Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. (English) Zbl 1335.45002

Summary: We first construct the second kind Chebyshev wavelet. Then we present a computational method based on the second kind Chebyshev wavelet for solving a class of nonlinear Fredholm integro-differential equations of fractional order. The second kind Chebyshev wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. The method is illustrated by applications and the results obtained are compared with the existing ones in open literature. Moreover, comparing the methodology with the known technique shows that the present approach is more efficient and more accurate.


45B05 Fredholm integral equations
34K07 Theoretical approximation of solutions to functional-differential equations
34K35 Control problems for functional-differential equations
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[1] He JH. Nonlinear oscillation with fractional derivative and its applications. In: International conference on vibrating engineering98. China: Dalian; 1998. p. 288-91.; He JH. Nonlinear oscillation with fractional derivative and its applications. In: International conference on vibrating engineering98. China: Dalian; 1998. p. 288-91.
[2] He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bull Sci Technol, 15, 2, 86-90 (1999)
[3] Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J Rheol, 27, 3, 201-210 (1983) · Zbl 0515.76012
[4] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and fractional calculus in continuum mechanics (1997), Springer, Verlag: Springer, Verlag New York), 291-348 · Zbl 0917.73004
[5] Mandelbrot, B., Some noises with \(1/f\) spectrum, a bridge between direct current and white noise, IEEE Trans Inform Theory, 13, 2, 289-298 (1967) · Zbl 0148.40507
[6] Rossikhin, Y. A.; Shitikova, M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl Mech Rev, 50, 15-67 (1997)
[7] Baillie, R. T., Long memory processes and fractional integration in econometrics, J Econometrics, 73, 5-59 (1996) · Zbl 0854.62099
[8] Panda, R.; Dash, M., Fractional generalized splines and signal processing, Signal Process, 86, 2340-2350 (2006) · Zbl 1172.65315
[9] Bohannan, G. W., Analog fractional order controller in temperature and motor control applications, J Vib Control, 14, 1487-1498 (2008)
[10] Chow, T. S., Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys Lett A, 342, 148-155 (2005)
[11] Momani, S. M., Local and global existence theorems on fractional integro-differential equations, J Fract Calc, 18, 81-86 (2000) · Zbl 0967.45004
[12] Boyadjiev, L.; Dobner, H. J.; KallaA, S. L., A fractional integro-differential equation of Volterra type, Math Comput Model, 28, 10, 103-113 (1998) · Zbl 0993.65153
[13] Momani, S.; Noor, M., Numerical methods for fourth order fractional integro-differential equations, Appl Math Comput, 182, 754-760 (2006) · Zbl 1107.65120
[14] Ray, S. S., Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Commun Nonlinear Sci Numer Simulat, 14, 129-306 (2009)
[15] Nawaz, Y., Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Comput Math Appl, 61, 8, 2330-2341 (2011) · Zbl 1219.65081
[16] Saeedi, H.; Mohseni Moghadam, M.; Mollahasani, N.; Chuev, G. N., A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Commun Nonlinear Sci Numer Simulat, 16, 3, 1154-1163 (2011) · Zbl 1221.65354
[17] Huang, L.; Li, X. F.; Zhao, Y. L.; Duan, X. Y., Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput Math Appl, 62, 3, 1127-1134 (2011) · Zbl 1228.65133
[18] Beylkin, G.; Coifman, R.; Rokhlin, V., Fast wavelet transform and numerical algorithms I, Commun Pur Appl Math, 44, 141-183 (1991) · Zbl 0722.65022
[19] Yousefi, S.; Banifatemi, A., Numerical solution of Fredholm integral equations by using CAS wavelets, Appl Math Comput, 183, 1, 458-463 (2006) · Zbl 1109.65121
[20] Yousefi, S.; Razzaghi, M., Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math Comput Simulat, 70, 1, 1-8 (2005) · Zbl 1205.65342
[21] Maleknejad, K.; Sohrabi, S., Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets, Appl Math Comput, 186, 1, 836-843 (2007) · Zbl 1119.65126
[22] Singh, V. K.; Singh, O. P.; Rajesh Pandey, K., Numerical evaluation of the Hankel transform by using linear Legendre multi-wavelets, Comput Phys Commun, 179, 6, 424-429 (2008) · Zbl 1197.65237
[23] Zhu, L.; Han, H. L., Solving Hammerstein integral equations by multi-wavelets Galerkin method, J Hainan Normal Univ (Nat Sci), 22, 2, 142-145 (2009) · Zbl 1212.65537
[24] Lakestani, M.; Saray, B. N.; Dehgha, M., Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets, J Comput Appl Math, 235, 11, 3291-3303 (2011) · Zbl 1216.65185
[25] Hsiao, C. H.; Wu, S. P., Numerical solution of time-varying functional differential equations via Haar wavelets, Appl Math Comput, 188, 1, 1049-1058 (2007) · Zbl 1118.65077
[26] Babolian, E.; Fattahzadeh, F., Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl Math Comput, 188, 1, 417-426 (2007) · Zbl 1117.65178
[27] Li, Y. L., Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun Nonlinear Sci Numer Simulat, 15, 9, 2284-2292 (2010) · Zbl 1222.65087
[28] Saadatmandia, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput Math Appl, 59, 1326-1336 (2010) · Zbl 1189.65151
[29] Wu, J. L., A wavelet operational method for solving fractional partial differential equations numerically, Appl Math Comput, 214, 1, 31-40 (2009) · Zbl 1169.65127
[30] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999), Academic Press: Academic Press New York · Zbl 0924.34008
[31] Rivlin, T. J., Chebyshev polynomials (1990), Wiley: Wiley New York · Zbl 0734.41029
[32] Arfken, G., Mathematical methods for physicists (1985), Academic Press: Academic Press Orlando, FL · Zbl 0135.42304
[33] Fan, Q. B., Wavelet analysis (2008), Wuhan University Press: Wuhan University Press Wuhan
[34] Zhu, L.; Wang, Y. X.; Fan, Q. B., Numerical computation method in solving integral equation by using the second Chebyshev wavelets, (The 2011 international conference on scientific computing (2011), Las Vegas: Las Vegas USA), 126-130
[35] Kilicman, A.; Al Zhour, Z. A.A., Kronecker operational matrices for fractional calculus and some applications, Appl Math Comput, 187, 1, 250-265 (2007) · Zbl 1123.65063
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