A NEW NUMERICAL METHOD FOR SOLVING DELAY INTEGRAL EQUATIONS WITH VARIABLE BOUNDS BY USING GENERALIZED MOTT POLYNOMIALS

Year 2018, Volume: 19 Issue: 4, 844 - 857, 31.12.2018

Ömür Kürkçü

https://doi.org/10.18038/aubtda.409056

Cited By: 1

Abstract

In this study, the functional delay integral equations with variable bounds are considered. Their approximate solutions are obtained by using a new method based on matrix, collocation points and the generalized Mott polynomials with the parameter-$\beta$. An error analysis technique consisting of the residual function is performed. The numerical examples are illustrated for the practicability and usability of the method. The behavior of the solutions is monitored in terms of the parameter-$\beta$. The accuracy of the method is scrutinized for different values of N. In addition, the numerical results are discussed in figures and tables.

Keywords

Mott polynomials, Matrix method, Collocation points, Error analysis

References

• [1] Iwasaki, H, Gyoubu, S, Kawatsu, T, Miura, S. A 3D-RISM integral equation study of a hydrated dipeptide. Mol Simulat 2015; 41: 1015–1020.
• [2] Wang, X, Wildman, RA, Weile, DS, Monk, P. A finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics. IEEE Trans Antennas Propag 2008; 56: 2442–2452.
• [3] Ojala, R, Tornberg, A-K. An accurate integral equation method for simulating multi-phase Stokes flow. J Comput Phys 2015; 298: 145–160.
• [4] Ioannou, Y, Fyrillas, MM, Doumanidis, C. Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer. Eng Anal Bound Elem 2012; 36: 1278–1283.
• [5] Buryachenko, V.A. Solution of general integral equations of micromechanics of heterogeneous materials. Int J Solids Struct 2014; 51: 3823–3843.
• [6] Sezer, M. Taylor polynomial solution of Volterra integral equations. Int J Math Educ Sci Technol 1994; 25: 625–633.
• [7] Kürkçü, ÖK, Aslan, E, Sezer, M. A numerical approach with error estimation to solve general integro-differential–difference equations using Dickson polynomials. Appl Math Comput 2016; 276: 324–339.¬¬
• [8] Kürkçü, ÖK, Aslan, E, Sezer, M. A novel collocation method based on residual error analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials. Sains Malays 2017; 46: 335–347.[9] Kürkçü, ÖK, Aslan, E, Sezer, M. A numerical method for solving some model problems arising in science and convergence analysis based on residual function. Appl Numer Math 2017; 121: 134–148.
• [10] Yüzbaşı, Ş, Şahin, N, Sezer, M. Bessel polynomial solutions of high-order linear Volterra integro-differential equations. Comput Math Appl 2011; 62: 1940–1956.
• [11] Gülsu, M, Öztürk, Y, Sezer, M. A new collocation method for solution of mixed linear integro-differential-difference equations. Appl Math Comput 2010; 216: 2183–2198.
• [12] Gokmen, E, Yuksel, G, Sezer, M. A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays. J Comput Appl Math 2017; 311: 354–363.
• [13] Çelik, İ. Collocation method and residual correction using Chebyshev series. Appl Math Comput 2006; 174: 910–920.
• [14] Nemati, S. Numerical solution of Volterra–Fredholm integral equations using Legendre collocation method, J Comput Appl Math 2015; 278: 29–36.
• [15] Adomian, G. A review of the decomposition method in applied mathematics, J Math Anal Appl 1988; 135: 501–544.
• [16] Babolian, E, Davary, A. Numerical implementation of Adomian decomposition method for linear Volterra integral equations for the second kind, Appl Math Comput 2005; 165: 223–227.
• [17] Xu, L. Variational iteration method for solving integral equations, Comput Math Appl 2077; 54: 1071–1078.
• [18] Ebrahimi, N, Rashidinia, J. Collocation method for linear and nonlinear Fredholm and Volterra integral equations. Appl Math Comput 2015; 270: 156–164.
• [19] Rashidinia, J, Mahmoodi, Z. Collocation method for Fredholm and Volterra integral equations. Kybernetes 2013; 42: 400–412.
• [20] Blyth, WF, May, RL, Widyaningsih, P. Volterra integral equations solved in Fredholm form using Walsh functions. Anziam J 2004; 45: C269–C282.
• [21] Saberi-Nadjafi, J, Mehrabinezhad, M, Diogo, T. The Coiflet–Galerkin method for linear Volterra integral equations. Appl Math Comput 2013; 221: 469–483.
• [22] Mott, N.F. The Polarisation of Electrons by Double Scattering, Proc R Soc Lond A 1932; 135: 429–458.
• [23] Erdélyi, A, Magnus, W, Oberhettinger, F, Tricomi, F.G. Higher transcendental functions. Vol. III, New York-Toronto-London: McGraw-Hill Book Company, 1955.
• [24] Roman, S. The umbral calculus. Pure and Applied Mathematics, 111, London: Academic Press, 1984.
• [25] Solane, NJ. The On-Line Encyclopedia of integer sequences. published electronically at https://oeis.org/A137378.
• [26] Kruchinin, DV. Explicit formula for generalized Mott polynomials. Adv Stud Contemp Math. 2014; 24: 327–322.