Sains Malaysiana 40(5)(2011): 515–519

A New Hybrid Non-standard Finite Difference-Adomian Scheme for Solution of Nonlinear Equations

(Skim Hibrid Baru Beza-terhingga Tak Piawai-Adomian bagi Penyelesaian Persamaan Tak Linear)

K. Moaddy & I. Hashim*

School of Mathematical Sciences , Universiti Kebangsaan Malaysia

43600 UKM Bangi Selangor D.E., Malaysia

A.K. Alomari

Department of Sciences, Faculty of Nursing and Science

Jerash Private University, 26150 Jerash, Jordan

S. Momani

Department of Mathematics, Faculty of Science

The University of Jordan, Amman 11942 , Jordan

Received: 20 May 2010 / Accepted: 7 July 2010

ABSTRACT

This research develops a new non-standard scheme based on the Adomian decomposition method (ADM) to solve nonlinear equations. The ADM was adopted to solve the nonlinear differential equation resulting from the discretization of the differential equation. The new scheme does not need to linearize or non-locally linearize the nonlinear term of the differential equation. Two examples are given to demonstrate the efficiency of this scheme.

Keywords: Adomian decomposition method; Logistic equation; Lotka-Volterra system; non-standard schemes

ABSTRAK

Penyelidikan ini membangunkan satu skim tak piawai baru berdasarkan pada kaedah penguraian Adomian (KPA) bagi menyelesaikan persamaan tak linear. KPA ini diadaptasi untuk menyelesaikan persamaan tak linear yang terhasil daripada pendiskretan persamaan terbitan. Skim baru ini tidak perlu melinearkan atau melinearkan secara tak setempat sebutan tak linear persamaan terbitan itu. Dua contoh diberi untuk medemonstrasikan keefisienan skim ini.

Kata kunci: Kaedah penguraian Adomian; persamaan logistik; skim tak piawai; sistem Lotka-Volterra

REFERENCES

Adomian, G. 1988. A review of the decomposition method in applied mathematics. J. Math. Analy. Appl. 135: 501-544.

Cherruault, Y. & Adomian, G. 1993. Decomposition method: A new proof ofconvergence. Numer. Mathl. Comput. Model., 18: 103-106.

Lubuma, J.M.S. & Patidar, K.C. 2005. Numerical Contributions to the theory of non-standard finite difference method and applications to singular perturbation problems. In Advances In the Applications of Nonstandard Finite Difference Schemes, edited by R.E. Mickens. Singapore: World Scientific 513-560.

Ibijola, E.A., Lubuma, J.M.S. & Ade-Ibijola, O.A. 2008. On nonstandard finite difference schemes for initial value problems in ordinary differential equations, Int. J. Phys. Sci. 3(2): 59-64.

Mickens, R.E. 1989. Exact solutions to a finite difference model of anonlinear reaction-advection equation: Implications for numerical analysis, Numer. Meth. Partial Diff. Eq. 5: 313-325.

Mickens, R.E. & Smith, A. 1990. Finite difference models of ordinary differential equations: Influence of denominator models, J. Franklin Institute 327: 143-145.

Mickens, R.E. 1994. Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific.

Mickens, R.E. 1999. Nonstandard finite difference schemes for reaction-diffusion equations, Numer. Meth. Partial Diff. Eq. 15: 201-214.

Mickens, R.E. 2000. Applications of Nonstandard Finite Difference Schemes. Singapore: World Scientific.

Mickens, R.E. 2003. Non-standard finite-difference schemes for the Lokta-Volterra system. Applied Numerical Mathematics 45: 309-314.

Mickens, R.E. 2005. Advances in the Applications of Nonstandard Finite Difference Schemes. Singapore: World Scientific.

Mickens, R.E. 2006. Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numer. Meth. Partial Diff. Eq. 23(3): 672-691.

Momani, S., Moadi, K. & Noor, M.A. 2006. Decomposition method for solving a system of fourth-order obstacle boundary value problems. Appl. Math. Comput. 175: 923-931.

Vahidi, A.R., Babobian, E., Asadi, Cordshooli, G.H. & Mirzaie, M. 2009. Adomian decomposition method to systems of nonlinear algebraic equations. Appl. Math. Sci. 3: 883-889.

*Corresponding author; email: ishak_h@ukm.my