# zbMATH — the first resource for mathematics

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.

##### MSC:
 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations
Full Text:
##### References:
 [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
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.