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June  2019, 12(3): 615-624. doi: 10.3934/dcdss.2019039

## The discrete homotopy perturbation Sumudu transform method for solving partial difference equations

 1 Bolvadin Vocational School, Afyon Kocatepe University, Afyonkarahisar, Turkey 2 Department of Mathematics, Faculty of Basic Education, PAAET, Al-Ardhiya, Kuwait

* Corresponding author: Figen Özpinar

Received  April 2017 Revised  August 2017 Published  September 2018

In this paper, we introduce a combined form of the discrete Sumudu transform method with the discrete homotopy perturbation method to solve linear and nonlinear partial difference equations. This method is called the discrete homotopy perturbation Sumudu transform method(DHPSTM). The results reveal that the introduced method is very efficient, simple and can be applied to other linear and nonlinear difference equations. The nonlinear terms can be easily handled by use of He's polynomials.

Citation: Figen Özpinar, Fethi Bin Muhammad Belgacem. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 615-624. doi: 10.3934/dcdss.2019039
##### References:
 [1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, Newyork, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [2] M. A. Asiru, Further properties of the Sumudu transform and its applications, International Journal of Mathematical Education in Science and Technology, 33 (2002), 441-449. doi: 10.1080/002073902760047940. Google Scholar [3] M. A. Asiru, Clasroom note: application of the Sumudu to discrete dynamic systems, International Journal of Mathematical Education in Science and Technology, 34 (2003), 944-949. Google Scholar [4] A. Atangana and E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of CD$4^+$ cells and attractor one-dimensional Keller-Segel equations, Advances in Difference Equations, 2013 (2013), 14pp. doi: 10.1186/1687-1847-2013-94. Google Scholar [5] A. Atangana and A. Kılıçman, The use of Sumudu transform for solving certain nonlinear fractional heat-like equations, Abstract and Applied Analysis, 2013 (2013), Article ID 737481, 12 pages. Google Scholar [6] A. Atangana, Extension of the Sumudu homotopy perturbation method to an attractor for one-dimensional Kelleregel equations, Applied Mathematical Modelling, 39 (2015), 2909-2916. doi: 10.1016/j.apm.2014.09.029. Google Scholar [7] A. Atangana, On the new fractional derivative and application to nonlinear Fisher reaction iffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar [8] F. B. M. Belgacem, A. Karaballi and S.L. Kalla, Analytical Investigations of the Sumudu Transform and Applications to Integral Production Equations, Journal of Mathematical Problems in Engineering, 3 (2003), 103-118. doi: 10.1155/S1024123X03207018. Google Scholar [9] F. B. M. Belgacem and A. Karaballi, Sumudu transform fundamental properties investigations and applications, Journal of Applied Mathematics and Stochastic Analysis, (2006), Article ID 91083, 23 pages. doi: 10.1155/JAMSA/2006/91083. Google Scholar [10] F. B. M. Belgacem, Introducing and analysing deeper Sumudu properties, Nonlinear Studies, 13 (2006), 23-41. Google Scholar [11] F. B. M. Belgacem, Sumudu Applications to Maxwells Equations, PIERS Online, 5(9) (2009), 355-360. Google Scholar [12] F. B. M. Belgacem, Applications with the Sumudu Transform to Bessel Functions and Equ, App. Math. Sci. (AMS), 4(74) (2010), 3665-3686. Google Scholar [13] J. Biazar and H. Aminikhah, Exact and numerical solutions for non-linear Burger's equation by VIM, Mathematical and Computer Modelling, 49 (2009), 1394-1400. doi: 10.1016/j.mcm.2008.12.006. Google Scholar [14] H. Bulut, H. M. Baskonus and S. Tuluce, Homotopy perturbation Sumudu transform method for one and two dimensional homogeneous heat equations, International Journal of Basic and Applied Sciences IJBAS-IJEMS, 12 (2012), 1-16. Google Scholar [15] H. Bulut, H. M. Baskonus and S. Tuluce, Homotopy perturbation Sumudu transform method for one-two-three dimensional initial value problems, New World Sciences Academy, 7 (2012), 55-65. Google Scholar [16] H. Bulut, H. M. Baskonus and S. Tuluce, Homotopy perturbation Sumudu transform method for heat equations, Mathematics in Engineering Science and Aerospace Mesa, 4 (2013), 49-60. Google Scholar [17] H. Bulut, H. M. Baskonus and F. B. M. Belgacem, The Analytical solutions of some fractional ordinary differential equations by the Sumudu transform method, Abstract and Applied Analysis, (2013), Article ID 203875, 6 pages. Google Scholar [18] J. M. Burgers, A Mathematical model illustration the theory of turbulence, Adv. in Appl. Mech., 1 (1948), 171-199. Google Scholar [19] J. H. He, An approximate solution technique depending on an artificial parameter: a special example, Commun. Nonlinear Sci. Numer. Simulat., 3 (1998), 92-97. doi: 10.1016/S1007-5704(98)90070-3. Google Scholar [20] J. H. He, A Coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., 35 (2000), 37-43. doi: 10.1016/S0020-7462(98)00085-7. Google Scholar [21] J. H. He, Homotopy perturbation method: A new nonlinear analytic technique, Appl. Math. Comput., 135 (2003), 73-79. doi: 10.1016/S0096-3003(01)00312-5. Google Scholar [22] J. H. He, An elementary introduction to the homotopy perturbation method, Comput. Math. Appl., 57 (2009), 410-412. doi: 10.1016/j.camwa.2008.06.003. Google Scholar [23] F. Jarad, K. Bayram, T. Abdeljawad and D. Beleanu, On the discerete Sumudu transform, Romanian Reports in Physics, 64 (2012), 347-356. Google Scholar [24] F. Jarad and K. Taş, On Sumudu transform method in discrete fractional calculus, Abstract and Applied Analysis, 2012 (2012), Article ID 270106, 16 pages. Google Scholar [25] F. Jarad, B. Kaymakçalan and K. Taş, A New transform method in nabla discrete fractional calculus, Advances in Difference Equations, 2012 (2012), 1-17. doi: 10.1186/1687-1847-2012-190. Google Scholar [26] Q. K. Katatbeh and F. B. M. Belgacem, Applications of the Sumudu Transform to Fractional Diff. Equations, Nonlinear Studies (NSJ), 18(1) (2011), 99-112. Google Scholar [27] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Publ. Co., Singapore, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [28] J. J. Mohan and G. V. S. R. Deekshitulu, Solutions of fractional difference equations using S-transform, Malaya Journal of Matematik, 3 (2013), 7-13. Google Scholar [29] J. Singh and D. Kumar, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Theor. Appl. Mech., 4 (2011), 165-175. Google Scholar [30] G. K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology, 24 (1993), 35-43. doi: 10.1080/0020739930240105. Google Scholar [31] G. K. Watugala, Sumudu transform new integral transform to solve differential equations and control engineering problems, Mathematical Engineering in Industry, 6 (1998), 319-329. Google Scholar [32] G. K. Watugala, The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 8 (2002), 293-302. Google Scholar [33] H. Zhu and M. Ding, The Discrete homotopy perturbation method for solving Burgers' and heat equations, J. Inf. and Comput. Sci., 11 (2014), 1647-1657. doi: 10.12733/jics20103159. Google Scholar

show all references

##### References:
 [1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, Newyork, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [2] M. A. Asiru, Further properties of the Sumudu transform and its applications, International Journal of Mathematical Education in Science and Technology, 33 (2002), 441-449. doi: 10.1080/002073902760047940. Google Scholar [3] M. A. Asiru, Clasroom note: application of the Sumudu to discrete dynamic systems, International Journal of Mathematical Education in Science and Technology, 34 (2003), 944-949. Google Scholar [4] A. Atangana and E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of CD$4^+$ cells and attractor one-dimensional Keller-Segel equations, Advances in Difference Equations, 2013 (2013), 14pp. doi: 10.1186/1687-1847-2013-94. Google Scholar [5] A. Atangana and A. Kılıçman, The use of Sumudu transform for solving certain nonlinear fractional heat-like equations, Abstract and Applied Analysis, 2013 (2013), Article ID 737481, 12 pages. Google Scholar [6] A. Atangana, Extension of the Sumudu homotopy perturbation method to an attractor for one-dimensional Kelleregel equations, Applied Mathematical Modelling, 39 (2015), 2909-2916. doi: 10.1016/j.apm.2014.09.029. Google Scholar [7] A. Atangana, On the new fractional derivative and application to nonlinear Fisher reaction iffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar [8] F. B. M. Belgacem, A. Karaballi and S.L. Kalla, Analytical Investigations of the Sumudu Transform and Applications to Integral Production Equations, Journal of Mathematical Problems in Engineering, 3 (2003), 103-118. doi: 10.1155/S1024123X03207018. Google Scholar [9] F. B. M. Belgacem and A. Karaballi, Sumudu transform fundamental properties investigations and applications, Journal of Applied Mathematics and Stochastic Analysis, (2006), Article ID 91083, 23 pages. doi: 10.1155/JAMSA/2006/91083. Google Scholar [10] F. B. M. Belgacem, Introducing and analysing deeper Sumudu properties, Nonlinear Studies, 13 (2006), 23-41. Google Scholar [11] F. B. M. Belgacem, Sumudu Applications to Maxwells Equations, PIERS Online, 5(9) (2009), 355-360. Google Scholar [12] F. B. M. Belgacem, Applications with the Sumudu Transform to Bessel Functions and Equ, App. Math. Sci. (AMS), 4(74) (2010), 3665-3686. Google Scholar [13] J. Biazar and H. Aminikhah, Exact and numerical solutions for non-linear Burger's equation by VIM, Mathematical and Computer Modelling, 49 (2009), 1394-1400. doi: 10.1016/j.mcm.2008.12.006. Google Scholar [14] H. Bulut, H. M. Baskonus and S. Tuluce, Homotopy perturbation Sumudu transform method for one and two dimensional homogeneous heat equations, International Journal of Basic and Applied Sciences IJBAS-IJEMS, 12 (2012), 1-16. Google Scholar [15] H. Bulut, H. M. Baskonus and S. Tuluce, Homotopy perturbation Sumudu transform method for one-two-three dimensional initial value problems, New World Sciences Academy, 7 (2012), 55-65. Google Scholar [16] H. Bulut, H. M. Baskonus and S. Tuluce, Homotopy perturbation Sumudu transform method for heat equations, Mathematics in Engineering Science and Aerospace Mesa, 4 (2013), 49-60. Google Scholar [17] H. Bulut, H. M. Baskonus and F. B. M. Belgacem, The Analytical solutions of some fractional ordinary differential equations by the Sumudu transform method, Abstract and Applied Analysis, (2013), Article ID 203875, 6 pages. Google Scholar [18] J. M. Burgers, A Mathematical model illustration the theory of turbulence, Adv. in Appl. Mech., 1 (1948), 171-199. Google Scholar [19] J. H. He, An approximate solution technique depending on an artificial parameter: a special example, Commun. Nonlinear Sci. Numer. Simulat., 3 (1998), 92-97. doi: 10.1016/S1007-5704(98)90070-3. Google Scholar [20] J. H. He, A Coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., 35 (2000), 37-43. doi: 10.1016/S0020-7462(98)00085-7. Google Scholar [21] J. H. He, Homotopy perturbation method: A new nonlinear analytic technique, Appl. Math. Comput., 135 (2003), 73-79. doi: 10.1016/S0096-3003(01)00312-5. Google Scholar [22] J. H. He, An elementary introduction to the homotopy perturbation method, Comput. Math. Appl., 57 (2009), 410-412. doi: 10.1016/j.camwa.2008.06.003. Google Scholar [23] F. Jarad, K. Bayram, T. Abdeljawad and D. Beleanu, On the discerete Sumudu transform, Romanian Reports in Physics, 64 (2012), 347-356. Google Scholar [24] F. Jarad and K. Taş, On Sumudu transform method in discrete fractional calculus, Abstract and Applied Analysis, 2012 (2012), Article ID 270106, 16 pages. Google Scholar [25] F. Jarad, B. Kaymakçalan and K. Taş, A New transform method in nabla discrete fractional calculus, Advances in Difference Equations, 2012 (2012), 1-17. doi: 10.1186/1687-1847-2012-190. Google Scholar [26] Q. K. Katatbeh and F. B. M. Belgacem, Applications of the Sumudu Transform to Fractional Diff. Equations, Nonlinear Studies (NSJ), 18(1) (2011), 99-112. Google Scholar [27] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Publ. Co., Singapore, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [28] J. J. Mohan and G. V. S. R. Deekshitulu, Solutions of fractional difference equations using S-transform, Malaya Journal of Matematik, 3 (2013), 7-13. Google Scholar [29] J. Singh and D. Kumar, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Theor. Appl. Mech., 4 (2011), 165-175. Google Scholar [30] G. K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology, 24 (1993), 35-43. doi: 10.1080/0020739930240105. Google Scholar [31] G. K. Watugala, Sumudu transform new integral transform to solve differential equations and control engineering problems, Mathematical Engineering in Industry, 6 (1998), 319-329. Google Scholar [32] G. K. Watugala, The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 8 (2002), 293-302. Google Scholar [33] H. Zhu and M. Ding, The Discrete homotopy perturbation method for solving Burgers' and heat equations, J. Inf. and Comput. Sci., 11 (2014), 1647-1657. doi: 10.12733/jics20103159. Google Scholar
Numerical illustration of solution $U_{m,n}$ by DHPSTM
Numerical illustration of solution $U_{m,n}$ by DHPSTM
Numerical illustration of approximate solution $U_{m,n}$ by DHPSTM
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