A GALERKIN-TYPE METHOD FOR SOLUTIONS OF PANTOGRAPH-TYPE VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS WITH FUNCTIONAL UPPER LIMIT

Yıl 2020, Cilt: 38 Sayı: 2, 995 - 1005, 01.06.2021

Şuayip Yüzbaşı Murat Karaçayır

Öz

In this study, we present a Galerkin-type method for obtaining approximate solutions of linear Volterra-Fredholm delay integro-differential equations with a functional upper limit under mixed conditions. The method gives an approximate solution of the problem in power series form truncated after a certain term. Using an integer value N as the truncation point and making use of the matrix representations of a polynomial and its derivatives, we obtain the matrix form of the problem expressed in terms of the approximate solution polynomial. By applying inner product to these relations with monomials up to degree N and incorporating the mixed conditions, the problem is reduced to a system of linear algebraic equations. The approximate solution of the problem is then determined from this linear system. In addition, we discuss a way of improving an obtained approximate solution by means of its estimated error function. The presented scheme has the advantages of (1) being applicable to a wide range of problems including pantograph-type equations with or without Fredholm and Volterra integral terms, and (2) giving accurate results as demonstrated by applications to example problems taken from existing studies.

Anahtar Kelimeler

Volterra-Fredholm integro-differential equations, pantograph-type integro-differential equations, Galerkin type method, numerical solutions, error estimation, residual correction

Kaynakça

• [1] Kapur J.N. (2005) Mathematical Modelling, New Age International Ltd., New Delhi, India.
• [2] Jackiewicz Z., Rahman M. and Welfert B.D. (2006) Numerical solution of a Fredholm integro-differential equation modelling neural networks, Appl. Numer. Math. 56, 423-432.
• [3] Bates A.P., Khalid Z. and Kennedy R.A. (2017) Slepian Spatial-Spectral Concentration Problem on the Sphere: Analytical Formulation for Limited Colatitude-Longitude Spatial Region, IEEE T. Signal Proces. 65 (6), 1527-1537.
• [4] Scudo F.M. (1971) Vito Volterra and theoretical ecology, Theor. Popul. Biol. 2, 1-23.
• [5] MacCamy R.C. (1977) An integro-differential equation with application in heat flow, Q. Appl. Math. 35 (1), 1-19.
• [6] Diekman O. (1978) Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol. 6, 109-130.
• [7] Abdou M.A. (2003) On asymptotic methods for Fredholm-Volterra integral equation of the second kind in contact problems, J. Comput. Appl. Math. 154, 431-446.
• [8] Gülsu M. and Sezer M. (2016) Taylor collocation method for solution of systems of high-order linear Fredholm-Volterra integro-differential equations, Int. J. Comput. Math. 83 (4), 429-448.
• [9] Yüzbaşı Ş., Şahin N. and Yıldırım A. (2012) A collocation approach for solving high-order linear Fredholm-Volterra integro-differential equations, Math. Comput. Model. 55, 547-563.
• [10] Yüzbaşı Ş. and Ismailov N. (2018) An operational matrix method for solving linear Fredholm-Volterra integro-differential equations, Turk. J. Math. 42, 243-256.
• [11] Taiwo O.A. and Adio A.K. (2014) Variational Iteration and Homotopy Perturbation Methods for Solving Fredholm-Volterra Integro-Differential Equations, IJMSI 2 (1), 49-55.
• [12] Yüzbaşı Ş. (2014) Laguerre approach for solving pantograph-type Volterra integro-differential equations, Appl. Math. Comput. 232, 1183-1199.
• [13] Wang K. and Wang Q. (2014) Taylor polynomial method and error estimation for a kind of mixed Volterra-Fredholm integral equations, Appl. Math. Comput. 229, 53-59.
• [14] Oğuz C. And Sezer M. (2015) Chelyshkov collocation method for a class of mixed functional integro-differential equations, Appl. Math. Comput. 259, 943-954.
• [15] Kürkçü Ö.K., Aslan E. And Sezer M. (2016) A numerical approach with error estimation to solve general integro-differential-difference equations using Dickson polynomials, Appl. Math. Comput. 259, 324-339.
• [16] Saaedi L., Tari A. and Babolian E. (2019) Numerical Solution of Functional Volterra-Hammerstein Integro-differential Equations of Order Two, Iranian Journal of Science and Technology Transactions A – Science 43 (3), 2981-2992.
• [17] Türkyılmazoğlu M. (2014) An effective approach for numerical solutions of high-order Fredholm integro-differential equations, Appl. Math. Comput. 227, 384-398.
• [18] Yüzbaşı Ş. and Karaçayır M. (2016) A Galerkin-like approach to solve high-order integro-differential equations with weakly singular kernel, Kuwait J. Sci. 43 (2), 106-120.
• [19] Yüzbaşı Ş. and Karaçayır M. (2018) A Galerkin-Type Method for Solving a Delayed Model on HIV Infection of CD4+ T-cells, Iranian Journal of Science and Technology Transactions A – Science 42 (2), 1087-1095.
• [20] Türkyılmazoğlu M. (2013) Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type, Appl. Math. Model. 37, 7539-7548.
• [21] Türkyılmazoğlu M. (2014) Effective Computation of Solutions for Nonlinear Heat Transfer Problems in Fins, J. Heat Transfer 136 (9), 091901.
• [22] Du Croz J.J. and Higham N.J. (1992) Stability of Methods for Matrix Inversion, IMA Journal of Numerical Analysis 12 (1), 1-19.
• [23] Amat S., Busquier S. and Gutiérrez J.M. (2003) Geometric construction of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 153, 197-205.
• [24] Li H.B., Huang T.Z., Zhang Y., Liu X.P. and Gu T.X. (2011) Chebyshev-type methods and preconditioning techniques, Appl. Math. Comput. 218 (2), 260-270.