• Previous Article
    Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models
  • DCDS-S Home
  • This Issue
  • Next Article
    Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics
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. BelgacemA. 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. BulutH. 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. BulutH. 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. BulutH. 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. JaradK. BayramT. 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. JaradB. 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. BelgacemA. 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. BulutH. 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. BulutH. 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. BulutH. 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. JaradK. BayramT. 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. JaradB. 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

Figure 1.  Numerical illustration of solution $U_{m,n}$ by DHPSTM
Figure 2.  Numerical illustration of solution $U_{m,n}$ by DHPSTM
Figure 3.  Numerical illustration of approximate solution $U_{m,n}$ by DHPSTM
[1]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[2]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[3]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[4]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[5]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[6]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[7]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[8]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[9]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[10]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[11]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[12]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[13]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[14]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269

[15]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[16]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[17]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[18]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012

[19]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[20]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (77)
  • HTML views (169)
  • Cited by (1)

Other articles
by authors

[Back to Top]