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Bushnaq, Samia; Momani, Shaher; Zhou, Yong

A reproducing kernel Hilbert space method for solving integro-differential equations of fractional order. (English) Zbl 1273.65194

J. Optim. Theory Appl. 156, No. 1, 96-105 (2013).
The authors, using the notion of the fractional derivative of Caputo, construct an approximate method to solve the integro-differential equation of fractional order of the special form \[ D^{\alpha}u(x)= F(x, u(x), Tu(x)),\qquad m-1<\alpha\leq m,\qquad 0\leq x\leq 1, \] with \( Tu(x)=\int^{x}_{0}h(x,t)u(t)dt.\) The solution \( u(x)\) is presented as convergent power series. The algorithm of the numerical computations is not strictly formulated.
Reviewer: Ivan Secrieru (Chişinău)

Cited in 1 Review
Cited in 9 Documents

MSC:

65R20 Numerical methods for integral equations
26A33 Fractional derivatives and integrals
45G10 Other nonlinear integral equations
45J05 Integro-ordinary differential equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Keywords:

reproducing kernel Hilbert space method; iterative method; nonlinear integro-differential equations of fractional order; algorithm
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\textit{S. Bushnaq} et al., J. Optim. Theory Appl. 156, No. 1, 96--105 (2013; Zbl 1273.65194)
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References:

[1] Leszczynski, J.S.: An Introduction to Fractional Mechanics. Czestochowa University of Technology, Czestochowa (2011)
[2] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997) · Zbl 0917.73004
[3] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13(5), 529–539 (1967)
[4] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[5] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[6] Momani, S.: Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl. Math. Comput. 165(2), 459–472 (2005) · Zbl 1070.65105
[7] Momani, S.: Analytic and approximate solutions of the space- and time-fractional telegraph equations. Appl. Math. Comput. 170, 1126–1134 (2005) · Zbl 1103.65335
[8] Momani, S., Odibat, Z., Alawneh, A.: Variational iteration method for solving the space- and time-fractional KdV equation. Numer. Methods Partial Differ. Equ. 24(1), 262–271 (2008) · Zbl 1130.65132
[9] Odibat, Z., Momani, S.: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput. Math. Appl. 58, 2199–2208 (2009) · Zbl 1189.65254
[10] Odibat, Z., Momani, S., Ertürk, V.S.: Generalized differential transform method: application to differential equations of fractional order. Appl. Math. Comput. 197, 467–477 (2008) · Zbl 1141.65092
[11] Odibat, Z., Momani, S.: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula. J. Comput. Appl. Math. 220, 85–95 (2008) · Zbl 1148.65099
[12] Radwan, A.G., Moaddy, K., Momani, S.: Stability and nonstandard finite difference method of the generalized Chua’s circuit. Comput. Math. Appl. (in press) · Zbl 1228.65120
[13] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) · Zbl 1014.34003
[14] Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 26, 31–52 (2004) · Zbl 1055.65098
[15] Zurigat, M., Momani, S., Odibat, Z., Alawneh, A.: The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Model. 34, 24–35 (2010) · Zbl 1185.65140
[16] Odibat, Z., Momani, S., Xu, H.: A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. Appl. Math. Model. 34, 593–600 (2010) · Zbl 1185.65139
[17] Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 674–684 (2009) · Zbl 1221.65277
[18] Cui, M.G., Lin, Y.Z.: Nonlinear Numerical Analysis in Reproducing Kernel Hilbert Space. Nova Science, New York (2009) · Zbl 1165.65300
[19] Zhou, Y.F., Lin, Y.Z., Cui, M.G.: A computational method for nonlinear 2m-th order boundary value problems. Math. Model. Anal. 15, 571–586 (2010) · Zbl 1220.34017
[20] Chen, Z., Lin, Y.Z.: The exact solution of a linear integral equation with weakly singular kernel. J. Math. Anal. Appl. 344, 726–734 (2008) · Zbl 1144.45002
[21] Lin, Y.Z., Zhou, Y.F.: Solving nonlinear pseudoparabolic equations with nonlocal boundary conditions in reproducing kernel space. Numer. Algorithms 52, 173–186 (2009) · Zbl 1180.65129
[22] Cui, M.G., Li, Ch.L.: The exact solution for solving a class nonlinear operator equations in the reproducing kernel space. Appl. Math. Comput. 143, 393–399 (2003) · Zbl 1034.47030
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.