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Shifted Jacobi collocation method for Volterra-Fredholm integral equation. (English) Zbl 1499.65761

Summary: In this paper, we compute the approximate numerical solution for the Volterra-Fredholm integral equation (VFIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the shifted Jacobi polynomials are introduced. These properties allow us to transform the VolterraFredholm integral equation into a system of algebraic equations in a nice form with the expansion coefficients of the solution. Also, the convergence and error analysis are studied extensively. Finally, some examples which verify the efficiency of the given method are supplied and compared with other methods.

MSC:

65R20 Numerical methods for integral equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A25 Rate of convergence, degree of approximation
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[1] O. R. Navid and E. Tohidi,The spectral method for solving systems of Volterra integral equations, J. Appl. Math. Comput.,40(2012), 477-497. · Zbl 1295.65128
[2] S. Nemati,Numerical solution of Volterra-Fredholm integral equations using Legendre collocation method, J. Comput. Appl. Math.,278(2015), 29-36. · Zbl 1304.65275
[3] S. Noeiaghdam, D. N. Sidorov, V. S. Sizikov, and N. A. Sidoro,Control of accuracy of Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the cestac method, Appl. Comput. Math.,19(1) (2020), 87-105. · Zbl 1463.65432
[4] E. L. Ortiz and H. Samara,Numerical solutions of differential eigen values problems with an operational approach to the tau method, Computing,31(163) (1983), 95-103. · Zbl 0508.65045
[5] E. D. Rainville,Special functions, Chelsea, New York, (1960). · Zbl 0092.06503
[6] S. S. Ray and R. K. Bera,Solution of an extraordinary differential equation by a domain decomposition method, J. Appl. Math.,2004(4) (2004), 331-338. · Zbl 1080.65069
[7] N. T. Shawagfeh,Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput.,131(2-3) (2002), 517-529. · Zbl 1029.34003
[8] N. Sweilam, A. M. Nagy, and A. A. El-Sayed,Sinc-Chebyshev collocation method for time-fractional order telegraph equation, Appl. Comput. Math.,19(2) (2020), 162-174. · Zbl 1480.65295
[9] G. SzegŐ,Orthogonal polynomials, A. M. S,23(1939).
[10] T. Tang, X. Xu, and J. Cheng,On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math.,26(6) (2008), 825-837. · Zbl 1174.65058
[11] K. Y. Wang and Q. S. Wang,Lagrange collocation method for solving Volterra-Fredholm integral equations, Appl. Math. Comput.,219(21) (2013), 10434-10440. · Zbl 1304.65280
[12] K. Y. Wang and Q. S. Wang,Taylor collocation method and convergence analysis for the Volterra-Fredholm integral equations, J. Comput. Appl. Math.,260(2014), 294-300. · Zbl 1293.65174
[13] K. Yasir, F. Naeem, Y. Ahmet, and W. Qingbiao,Fractional variational iteration method for fractional initialboundary value problems arising in the application of nonlinear science, Computers and Mathematics with applications,62(5) (2011), 2273-2278. · Zbl 1231.35288
[14] Y. H. Youssri and W. M. Abd-Elhameed,Numerical spectral Legendre-Galerkin algorithm for solving time fractional Telegraph equation, Romanian Journal of Physics,63(3-4) (2018), 107.
[15] Y. H. Youssri and R. M. Hafez,Chebyshev collocation treatment of Volterra-Fredholm integral equation with error analysis, Arab. J. Math.,9(2020), 471-480. · Zbl 1441.65130
[16] Y. H. Youssri and R. M. Hafez,Exponential Jacobi spectral method for hyperbolic partial differential equations, Mathematical Sciences,13(4) (2019), 347-354. · Zbl 1452.35088
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