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
Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.
📌 3 Key Takeaways
🎯
Objective & Hypothesis: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.
📊
Key Findings & Data: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.
💡
Conclusion & Impact:Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
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.
🔖
Cite This Article
SEZER, M., ÇAYAN, S. (2022). 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. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 6, 2073–2098. https://ajindex.com/articles/607593
Ayati Z, Biazar J. On the convergence of Homotopy perturbation method. Journal of the Egyptian Mathematical Society 2015; 23 (2): 424-428
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
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
Biçer KE, Yalcinbas S. A matrix approach to solving hyperbolic partial differential equations using Bernoulli polynomials. Filomat 2016; 30 (4): 993-1000
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
Branson D. An extension of stirling numbers. Fibonacci Quarterly 1996; 34 (3): 213-223
Brunner H. Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge, UK: Cambridge University Press, 2017
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
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
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
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
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
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