
| [37] | Y. Keskin, A. Kurnaz, M. E. Kiris, G. Oturanc, Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear Sci. 8 (2007), 159–164. |
| [38] | S. M. Abdelghany, K. M. Ewis, A. A. Mahmoud, Mohamed M. Nassar, Vibration of a circular beam with variable cross sections using differential transformation method, Beni – suef university journal of basic and applied sciences 4 (2015), 185–191. |
| [39] | R. Lal, N. Ahlawat, Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method, European Journal of Mechanics / A Solids (in press). DOI: 10.1016/j.euromechsol. 2015. 02. 004 |
| [40] | A. M. A. El-Sayed, H. M. Nour, W. E. Raslan, E. S. El-Shazly, A study of projectile motion in a quadratic resistant medium via fractional differential transform method, Appl. Math. Modelling (in press). DOI: 10.1016/j.apm.2014.10.018 |
| [41] | A. Arikoglu, I. Ozkol, Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput. 168 (2005), 1145–1158. |
| [42] | Z. M. Odibat, Differential transform method for solving Volterra integral equations with separable kernels, Math. Comput. Model. 48 (7-8) (2008), 1144-1149. |
| [43] | A. Arikoglu, I. Ozkol, Solutions of integral and integro-differential equation systems by using differential transform method, Computers and Mathematics with Applications 56 (2008), 2411–2417. |
APA Style
Mohammad Bagher Moghimi, Abdollah Borhanifar. (2016). Solving a Class of Nonlinear Delay Integro–differential Equations by Using Differential Transformation Method. Applied and Computational Mathematics, 5(3), 142-149. https://doi.org/10.11648/j.acm.20160503.18
ACS Style
Mohammad Bagher Moghimi; Abdollah Borhanifar. Solving a Class of Nonlinear Delay Integro–differential Equations by Using Differential Transformation Method. Appl. Comput. Math. 2016, 5(3), 142-149. doi: 10.11648/j.acm.20160503.18
AMA Style
Mohammad Bagher Moghimi, Abdollah Borhanifar. Solving a Class of Nonlinear Delay Integro–differential Equations by Using Differential Transformation Method. Appl Comput Math. 2016;5(3):142-149. doi: 10.11648/j.acm.20160503.18
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Download@article{10.11648/j.acm.20160503.18,
author = {Mohammad Bagher Moghimi and Abdollah Borhanifar},
title = {Solving a Class of Nonlinear Delay Integro–differential Equations by Using Differential Transformation Method},
journal = {Applied and Computational Mathematics},
volume = {5},
number = {3},
pages = {142-149},
doi = {10.11648/j.acm.20160503.18},
url = {https://doi.org/10.11648/j.acm.20160503.18},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160503.18},
abstract = {In this paper, differential transformation method is used to find exact solutions of nonlinear delay integro– differential equations. Many theorems are presented that required for applying differential transformation method for nonlinear delay integro–differential equation. The validity and efficiency of the proposed method are demonstrated through several tests.},
year = {2016}
}
TY - JOUR T1 - Solving a Class of Nonlinear Delay Integro–differential Equations by Using Differential Transformation Method AU - Mohammad Bagher Moghimi AU - Abdollah Borhanifar Y1 - 2016/07/13 PY - 2016 N1 - https://doi.org/10.11648/j.acm.20160503.18 DO - 10.11648/j.acm.20160503.18 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 142 EP - 149 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20160503.18 AB - In this paper, differential transformation method is used to find exact solutions of nonlinear delay integro– differential equations. Many theorems are presented that required for applying differential transformation method for nonlinear delay integro–differential equation. The validity and efficiency of the proposed method are demonstrated through several tests. VL - 5 IS - 3 ER -
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