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Hybrid Legendre polynomials and block-pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations. (English) Zbl 1221.65333
Summary: This paper introduces an approach for obtaining the numerical solution of the nonlinear Volterra-Fredholm integro-differential (NVFID) equations using hybrid Legendre polynomials and block-pulse functions. These hybrid functions and their operational matrices are used for representing matrix form of these equations. The main characteristic of this approach is that it reduces NVFID equations to a system of algebraic equations, which greatly simplifying the problem. Numerical examples illustrate the validity and applicability of the proposed method.
MSC:
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
References:
[1]Delves, L. M.; Mohamed, J. L.: Computational methods for integral equations, (1985)
[2]Wazwaz, A. M.: A first course in integral equations, (1997) · Zbl 0924.45001
[3]Rahman, M.; Jackiewicz, Z.; Welfert, B. D.: Stochastic approximations of perturbed Fredholm Volterra integro-differential equation arising in mathematical neurosciences, Appl. math. Comput. 186, 1173-1182 (2007) · Zbl 1117.65014 · doi:10.1016/j.amc.2006.07.137
[4]Abdou, M. A.: On asymptotic methods for Fredholm–Volterra integral equation of the second kind in contact problems, J. comput. Appl. math. 154, 431-446 (2003) · Zbl 1019.45003 · doi:10.1016/S0377-0427(02)00862-2
[5]Abdou, M. A.: Integral equation of mixed type and integrals of orthogonal polynomials, J. comput. Appl. math. 138, 273-285 (2002) · Zbl 0994.45001 · doi:10.1016/S0377-0427(01)00377-6
[6]Cattani, C.: Shannon wavelets for the solution of integrodifferential equations, mathematical problems in engineering, Math. probl. Eng., 1-22 (2010) · Zbl 1191.65174 · doi:10.1155/2010/408418
[7]Cattani, C.; Kudreyko, A.: Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. math. Comput. 215, 4164-4171 (2010) · Zbl 1186.65160 · doi:10.1016/j.amc.2009.12.037
[8]Wazwaz, A. M.: The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Appl. math. Comput. 216, 1304-1309 (2010) · Zbl 1190.65199 · doi:10.1016/j.amc.2010.02.023
[9]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 · doi:10.1016/S0096-3003(03)00152-8
[10]Maleknejad, K.; Kajani, M. Tavassoli: Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. math. Comput. 159, 603-612 (2004) · Zbl 1063.65145 · doi:10.1016/j.amc.2003.10.046
[11]Maleknejad, K.; Shahrezaee, M.; Khatami, H.: Numerical solution of integral equations system of the second kind by block-pulse functions, Appl. math. Comput. 166, 15-24 (2005) · Zbl 1073.65149 · doi:10.1016/j.amc.2004.04.118
[12]Maleknejad, K.; Mahmoudi, Y.: Numerical solution of linear Fredholm integral equations by using hybrid Taylor and block-pulse functions, Appl. math. Comput. 149, 799-806 (2004) · Zbl 1038.65147 · doi:10.1016/S0096-3003(03)00180-2
[13]Hsiao, C. H.: Hybrid function method for solving Fredholm and Volterra integral equations of the second kind, J. comput. Appl. math. 230, 59-68 (2009) · Zbl 1167.65473 · doi:10.1016/j.cam.2008.10.060
[14]Marzban, H. R.; Razzaghi, M.: Hybrid functions approach for linearly constrained quadratic optimal control problems, Appl. math. Model. 27, 471-485 (2003) · Zbl 1020.49025 · doi:10.1016/S0307-904X(03)00050-7
[15]Marzban, H. R.; Razzaghi, M.: Numerical solution of the controlled Duffing oscillator by hybrid functions, Appl. math. Comput. 140, 179-190 (2003) · Zbl 1027.65085 · doi:10.1016/S0096-3003(02)00112-1
[16]Hsiao, C. H.: Numerical solutions of linear time-varying descriptor systems via hybrid functions, Appl. math. Comput. 216, 1363-1374 (2010) · Zbl 1189.65131 · doi:10.1016/j.amc.2010.03.004
[17]Maleknejad, K.; Kajani, M. Tavassoli: Solving integro-differential equation by using hybrid Legendre and block-pulse functions, Int. J. Appl. math. 11, No. 1, 67-76 (2002) · Zbl 1029.65147
[18]Chang, R. Y.; Wang, M. L.: Shifted Legendre direct method for variational problems, J. optim. Theory appl. 39, 299-307 (1983) · Zbl 0481.49004 · doi:10.1007/BF00934535
[19]Babolian, E.; Masouri, Z.; Hatamzadeh-Varmazyar, S.: Numerical solution of nonlinear Volterra–Fredholm integro-differential equations via direct method using triangular functions, Comput. math. Appl. 58, 239-247 (2009) · Zbl 1189.65306 · doi:10.1016/j.camwa.2009.03.087