The numerical solution of the singularly perturbed differential-difference equations based on the Meshless method
In this paper, we describe a meshless approach to solve singularly perturbed differential- difference equations of the second order with boundary layer at one end of the interval. In the numerical treatment for such type of problems, first we approximate the terms containing negative and positive shifts which converts it to a singularly perturbed differential equation. Next, a numerical scheme based on the moving least squares (MLS) method is used for solving singularly perturbed differential equation. The MLS methodology is a meshless method, because it does not need any background mesh or cell structures. The proposed scheme is simple and efficient to approximate the unknown function. Several examples are presented to demonstrate the efficiency and validity of the numerical scheme presented in the paper.
Keywords: Differential-Difference Equation; Singular Perturbations; Boundary Layer; Meshless Method.
A Longtin, J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988), pp. 183-199.
E.L. Elsgolts, Qualitative method in mathematical analysis: Translations of mathematical monographs, vol.12, American mathematical society, Providence, RI, 1964.
J. D. Murray, Mathematical biology I: An introduction, 3rd ed., Springer-verlag, Berlin, 2001.
M.W. Gerstein, H.M. Gibbs, F.A. Hopf, D.L. Kaplan, Bifurcation gap in a hybrid optical system, Physs. Rev. A 26 (1982), pp. 3720-3722.
M. K. Kadalbajoo, K. K. sharma , "Numerical treatment of mathematical model arising from a model of neuronal variability ," J. math. Anal. Appl., 307(2005), pp. 606-627.
M. K. Kadalbajoo, K. K. Sharma, "Numerical Analysis of singularly perturbed delay differential equations with layer behavior," Appl. Math. Compute, 157(2004), pp. 11-28.
M. K. Kadalbajoo, K. K. Sharma, “Numerical Analysis of boundary value problem for singularly perturbed delay differential equations with layer behavior," Appl. Math. Comput, 151(1) (2004), pp. 11-28.
M. K. Kadalbajoo, K. K. Sharma, "A Numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equation ," Appl. Math. Comput., 197(2008), pp. 692-707.
M. K. Kadalbajoo,V.P. Ramesh. Hybrid method for numerical solution of singularly perturbed delay differential equations, Appl. Math. Comput. 187(2007), pp. 797-814.
I. G. Amiraliyeva, F. Erdogan, "Uniform Numerical method for singularly perturbed delay differential equations," comput. Math. Appl. 53(2007), pp. 1251-1259.
I. G. Amiraliyeva, G. M. Amiraliyev, "Uniform difference method for parameterized singularly perturbed delay differential equations," Numer, Algo. 52(2009), pp. 509-521.
Rao, R. N., Chakravarthy, P. P. A fourth order finite difference method for singularly perturbed differential-difference equations, American journal of computational and applied mathematics. 1(2011), pp. 5-10.
S. N. Atluri, T. Zhv, "A new meshless local petrov-Galerkin (MLPG) approach in computational mechanics," Comput. Mech. 22(2) (1998), pp. 117-127.
M. Dehghan D. Mirzaei, "Meshless local petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic(MHD) flow through pipe with orbitrary wall conductivity," Appl. Numer. Math. 59(2009), pp. 1043-1058.
D. Mirzaei, M. Dehghan, "A meshless based method for solution of integral equations," Appl. Num. Math., 60(2010), pp. 245-262.
M. Dehghan, R. Salehi, The numerical solution of the non-linear integro-differential equations based on the meshless method, J. Comput. Appl. Math. 236(2012), pp. 2367-2377.
M. Dehghan D. Mirzaei, The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrodinger equation, Engineering Analysis with Boundary Elements 32(2008), pp. 747-756.
D. Shepard, "A two-dimensional interpolation function for irregularly space points," in: proc. 23rd Nat. conf. ACM. ACMpress, Newyork, 1968, pp. 517-524.
P. Lancaster, K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comput. 37 (1981), pp. 141-158.