Hybrid Bernstein block-pulse functions method for second kind integral equations with convergence analysis. (English) Zbl 1474.65487
Summary: We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval \([0, 1]\). These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45L05 | Theoretical approximation of solutions to integral equations |
| 45B05 | Fredholm integral equations |
References:
| [1] | Babolian, E.; Masouri, Z., Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Journal of Computational and Applied Mathematics, 220, 1-2, 51-57 (2008) · Zbl 1146.65082 · doi:10.1016/j.cam.2007.07.029 |
| [2] | Maleknejad, K.; Rahimi, B., Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind, Communications in Nonlinear Science and Numerical Simulation, 16, 6, 2469-2477 (2011) · Zbl 1221.65338 · doi:10.1016/j.cnsns.2010.09.032 |
| [3] | Mirzaee, F.; Piroozfar, S., Numerical solution of the linear two-dimensional Fredholm integral equations of the second kind via two-dimensional triangular orthogonal functions, Journal of King Saud University - Science, 22, 4, 185-193 (2010) · doi:10.1016/j.jksus.2010.04.007 |
| [4] | Ordokhani, Y., Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rationalized Haar functions, Applied Mathematics and Computation, 180, 2, 436-443 (2006) · Zbl 1102.65141 · doi:10.1016/j.amc.2005.12.034 |
| [5] | Marzban, H. R.; Tabrizidooz, H. R.; Razzaghi, M., A composite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1186-1194 (2011) · Zbl 1221.65340 · doi:10.1016/j.cnsns.2010.06.013 |
| [6] | Tavassoli Kajani, M.; Hadi Vencheh, A., Solving second kind integral equations with hybrid Chebyshev and block-pulse functions, Applied Mathematics and Computation, 163, 1, 71-77 (2005) · Zbl 1067.65151 · doi:10.1016/j.amc.2003.11.044 |
| [7] | Wang, X. T.; Li, Y. M., Numerical solutions of integrodifferential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials, Applied Mathematics and Computation, 209, 2, 266-272 (2009) · Zbl 1161.65099 · doi:10.1016/j.amc.2008.12.044 |
| [8] | Maleknejad, K.; Mahmoudi, Y., Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions, Applied Mathematics and Computation, 149, 3, 799-806 (2004) · Zbl 1038.65147 · doi:10.1016/S0096-3003(03)00180-2 |
| [9] | Asady, B.; Tavassoli Kajani, M.; Hadi Vencheh, A.; Heydari, A., Solving second kind integral equations with hybrid Fourier and block-pulse functions, Applied Mathematics and Computation, 160, 2, 517-522 (2005) · Zbl 1063.65144 · doi:10.1016/j.amc.2003.11.038 |
| [10] | Prasada Rao, G., Piecewise Constant Orthogonal Functions and their Application to Systems and Control (1983), Berlin, Germany: Springer, Berlin, Germany · Zbl 0518.93003 |
| [11] | Mohan, B. M.; Datta, K. B., Orthogonal Functions in Systems and Control (1995) · Zbl 0819.93036 |
| [12] | Doha, E. H.; Bhrawy, A. H.; Saker, M. A., Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations, Applied Mathematics Letters, 24, 4, 559-565 (2011) · Zbl 1236.65091 · doi:10.1016/j.aml.2010.11.013 |
| [13] | Mandal, B. N.; Bhattacharya, S., Numerical solution of some classes of integral equations using Bernstein polynomials, Applied Mathematics and Computation, 190, 2, 1707-1716 (2007) · Zbl 1122.65416 · doi:10.1016/j.amc.2007.02.058 |
| [14] | Yousefi, S. A.; Behroozifar, M., Operational matrices of Bernstein polynomials and their applications, International Journal of Systems Science, 41, 6, 709-716 (2010) · Zbl 1195.65061 · doi:10.1080/00207720903154783 |
| [15] | Alipour, M.; Rostamy, D., Solving nonlinear fractional differential equations by Bernstein polynomials operational matrices, The Journal of Mathematics and Computer Science, 5, 3, 185-196 (2012) |
| [16] | Rostamy, D.; Alipour, M.; Jafari, H.; Baleanu, D., Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis, Romanian Reports in Physics, 65, 2, 334-349 (2013) |
| [17] | Baleanu, D.; Alipour, M.; Jafari, H., The Bernstein operational matrices for solving the fractional quadratic Riccati differential equations with the Riemann-Liouville derivative, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.65241 · doi:10.1155/2013/461970 |
| [18] | Alipour, M.; Baleanu, D., Approximate analytical solution for nonlinear system of fractional differential equations by BPs operational matrices, Advances in Mathematical Physics, 2013 (2013) · Zbl 1273.34004 · doi:10.1155/2013/954015 |
| [19] | Alipour, M.; Rostamy, D.; Baleanu, D., Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices, Journal of Vibration and Control, 19, 16, 2523-2540 (2013) · Zbl 1358.93097 · doi:10.1177/1077546312458308 |
| [20] | Alipour, M.; Rostamy, D., BPs operational matrices for solving time varying fractional optimal control problems, The Journal of Mathematics and Computer Science, 6, 292-304 (2013) |
| [21] | Atkinson, K.; Han, W., Theoretical Numerical Analysis: A Functional Analysis Framework (2000), Springer |
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.
