Spectral Petrov-Galerkin Method for Solving the Two-Dimensional Integral Equations of the Second Kind
DOI:
https://doi.org/10.56714/bjrs.50.1.12Keywords:
Two-dimensional integral equation, Petrov-Galerkin method, Spectral method, Two-Dimensional Orthogonality Polynomials, Chebyshev polynomial, Laguerre polynomialAbstract
Using a spectral Petrov-Galerkin approach (SPGM), the main objective of this research is to propose a numerical solution for the two-dimensional (2D) linear Volterra and mixed Volterra-Fredholm integral equations (VIEs and MIEs, respectively) type. We solving four examples that illustrate the results of the method compared with other methods in terms of excellent accuracy, powerful method and less error it provides for this type of integral equations (IEs).
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H. O. Al-Humedi and A. Kadhim Munaty, "The spectral petrov-Galerkin method for solving integral equations of the first kind," Turkish Journal of Computer and Mathematics Education (TURCOMAT), vol. 12, no. 13, pp. 7856-7865, 2021.Doi:http://dx.doi.org/10.22075/ijnaa.2022.6074.
E. Babolian, K. Maleknejad, M. Roodaki, and H. Almasieh, "Two-dimensional triangular functions and their applications to nonlinear 2D Volterra–Fredholm integral equations," Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1711-1722, 2010.Doi:https://doi.org/10.1016/j.camwa.2010.07.002.
Z. Chen and Y. Xu, "The Petrov--Galerkin and Iterated Petrov--Galerkin Methods for Second-Kind Integral Equations," SIAM Journal on Numerical Analysis, vol. 35, no. 1, pp. 406-434, 1998.Doi:https://doi.org/10.1137/S0036142996297217.
E. H. Doha, R. Hafez, and Y. H. Youssri, "Shifted Jacobi spectral-Galerkin method for solving hyperbolic partial differential equations," Computers & Mathematics with Applications, vol. 78, no. 3, pp. 889-904, 2019.Doi:https://doi.org/10.1016/j.camwa.2019.03.011.
M. Erfanian and H. Zeidabadi, "Solving two-dimensional nonlinear mixed Volterra Fredholm integral equations by using rationalized Haar functions in the complex plane," Journal of Mathematical Modeling, vol. 7, no. 4, pp. 399-416, 2019.Doi:https://doi.org/10.22124/jmm.2019.13987.1300.
M. Eslahchi, M. Dehghan, and M. Parvizi, "Application of the collocation method for solving nonlinear fractional integro-differential equations," Journal of Computational and Applied Mathematics, vol. 257, pp. 105-128, 2014.Doi:https://doi.org/10.1016/j.cam.2013.07.044.
S. Fazeli, G. Hojjati, and H. Kheiri, "A piecewise approximation for linear two-dimensional volterra integral equation by chebyshev polynomials," Int. J. Nonlinear Sci, vol. 16, no. 3, pp. 255-261, 2013.Doi:https://www.researchgate.net/publication/260115266.
H. Guoqiang, K. Hayami, K. Sugihara, and W. Jiong, "Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations," Applied mathematics and computation, vol. 112, no. 1, pp. 49-61, 2000.Doi:https://doi.org/10.1016/S0096-3003(99)00036-3.
H. Guoqiang and Z. Liqing, "Asymptotic error expansion of two-dimensional Volterra integral equation by iterated collocation," Applied Mathematics and Computation, vol. 61, no. 2-3, pp. 269-285, 1994.Doi: https://doi.org/10.1016/0096-3003(94)90050-7.
R. Hafez, E. Doha, A. Bhrawy, and D. Baleanu, "Numerical solutions of two-dimensional mixed Volterra-Fredholm integral equations via Bernoulli collocation method," Rom. J. Phys, vol. 62, no. 111, pp. 1-11, 2017.
R. Hafez and Y. Youssri, "Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation," Computational and Applied Mathematics, vol. 37, pp. 5315-5333, 2018.Doi: http://dx.doi.org/10.1007/s40314-018-0633-3.
F. Khan, M. Omar, and Z. Ullah, "Discretization method for the numerical solution of 2D Volterra integral equation based on two-dimensional Bernstein polynomial," AIP Advances, vol. 8, no. 12, 2018.Doi:http://dx.doi.org/10.1063/1.5051113.
J. Khazaian, N. Parandin, F. Mohammadi Yaghoobi, and N. Karami Kabir, "Numerical Solution Two-Dimensional Volterra-Fredholm Integral Equations of the Second Kind with Block-Pulse Functions Based on Legendre Polynomials," International Journal of Mathematical Modelling & Computations, vol. 12, no. 1 (WINTER), pp. 1-14, 2022.Doi:https://dorl.net/dor/20.1001.1.22286225.2022.12.1.1.5.
H. Laib, A. Boulmerka, A. Bellour, and F. Birem, "Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method," Journal of Computational and Applied Mathematics, vol. 417, p. 114537, 2023.Doi:https://doi.org/10.1016/j.cam.2022.114537.
S. Micula, "Numerical solution of two-dimensional Fredholm–Volterra integral equations of the second kind," Symmetry, vol. 13, no. 8, p. 1326, 2021.Doi:https://doi.org/10.3390/sym13081326.
F. Mirzaee and N. Samadyar, "Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed Volterra–Fredholm integral equations," Transactions of A. Razmadze Mathematical Institute, vol. 172, no. 3, pp. 631-641, 2018.Doi:https://doi.org/10.1016/j.trmi.2017.09.006.
P. Mokhtary and F. Ghoreishi, "The L 2-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations," Numerical Algorithms, vol. 58, pp. 475-496, 2011.Doi:http://dx.doi.org/10.1007/s11075-011-9465-6.
S. Nemati and Y. Ordokhani, "Numerical solution of two-dimensional nonlinear Volterra integral equations by the Legendre polynomials," 2011.Doi:https://doi.org/10.1016/j.cam.2012.10.021.
Y. Pan and J. Huang, "Extrapolation method for solving two-dimensional volterral integral equations of the second kind," Applied Mathematics and Computation, vol. 367, p. 124784, 2020.Doi:https://doi.org/10.1016/j.amc.2019.124784.
J. N. Reddy, An introduction to the finite element method. McGraw-Hill New York, 2005.
A. Tari, M. Rahimi, S. Shahmorad, and F. Talati, "Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method," Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 70-76, 2009.Doi:http://dx.doi.org/10.1016/j.cam.2008.08.038.
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