Mehmet SEZER, Seda ÇAYAN

Lerch matrix collocation method for 2D and 3D Volterra type integral and second order partial integro differential equations together with an alternative error analysis and convergence criterion based on residual functions

In this study, second order linear Volterra partial integro-differential equation with two- and three-dimensional are solved by collocation method based on Lerch polynomials. This method is composed of the operational matrix and collocation methods, which are based upon the matrix forms of the Lerch polynomials with the parameter λ and Taylor polynomials, and their derivatives and integrals. The approximate solutions of the mentioned equations are investigated in terms of the Lerch polynomials the different values of λ. Also, to verify the accuracy and efficiency of the present method, an alternative convergence criterion along with error analysis depending on residual function is enhanced. Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.

PDF

___

[1] Ayati Z, Biazar J. On the convergence of Homotopy perturbation method. Journal of the Egyptian Mathematical Society 2015; 23 (2): 424-428.
[2] Bakhshi M, Asghari-Larimi M, Asghari-Larimi M. Three-dimensional differential transform method for solving nonlinear three-dimensional Volterra integral equations. The Journal of Mathematics and Computer Science 2012; 4 (2): 246-256.
[3] Behzadi SS. The use of iterative methods to solve two-dimensional nonlinear Volterra-Fredholm integro-differential equations. Communications in Numerical Analysis 2012; 2012: 1-20.
[4] Biçer KE, Yalcinbas S. A matrix approach to solving hyperbolic partial differential equations using Bernoulli polynomials. Filomat 2016; 30 (4): 993-1000.
[5] Borhanifar A, Sadri K. A generalized operational method for solving integro-partial differential equations based on Jacobi polynomials. Hacettepe Journal of Mathematics and Statistics 2016; 45 (2): 311-335.
[6] Branson D. An extension of stirling numbers. Fibonacci Quarterly 1996; 34 (3): 213-223.
[7] Brunner H. Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge, UK: Cambridge University Press, 2017.
[8] Darania P, Shali JA, Ivaz K. New computational method for solving some 2-dimensional nonlinear Volterra integrodifferential equations. Numerical Algorithms 2011; 57 (1): 125-147.
[9] Doha EH, Abdelkawy MA, Amin AZM, Baleanu D. Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations. Nonlinear Analysis: Modelling and Control 2019; 24 (3): 332-352.
[10] Gürbüz B, Sezer M. Laguerre polynomial solutions of a class of delay partial functional differential equations. Acta Physica Polonica Series A 2017; 132 (3): 558-560.
[11] Gürbüz B, Sezer M. A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method. International Journal of Applied Physics and Mathematics 2017; 7 (1): 49-58.
[12] Hashim HE, Elzaki TM. Solving singular partial integro-differential equations using Taylor series. International Journal of Innovative Science, Engineering & Technology 2015; 2 (4): 501-506.
[13] Hendi FA, Al-Qarni MM. The variational Adomian decomposition method for solving nonlinear two-dimensional Volterra-Fredholm integro-differential equation. Journal of King Saud University - Science 2019; 31 (1): 110-113.
[14] Illie S, Jeffrey DJ, Corless RM, Zhang X. Computation of Stirling Numbers and Generalizations. In: 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing; Timisoara, Romania; 2015. pp. 57-60.
[15] Khajehnasiri AA. Numerical solution of nonlinear 2D Volterra-Fredholm integro-differential equations by twodimensional triangular function. International Journal of Applied and Computational Mathematics 2016; 2 (4): 575-591.
[16] Kruchinin V, Kruchinin D. Explicit formulas for some generalized polynomials. Applied Mathematics & Information Sciences 2013; 7 (5): 2083-2088.
[17] Kürkçü ÖK, Aslan E, Sezer M. A numerical method for solving some model problems arising in science and convergence analysis based on residual function. Applied Numerical Mathematics 2017; 121: 134-148.
[18] Kürkçü ÖK, Aslan E, Sezer M. An advanced method with convergence analysis for solving space-time fractional partial differential equations with multi delays. The European Physical Journal Plus 2019; 134 (8): 393.
[19] Kürkçü ÖK, Aslan E, Sezer M. A novel collocation method based on residual error analysis for solving integrodifferential equations using by hybrid Dickson and Taylor polynomials. Sains Malays 2017; 46: 335-347.
[20] Kythe PK, Puri P. Computational Methods for Linear Integral Equations. Boston, MA, USA: Birkhăuser, 2002.
[21] Maleknejad K, Sohrabi S, Baranji B. Application of 2D-BPFs to nonlinear integral equations. Communications in Nonlinear Science and Numerical Simulation 2010; 15 (3): 527-535.
[22] Mckee S, Tang T, Diogo T. An Euler-type method for two-dimensional Volterra integral equations of the first kind. IMA Journal of Numerical Analysis 2000; 20 (3): 423-440.
[23] Mirzaee F, Hadadiyan E, Bimesl S. Numerical solution for three dimensional nonlinear mixed Volterra-Fredholm integral equations via three-dimensional block-pulse functions. Applied Mathematics and Computation 2014; 237: 168-175.
[24] Poorfettah E, Shaerlar AJ. Direct method for solving nonlinear two dimensional Volterra-Fredholm integrodifferential equations by block-pulse functions. IJISSM 2015; 4 (1): 418-423.
[25] Rahman M. Integral Equations and their Applications. Boston, MA, USA: WIT Press, 2007.
[26] Rivaz A, Jahan S, Yousefi F. Two-dimensional Chebyshev polynomials for solving two-dimensional integrodifferential equations. Çankaya University Journal of Science and Engineering 2015; 12 (2): 1-11.
[27] Savaşaneril NB. Bernstein series solution of the heat equation in 2-D. New Trends in Mathematical Sciences 2017; 5 (4): 220-231.
[28] Singh S, Patel VK, Singh VK. Operational matrix approach for the solution of partial integro-differential equation. Applied Mathematics and Computation 2016; 283: 195-207.
[29] Tari A, Rahimi MY, Shahmorad S, Talati F. Development of the Tau method for the numerical solution of twodimensional linear Volterra integro-differential equations. Computational Methods in Applied Mathematics 2009; 9 (4): 421-435.
[30] Wazwaz A. A First Course in Integral Equations. Singapore: World Scientific Publishing, 2015.
[31] Wazwaz A. Linear and Nonlinear Integral Equations: Methods and Applications. Heidelberg, Germany: Springer Verlag GmbH, 2011.
[32] Yalçın E, Sezer M. Hermite polynomial solutions of one dimensional parabolic-type Volterra partial integrodifferential equations with mixed delays by using matrix-collocation method. In: 2nd International Symposium on Multidisciplinary Academic Studies Proceedings; İstanbul, Turkey; 2018. pp. 423-433.
[33] Yalçın E, Sezer M. A new computational method for solving two-dimensional integral and partial integro differential equations by means of Hermite and Taylor polynomials. In: 2nd International Symposium on Multidisciplinary Academic Studies Proceedings; İstanbul, Turkey; 2018. pp. 397-410.
[34] Yalçın E. Numerical solutions based on Hermite polynomials of partial integro differential equations with two independent variables and their applications. PhD, Manisa Celal Bayar University, Manisa, Turkey, 2019.
[35] Yüzbaşı Ş. A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions. International Journal of Computational Methods 2017; 14 (1750015): 1-20.
[36] Yüzbaşı Ş. A collocation approach for solving two-dimensional second-order linear hyperbolic equations. Applied Mathematics and Computation 2018; 338: 101-114.
[37] Yüzbaşı Ş. Bessel collocation approach for solving one-dimensional wave equation with Dirichlet, Neumann boundary and integral conditions. Research and Communications in Mathematics and Mathematical Sciences 2019; 11 (1): 63-87.
[38] Yüzbaşı Ş. An exponential method to solve linear Fredholm-Volterra integro-differential equations and residual improvement. Turkish Journal of Mathematics 2018; 42: 2546-2562.
[39] Yüzbaşı Ş, Karaçayir M. A numerical approach for solving high-order linear delay Volterra integro-differential equations. International Journal of Computational Methods 2018; 15 (5): 1850042.
[40] Yüzbaşı Ş, Karaçayir M. An approximation technique for solutions of singularly perturbed one-dimensional convection-diffusion problem. International Journal of Modelling: Electronic Networks, Devices and Fields 2020; 33 (1): e2686.
[41] Ziqan A, Armiti S, Suwan I. Solving three-dimensional Volterra integral equation by the reduced differential transform method. International Journal of Applied Mathematical Research 2016; 5 (2): 103-106.