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Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. (English) Zbl 1102.65136
Summary: We use the Adomian decomposition method for solving the fractional nonlinear two-point boundary value problem \[ D^\alpha u(x)+\mu F\bigl (x,u(x)\bigr)=0,\quad 0<x<1,\;1<\alpha\leq 2, \] \[ u(0)=0,\;u(1)=c, \] where \(D^\alpha\) is Caputo fractional derivative, \(c\) is a constant \(\mu >0\), and \(F:[0,1]\times[0,\infty) \to[0,\infty)\) a continuous function. The fractional Bratu problem is solved as an illustrative example.

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
Full Text: DOI
[1] Abboui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. appl. math., 29, 7, 103-105, (1995) · Zbl 0832.47051
[2] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Boston, MA · Zbl 0802.65122
[3] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1988) · Zbl 0671.34053
[4] Adomian, G.; Rach, R., Analytic solution of nonlinear boundary-value problems in several dimensions, J. math. anal. appl., 174, 118-127, (1993) · Zbl 0796.35017
[5] Adomian, G.; Elrod, M.; Rach, R., New approach to boundary value equations and application to a generalization of airy’s equation, J. math. anal. appl., 140, 554-568, (1989) · Zbl 0678.65057
[6] Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dyn., 29, 145-155, (2002) · Zbl 1009.65085
[7] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. anal. appl., 311, 2, 495-505, (2005) · Zbl 1079.34048
[8] Bellomo, N.; Monaco, R., A comparison between adomian’s decomposition method and perturbation techniques for nonlinear random differential equations, J. math. anal. appl., 110, 495-502, (1985) · Zbl 0575.60064
[9] Deeba, E.; Khuri, S.; Xie, S., An algorithm for solving boundary value problems, J. comput. phys., 159, 2, 125-138, (2000) · Zbl 0959.65091
[10] Hon, Y.C., A decomposition method for the Thomas-Fermi equation, SEA bull. math., 20, 3, 55-58, (1996) · Zbl 0858.34017
[11] H. Jafari, V. Daftardar-Gejji, Solving linear and non-linear fractional diffusion and wave equations by Adomian decomposition, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.12.031. · Zbl 1102.65135
[12] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Physica A, 278, 107-125, (2000)
[13] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[14] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[15] Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. comput., 131, 517-529, (2002) · Zbl 1029.34003
[16] Wazwaz, A., Adomian decomposition method for a reliable treatment of the bratu-type equations, Appl. math. comput., 166, 3, 638-651, (2005) · Zbl 1073.65105
[17] Wazwaz, A., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput. math. appl., 41, 1237-1244, (2001) · Zbl 0983.65090
[18] Wazwaz, A.M., A reliable algorithm for solving boundary value problems for higher-order integro-differential equations, Appl. math. comput., 118, 2/3, 327-342, (2001) · Zbl 1023.65150
[19] Wazwaz, A.M., The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth-order and 12th-order, Int. J. nonlinear sci. numer. simul., 1, 17-24, (2000) · Zbl 0966.65058
[20] Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl. math. comput., 102, 77-86, (1999) · Zbl 0928.65083
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