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Numerical solution of nonlinear fractional Volterra integro-differential equations via Bernoulli polynomials. (English) Zbl 1470.65145

Summary: This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34K37 Functional-differential equations with fractional derivatives
45J05 Integro-ordinary differential equations
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[1] El-Mesiry, A. E. M.; El-Sayed, A. M. A.; El-Saka, H. A. A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Applied Mathematics and Computation, 160, 3, 683-699 (2005) · Zbl 1062.65073
[2] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), New York, NY, USA: Elsevier, New York, NY, USA · Zbl 1092.45003
[3] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[4] Podlubny, I., Fractional Differential Equations (1999), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0918.34010
[5] El-Wakil, S. A.; Elhanbaly, A.; Abdou, M. A., Adomian decomposition method for solving fractional nonlinear differential equations, Applied Mathematics and Computation, 182, 1, 313-324 (2006) · Zbl 1106.65115
[6] Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Computers & Mathematics with Applications, 57, 3, 483-487 (2009) · Zbl 1165.35398
[7] Erturk, V. S.; Momani, S.; Odibat, Z., Application of generalized differential transform method to multi-order fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 13, 8, 1642-1654 (2008) · Zbl 1221.34022
[8] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56, 1, 80-90 (2006) · Zbl 1086.65087
[9] Odibat, Z. M.; Shawagfeh, N. T., Generalized Taylor’s formula, Applied Mathematics and Computation, 186, 1, 286-293 (2007) · Zbl 1122.26006
[10] Gülsu, M.; Gürbüz, B.; Öztürk, Y.; Sezer, M., Laguerre polynomial approach for solving linear delay difference equations, Applied Mathematics and Computation, 217, 15, 6765-6776 (2011) · Zbl 1211.65166
[11] Samadi, O. R. N.; Tohidi, E., The spectral method for solving systems of Volterra integral equations, Journal of Applied Mathematics and Computing, 40, 1-2, 477-497 (2012) · Zbl 1295.65128
[12] Tohidi, E.; Soleymani, F.; Kilicman, A., Robustness of operational matrices of differentiation for solving state-space analysis and optimal control problems, Abstract and Applied Analysis, 2013 (2013) · Zbl 1272.49066
[13] Toutounian, F.; Tohidi, E.; Kilicman, A., Fourier operational matrices of differentiation and transmission: introduction and applications, Abstract and Applied Analysis, 2013 (2013) · Zbl 1275.65036
[14] Yalçinbaş, S.; Aynigül, M.; Sezer, M., A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute, 348, 6, 1128-1139 (2011) · Zbl 1221.65187
[15] Yüzbaşı, S., A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential-difference equations, Computers & Mathematics with Applications, 64, 6, 1691-1705 (2012) · Zbl 1268.65090
[16] Tohidi, E.; Kılıçman, A., A collocation method based on the Bernoulli operational matrix for solving nonlinear BVPs which arise from the problems in calculus of variation, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.49043
[17] Tohidi, E.; Erfani, Kh.; Gachpazan, M.; Shateyi, S., A new Tau method for solving nonlinear Lane-Emden type equations via Bernoulli operational matrix of differentiation, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.65138
[18] Toutounian, F.; Tohidi, E.; Shateyi, S., A collocation method based on the Bernoulli operational matrix for solving high-order linear complex differential equations in a rectangular domain, Abstract and Applied Analysis, 2013 (2013) · Zbl 1275.65041
[19] Saadatmandi, A.; Dehghan, M., A Legendre collocation method for fractional integro-differential equations, Journal of Vibration and Control, 17, 13, 2050-2058 (2011) · Zbl 1271.65157
[20] Saeedi, H.; Moghadam, M. M., Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1216-1226 (2011) · Zbl 1221.65140
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