## Bernstein collocation method for solving nonlinear Fredholm-Volterra integrodifferential equations in the most general form.(English)Zbl 1437.65232

Summary: A collocation method based on the Bernstein polynomials defined on the interval $$[a, b]$$ is developed for approximate solutions of the Fredholm-Volterra integrodifferential equation (FVIDE) in the most general form. This method is reduced to linear FVIDE via the collocation points and quasilinearization technique. Some numerical examples are also given to demonstrate the applicability, accuracy, and efficiency of the proposed method.

### MSC:

 65R20 Numerical methods for integral equations 45B05 Fredholm integral equations 45D05 Volterra integral equations 45J05 Integro-ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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### References:

  Bellman, R. E.; Kalaba, R. E., Quasilinearization and Nonlinear Boundary Value Problems (1965), New York, NY. USA: Elsevier, New York, NY. USA  Stanley, E. L., Quasilinearization and Invariant Imbedding (1968), New York, NY, USA: Academic Press, New York, NY, USA  Agarwal, R. P.; Chow, Y. M., Iterative methods for a fourth order boundary value problem, Journal of Computational and Applied Mathematics, 10, 2, 203-217 (1984) · Zbl 0541.65055  Akyuz Dascıoglu, A.; Isler, N., Bernstein collocation method for solving nonlinear differential equations, Mathematical & Computational Applications, 18, 3, 293-300 (2013)  Charles, A.; Baird, J., Modified quasilinearization technique for the solution of boundary-value problems for ordinary differential equations, Journal of Optimization Theory and Applications, 3, 4, 227-242 (1969) · Zbl 0169.20001  Mandelzweig, V. B.; Tabakin, F., Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications, 141, 2, 268-281 (2001) · Zbl 0991.65065  Ramos, J. I., Piecewise quasilinearization techniques for singular boundary-value problems, Computer Physics Communications, 158, 1, 12-25 (2004) · Zbl 1196.65122  Ahmad, B.; Ali Khan, R.; Sivasundaram, S., Generalized quasilinearization method for nonlinear functional differential equations, Journal of Applied Mathematics and Stochastic Analysis, 16, 1, 33-43 (2003) · Zbl 1048.34105  Drici, Z.; McRae, F. A.; Devi, J. V., Quasilinearization for functional differential equations with retardation and anticipation, Nonlinear Analysis: Theory, Methods & Applications, 70, 4, 1763-1775 (2009) · Zbl 1162.34345  Caliò, F.; Munoz, F.; Marchetti, E., Direct and iterative methods for the numerical solution of mixed integral equations, Applied Mathematics and Computation, 216, 12, 3739-3746 (2010) · Zbl 1195.65223  Maleknejad, K.; Najafi, E., Numerical solution of nonlinear Volterra integral equations using the idea of quasilinearization, Communications in Nonlinear Science and Numerical Simulation, 16, 1, 93-100 (2011) · Zbl 1221.65336  Pandit, S. G., Quadratically converging iterative schemes for nonlinear volterra integral equations and an application, Journal of Applied Mathematics and Stochastic Analysis, 10, 2, 169-178 (1997) · Zbl 0881.45004  Ahmad, B., A quasilinearization method for a class of integro-differential equations with mixed nonlinearities, Nonlinear Analysis: Real World Applications, 7, 5, 997-1004 (2006) · Zbl 1111.45005  Ahmad, B.; Khan, R. A.; Sivasundaram, S., Generalized quasilinearization method for integro-differential equations, Nonlinear Studies, 8, 3, 331-341 (2001)  Wang, P.; Wu, Y.; Wiwatanapaphee, B., An extension of method of quasilinearization for integro-differential equations, International Journal of Pure and Applied Mathematics, 54, 1, 27-37 (2009)  Farouki, R. T.; Rajan, V. T., Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design, 5, 1, 1-26 (1988) · Zbl 0648.65007  Lorentz, G. G., Bernstein Polynomials (1986), New York, NY, USA: Chelsea, New York, NY, USA  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  Işık, O. R.; Sezer, M.; Güney, Z., A rational approximation based on Bernstein polynomials for high order initial and boundary values problems, Applied Mathematics and Computation, 217, 22, 9438-9450 (2011) · Zbl 1217.65150  Bhattacharya, S.; Mandal, B. N., Use of Bernstein polynomials in numerical solutions of Volterra integral equations, Applied Mathematical Sciences, 2, 33-36, 1773-1787 (2008)  Maleknejad, K.; Hashemizadeh, E.; Ezzati, R., A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation, Communications in Nonlinear Science and Numerical Simulation, 16, 2, 647-655 (2011) · Zbl 1221.65334  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  Shirin, A.; Islam, M. S., Numerical solutions of Fredholm integral equations using Bernstein polynomials, Journal of Scientific Research, 2, 2, 264-272 (2010)  Singh, V. K.; Pandey, R. K.; Singh, O. P., New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials, Applied Mathematical Sciences, 3, 5-8, 241-255 (2009)  Işık, O. R.; Sezer, M.; Güney, Z., Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Applied Mathematics and Computation, 217, 16, 7009-7020 (2011) · Zbl 1213.65153  Ordokhani, Y.; Davaei far, S., Application of the Bernstein polynomials for solving the nonlinear Fredholm integro-differential equations, Journal of Applied Mathematics & Bioinformatics, 1, 2, 13-31 (2011)  Pandya, B. M.; Joshi, D. C., Solution of a Volterra’s population model in a Bernstein polynomial basis, Applied Mathematical Sciences, 5, 69, 3403-3410 (2011)  Phillips, G. M., Interpolation and Approximation by Polynomials (2003), New York, NY, USA: Springer, New York, NY, USA  Marzban, H. R.; Hoseini, S. M., Solution of nonlinear Volterra-Fredholm integrodifferential equations via hybrid of block-pulse functions and Lagrange interpolating polynomials, Advances in Numerical Analysis, 2012 (2012) · Zbl 1268.65168  Berenguer, M. I.; Gámez, D.; López Linares, A. J., Fixed point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm—Volterra integro-differential equation, Journal of Computational and Applied Mathematics, 252, 52-61 (2013) · Zbl 1288.65183  Babolian, E.; Masouri, Z.; Hatamzadeh-Varmazyar, S., Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions, Computers & Mathematics with Applications, 58, 2, 239-247 (2009) · Zbl 1189.65306  Shidfar, A.; Molabahrami, A.; Babaei, A.; Yazdanian, A., A series solution of the nonlinear Volterra and Fredholm integro-differential equations, Communications in Nonlinear Science and Numerical Simulation, 15, 2, 205-215 (2010) · Zbl 1221.65343
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