Abazari, Reza; Kılıçman, Adem Numerical study of two-dimensional Volterra integral equations by RDTM and comparison with DTM. (English) Zbl 1470.65207 Abstr. Appl. Anal. 2013, Article ID 929478, 10 p. (2013). Summary: The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM), and compared with the differential transform method (DTM). The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences. Cited in 2 Documents MSC: 65R20 Numerical methods for integral equations 45D05 Volterra integral equations × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Differential Equations. Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monographs on Applied and Computational Mathematics, 15, xiv+597 (2004), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1059.65122 · doi:10.1017/CBO9780511543234 [2] Huabsomboon, P.; Novaprateep, B.; Kaneko, H., On Taylor-series expansion methods for the second kind integral equations, Journal of Computational and Applied Mathematics, 234, 5, 1466-1472 (2010) · Zbl 1190.65195 · doi:10.1016/j.cam.2010.02.023 [3] Kaneko, H.; Xu, Y., Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Mathematics of Computation, 62, 206, 739-753 (1994) · Zbl 0799.65023 · doi:10.2307/2153534 [4] Tari, A.; Rahimi, M. Y.; Shahmorad, S.; Talati, F., Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method, Journal of Computational and Applied Mathematics, 228, 1, 70-76 (2009) · Zbl 1176.65164 · doi:10.1016/j.cam.2008.08.038 [5] Jang, B., Comments on “Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method”, Journal of Computational and Applied Mathematics, 233, 2, 224-230 (2009) · Zbl 1175.65149 · doi:10.1016/j.cam.2009.07.012 [6] jiang, Y.-J., On spectral methods for Volterra-type integro-differential equations, Journal of Computational and Applied Mathematics, 230, 2, 333-340 (2009) · Zbl 1202.65170 · doi:10.1016/j.cam.2008.12.001 [7] Akyüz-Daşcıoğlu, A.; Sezer, M., Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations, Journal of the Franklin Institute. Engineering and Applied Mathematics, 342, 6, 688-701 (2005) · Zbl 1086.65121 · doi:10.1016/j.jfranklin.2005.04.001 [8] Hosseini, S. M.; Shahmorad, S., Numerical solution of a class of integro-differential equations by the tau method with an error estimation, Applied Mathematics and Computation, 136, 2-3, 559-570 (2003) · Zbl 1027.65182 · doi:10.1016/S0096-3003(02)00081-4 [9] Kajani, M. T.; Ghasemi, M.; Babolian, E., Numerical solution of linear integro-differential equation by using sine-cosine wavelets, Applied Mathematics and Computation, 180, 2, 569-574 (2006) · Zbl 1102.65137 · doi:10.1016/j.amc.2005.12.044 [10] Farnoosh, R.; Ebrahimi, M., Monte Carlo method for solving Fredholm integral equations of the second kind, Applied Mathematics and Computation, 195, 1, 309-315 (2008) · Zbl 1131.65109 · doi:10.1016/j.amc.2007.04.097 [11] Maleknejad, K.; Mirzaee, F., Numerical solution of integro-differential equations by using rationalized Haar functions method, Kybernetes, 35, 10, 1735-1744 (2006) · Zbl 1160.45303 · doi:10.1108/03684920610688694 [12] Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind. The Numerical Solution of Integral Equations of the Second Kind, Cambridge Monographs on Applied and Computational Mathematics, 4, xvi+552 (1997), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0899.65077 · doi:10.1017/CBO9780511626340 [13] Guoqiang, H.; Jiong, W., Extrapolation of Nystrom solution for two dimensional nonlinear Fredholm integral equations, Journal of Computational and Applied Mathematics, 134, 1-2, 259-268 (2001) · Zbl 0989.65150 · doi:10.1016/S0377-0427(00)00553-7 [14] Han, G.; Wang, R., Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations, Journal of Computational and Applied Mathematics, 139, 1, 49-63 (2002) · Zbl 1001.65142 · doi:10.1016/S0377-0427(01)00390-9 [15] Zhou, J. K., Differential Transformation and Its Application for Electrical CircuIts (1986), Wuhan, China: Huazhong University Press, Wuhan, China [16] Abazari, R.; Borhanifar, A., Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method, Computers & Mathematics with Applications, 59, 8, 2711-2722 (2010) · Zbl 1193.65178 · doi:10.1016/j.camwa.2010.01.039 [17] Borhanifar, A.; Abazari, R., Exact solutions for non-linear Schrödinger equations by differential transformation method, Journal of Applied Mathematics and Computing, 35, 1-2, 37-51 (2011) · Zbl 1211.35250 · doi:10.1007/s12190-009-0338-2 [18] Borhanifar, A.; Abazari, R., Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method, Optics Communications, 283, 10, 2026-2031 (2010) · doi:10.1016/j.optcom.2010.01.046 [19] Abazari, R.; Abazari, M., Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM, Communications in Nonlinear Science and Numerical Simulation, 17, 2, 619-629 (2012) · Zbl 1244.65153 · doi:10.1016/j.cnsns.2011.05.022 [20] Abazari, R., Solution of Riccati types matrix differential equations using matrix differential transform method, Journal of Applied Mathematics & Informatics, 27, 1133-1143 (2009) [21] Abazari, R.; Abazari, R., Numerical study of some coupled PDEs by using differential transformation method, World Academy of Science, Engineering and Technology, 66, 52-59 (2010) [22] Keskin, Y.; Oturanç, G., Reduced differential transform method for partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 6, 741-749 (2009) [23] Keskin, Y.; Oturanç, G., The reduced differential transform method: a new approach to factional partial differential equations, Nonlinear Science Letters A, 1, 207-217 (2010) [24] Abazari, R.; Ganji, M., Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay, International Journal of Computer Mathematics, 88, 8, 1749-1762 (2011) · Zbl 1232.35012 · doi:10.1080/00207160.2010.526704 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.