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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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] F. B. M. Belgacem, Introducing and analysing deeper Sumudu properties, Nonlinear Studies, 13 (2006), 23-41. [11] F. B. M. Belgacem, Sumudu Applications to Maxwells Equations, PIERS Online, 5(9) (2009), 355-360. [12] F. B. M. Belgacem, Applications with the Sumudu Transform to Bessel Functions and Equ, App. Math. Sci. (AMS), 4(74) (2010), 3665-3686. [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. [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. [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. [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. [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. [18] J. M. Burgers, A Mathematical model illustration the theory of turbulence, Adv. in Appl. Mech., 1 (1948), 171-199. [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. [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. [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. [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. [23] F. Jarad, K. Bayram, T. Abdeljawad and D. Beleanu, On the discerete Sumudu transform, Romanian Reports in Physics, 64 (2012), 347-356. [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. [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. [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. [27] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Publ. Co., Singapore, 1994. doi: 10.1007/978-1-4612-0873-0. [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. [29] J. Singh and D. Kumar, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Theor. Appl. Mech., 4 (2011), 165-175. [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. [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. [32] G. K. Watugala, The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 8 (2002), 293-302. [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.

show all references

##### References:
 [1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, Newyork, 1994. doi: 10.1007/978-1-4612-0873-0. [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. [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. [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. [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. [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. [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. [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. [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. [10] F. B. M. Belgacem, Introducing and analysing deeper Sumudu properties, Nonlinear Studies, 13 (2006), 23-41. [11] F. B. M. Belgacem, Sumudu Applications to Maxwells Equations, PIERS Online, 5(9) (2009), 355-360. [12] F. B. M. Belgacem, Applications with the Sumudu Transform to Bessel Functions and Equ, App. Math. Sci. (AMS), 4(74) (2010), 3665-3686. [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. [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. [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. [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. [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. [18] J. M. Burgers, A Mathematical model illustration the theory of turbulence, Adv. in Appl. Mech., 1 (1948), 171-199. [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. [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. [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. [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. [23] F. Jarad, K. Bayram, T. Abdeljawad and D. Beleanu, On the discerete Sumudu transform, Romanian Reports in Physics, 64 (2012), 347-356. [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. [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. [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. [27] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Publ. Co., Singapore, 1994. doi: 10.1007/978-1-4612-0873-0. [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. [29] J. Singh and D. Kumar, Homotopy perturbation Sumudu transform method for nonlinear equations, Adv. Theor. Appl. Mech., 4 (2011), 165-175. [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. [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. [32] G. K. Watugala, The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 8 (2002), 293-302. [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.
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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