A Nonlinear Partial Integro-differential Equation Arising in Population Dynamic Via Radial Basis Functions and Theta-method

Volume 13, Issue 1, pp 14-25 http://dx.doi.org/10.22436/jmcs.013.01.02

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Authors

M. Aslefallah - Department of Mathematics, Imam Khomeini International University, Qazvin, Iran. E. Shivanian - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

Abstract

This paper proposes a numerical method to deal with the integro-differential reaction-diffusion equation. In the proposed method, the time variable is eliminated by using finite difference \(\theta\)−method to enjoy the stability condition. The method benefits from collocation radial basis function method, the generallized thin plate splines (GTPS) radial basis functions are used. Therefore, it does not require any struggle to determine shape parameter. The obtained results for some numerical examples reveal that the proposed technique is very effective, convenient and quite accurate to such considered problems.

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Keywords

• Integro-differential equation
• Radial basis functions
• Kansa method
• Finite differences \(\theta\)-method.

MSC

•  92D25
•  45K05
•  35R11

References

• [1] L. Sabatelli, S. Keating, J. Dudley, P. Richmond, Waiting time distributions in financial markets, European Physical Journal B, 27 (2002), 273-275.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] R. Schumer, D. A. Benson, M. M. Meerschaert, S. W. Wheatcraft, Eulerian derivation of the fractional advection-dispersion equation, Journal of Contaminant Hydrology, 48 (2001), 69-88.
  • View Article
  • Google Scholar


• [3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000)
  • View Article
  • MathSciNet
  • Google Scholar


• [4] M. M. Meerschaert, C. Tadjeran, Finite Difference Approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56 (2006), 80-90.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] S. Samko, A. Kibas, O. Marichev, Fractional Integrals and derivatives: Theory and Applications, Gordon and Breach , London (1993)
  • Google Scholar


• [6] MD. Buhmann, Radial basis functions: theory and implementations, Cambridge University Press, (2003)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] V. R. Hosseini, W. Chen, Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Boundary. Elem, 38 (2014), 31-39.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] S. Abbasbandy, H. R. Ghehsareh, I. Hashim, Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function, Eng. Anal. Boundary. Elem. , 36(12) (2012), 1811-1818.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] S. Abbasbandy, H. R. Ghehsareh, I. Hashim, A meshfree method for the solution of two-dimensional cubic nonlinear schrödinger equation, Eng. Anal. Boundary. Elem. , 37(6) (2013), 885-898.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] B. Nayroles, G. Touzot, P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. Mech., 10 (1992), 307-318.
  • View Article
  • Google Scholar
  • MATH


• [11] C. Duarte, J. Oden, An h-p adaptative method using clouds , Comput. Methods Appl. Mech. Engrg., 139 (1996), 237-262.
  • View Article
  • Google Scholar


• [12] S. N. Atluri, T. L. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. , 22 (2) (1998), 117-127.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] S. Abbasbandy, V. Sladek, A. Shirzadi, J. Sladek, Numerical Simulations for Coupled Pair of Diffusion Equations by MLPG Method, CMES Compt. Model. Eng. Sci. , 71 (1) (2011), 15-37.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] A. Shirzadi, L. Ling, S. Abbasbandy, Meshless simulations of the two-dimensional fractional-time convectiondiffusion- reaction equations, Eng. Anal. Bound. Elem. , 36 (2012), 1522-1527.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] T. Zhu, J. D. Zhang, S. N. Atluri, A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach, Comput. Mech. , 21 (1998), 223-235.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] J. M. Melenk, I. Babu ska, The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg. , 139 (1996), 289-314.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] E. J. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II, J. Comput. Math. Appl. , 19 (1990), 147-161.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] M. Aslefallah, D. Rostamy, A Numerical Scheme For Solving Space-Fractional Equation By Finite Differences Theta-Method, Int. J. of Adv. in Aply. Math. and Mech., 1(4) (2014), 1-9.
  • Google Scholar
  • MATH


• [19] L. B. Lucy, A numerical approach to the testing of fusion process, Astron. J. , 88 (1977), 1013-1024.


• [20] W. K. Liu, S. Jun, Y. F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 21 (1995), 1081-1106.
  • View Article
  • Google Scholar


• [21] M. Aslefallah, D. Rostamy, K. Hosseinkhani, Solving time-fractional differential diffusion equation by theta-method, Int. J. of Adv. in Aply. Math. and Mech., 2(1) (2014), 1-8.
  • MathSciNet
  • Google Scholar
  • MATH


• [22] G. R. Liu , Mesh Free Methods: Moving beyond the Finite Element Method, CRC Press., (2003)
  • Google Scholar
  • MATH


• [23] E. Shivanian, Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics, Eng. Anal. Boundary. Elem., 37 (2013), 1693-1702.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] M. Dehghan, A. Ghesmati, Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM), Comput. Phys. Commun., 181 (2010), 772-786.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] N. Apreutesei, N. Bessonov, V. Volpert, V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 537-557.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] F. Fakhar-Izadi, M. Dehghan, An efficient pseudo-spectral LegendreGalerkin method for solving a nonlinear partial integro-differential equation arising in population dynamics, Math. Methods Appl. Sci., 36(12) (2013), 1485-1511.
  • View Article
  • Google Scholar
  • MATH


• [27] M. Aslefallah, D. Rostamy , Numerical solution for Poisson fractional equation via finite differences theta-method, J. Math. Computer Sci. , TJMCS, 12(2) (2014), 132-142.
  • Google Scholar


• [28] C. A. Micchell, Interpolation of scattered data; distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11-22.
  • View Article
  • MathSciNet