Numerical solution of Volterra-Fredholm integral equations using Legendre collocation method. (English) Zbl 1304.65275
Summary: A numerical method for solving the Volterra-Fredholm integral equations is presented. The method is based upon shifted Legendre polynomials approximation. The properties of shifted Legendre polynomials are first presented. These properties together with the shifted Gauss-Legendre nodes are then utilized to reduce the Volterra-Fredholm integral equations to the solution of a matrix equation. An estimation of the error is given. Illustrative examples are included to demonstrate the validity and applicability of the technique.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45B05 | Fredholm integral equations |
| 45D05 | Volterra integral equations |
Keywords:
Volterra-Fredholm integral equations; shifted Legendre polynomials; Gauss-Legendre nodes; matrix equation; numerical examplesReferences:
| [1] | Bloom, F., Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory, J. Math. Anal. Appl., 73, 524-542 (1980) · Zbl 0434.45018 |
| [2] | Abdou, M. A., Fredholm-Volterra integral equation of the first kind and contact problem, Appl. Math. Comput., 125, 177-193 (2002) · Zbl 1028.45003 |
| [3] | Isaacson, S. A.; Kirby, R. M., Numerical solution of linear Volterra integral equations of the second kind with sharp gradients, J. Comput. Appl. Math., 235, 4283-4301 (2011) · Zbl 1219.65162 |
| [4] | Frankel, J., A Galerkin solution to regularized Cauchy singular integro-differential equations, Quart. Appl. Math., 24, 145-258 (1995) · Zbl 0823.65145 |
| [5] | Brunner, H., On the numerical solution of Volterra-Fredholm integral equation by collocation methods, SIAM J. Numer. Anal., 27, 4, 87-96 (1990) |
| [6] | Maleknejad, K.; Almasieh, H.; Roodaki, M., Triangular functions (TFs) method for the solution of nonlinear Volterra-Fredholm integral equations, Commun. Nonlinear Sci. Numer. Simul., 15, 3293-3298 (2010) · Zbl 1222.65147 |
| [7] | Ordokhani, Y., Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rationalized Harr functions, Appl. Math. Comput., 180, 436-443 (2006) · Zbl 1102.65141 |
| [8] | Ordokhani, Y.; Razzaghi, M., Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Harr functions, Appl. Math. Lett., 21, 4-9 (2008) · Zbl 1133.65117 |
| [9] | Yalcinbas, S., Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 127, 195-206 (2002) · Zbl 1025.45003 |
| [10] | Maleknejad, K.; Mahmoudi, Y., Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput., 145, 641-653 (2003) · Zbl 1032.65144 |
| [11] | Yousei, S.; Razzaghi, M., Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simulation, 70, 419-428 (2005) · Zbl 1205.65342 |
| [12] | Wang, K. Y.; Wang, Q. S., Lagrange collocation method for solving Volterra-Fredholm integral equations, Appl. Math. Comput., 219, 10434-10440 (2013) · Zbl 1304.65280 |
| [13] | Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093 |
| [14] | Dickman, O., Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6, 2, 109-130 (1978) · Zbl 0415.92020 |
| [15] | Guoqiang, H.; Liqing, Z., Asymptotic expansion for the trapezoidal Nystrom method of linear Volterra-Fredholm equation, J. Comput. Appl. Math., 51, 3, 339-348 (1994) · Zbl 0822.65123 |
| [16] | Hacia, L., On approximate solution for integral equation of mixed type, ZAMM Z. Angew. Math. Mech., 76, 415-428 (1996) · Zbl 0900.65389 |
| [17] | Wang, K.; Wang, Q., Taylor collocation method and convergence analysis for the Volterra-Fredholm integral equations, J. Comput. Appl. Math., 260, 294-300 (2014) · Zbl 1293.65174 |
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.
