×

A reproducing kernel method for solving nonlocal fractional boundary value problems. (English) Zbl 1242.65144

Summary: We propose a reproducing kernel method for solving singular and nonsingular boundary value problems of integer order based on the reproducing kernel theory. In this letter, we expand the application of reproducing kernel theory to fractional differential equations and present an algorithm for solving nonlocal fractional boundary value problems. The results from numerical examples show that the present method is simple and effective.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integral and Derivatives, Theory and Applications (1993), Gordon and Breach: Gordon and Breach Switzerland · Zbl 0818.26003
[2] Kilbas, A. A.; Marzan, S., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. Equ., 1141, 84-89 (2005) · Zbl 1160.34301
[3] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[4] Lakshmikantham, V.; Vatsala, A. S., Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11, 395-402 (2007) · Zbl 1159.34006
[5] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal. TMA, 698, 2677-2683 (2008) · Zbl 1161.34001
[6] Ghorbani, A., Toward a new analytical method for solving nonlinear fractional differential equations, Comput. Methods Appl. Mech. Engrg., 197, 4173-4179 (2008) · Zbl 1194.65091
[7] Hu, Y. Z.; Luo, Y.; Lu, Z. Y., Analytical solution of the linear fractional differential equation by Adomian decomposition method, J. Comput. Appl. Math., 215, 220-229 (2008) · Zbl 1132.26313
[8] El-Sayed, A. M.A.; El-Kalla, I. L.; Ziada, E. A.A., Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations, Appl. Numer. Math., 60, 788-797 (2010) · Zbl 1192.65092
[9] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72, 916-924 (2010) · Zbl 1187.34026
[10] Salem, H. A.H., On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies, J. Comput. Appl. Math., 224, 565-572 (2009) · Zbl 1176.34070
[11] Zhong, W.; Lin, W., Nonlocal and multiple-point boundary value problem for fractional differential equations, Comput. Math. Appl., 59, 1345-1351 (2010) · Zbl 1189.34036
[12] Ur Rehman, M.; Khan, R. A., Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett., 23, 1038-1044 (2010) · Zbl 1214.34007
[13] Ahmad, B.; Sivasundaram, S., On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Appl. Math. Comput., 217, 480-487 (2010) · Zbl 1207.45014
[14] Cui, M. G.; Lin, Y. Z., Nonlinear Numerical Analysis in Reproducing Kernel Space (2009), Nova Science Pub. Inc. · Zbl 1165.65300
[15] Berlinet, A.; Thomas-Agnan, C., Reproducing Kernel Hilbert Space in Probability and Statistics (2004), Kluwer Academic Publishers · Zbl 1145.62002
[16] Geng, F. Z.; Cui, M. G., Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Appl. Math. Comput., 192, 389-398 (2007) · Zbl 1193.34017
[17] Geng, F. Z.; Cui, M. G., Solving a nonlinear system of second order boundary value problems, J. Math. Anal. Appl., 327, 1167-1181 (2007) · Zbl 1113.34009
[18] Geng, F. Z., Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. Math. Comput., 215, 2095-2102 (2009) · Zbl 1178.65085
[19] Li, X. Y.; Wu, B. Y., Reproducing kernel method for singular fourth order four-point boundary value problems, Bull. Malays. Math. Sci. Soc., 34, 147-151 (2011) · Zbl 1219.34032
[20] Bouboulis, P.; Mavroforakis, M., Reproducing kernel Hilbert Spaces and fractal interpolation, J. Comput. Appl. Math., 235, 3425-3434 (2011) · Zbl 1226.65009
[21] Mohammadi, M.; Mokhtari, R., Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math., 235, 4003-4014 (2011) · Zbl 1220.65143
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.