Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method

Abstract We use Adomian decomposition method for solving the fractional nonlinear two-point boundary value problem D α u ( x ) + μ F ( x , u ( x ) ) = 0 , 0 x 1 , 1 α ⩽ 2 , u ( 0 ) = 0 , u ( 1 ) = c , where D α is Caputo fractional derivative, c is a constant, μ  > 0, and F : [ 0 , 1 ] × [ 0 , ∞ ) → [ 0 , ∞ ) a continuous function. The fractional Bratu problem is solved as an illustrative example.

[1]  Abdul-Majid Wazwaz,et al.  A reliable modification of Adomian decomposition method , 1999, Appl. Math. Comput..

[2]  Abdul-Majid Wazwaz,et al.  A reliable algorithm for solving boundary value problems for higher-order integro-differential equations , 2001, Appl. Math. Comput..

[3]  A. Wazwaz.  The Modified Adomian Decomposition Method for Solving Linear and Nonlinear Boundary Value Problems of Tenth-order and Twelfth-order , 2000 .

[4]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

[5]  A. Wazwaz.  A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems , 2001 .

[6]  Nicola Bellomo,et al.  A comparison between Adomian's decomposition methods and perturbation techniques for nonlinear random differential equations , 1985 .

[7]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .

[8]  Abdul-Majid Wazwaz,et al.  Analytical solution for the time-dependent Emden-Fowler type of equations by Adomian decomposition method , 2005, Appl. Math. Comput..

[9]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[10]  I. Podlubny.  Fractional differential equations , 1998 .

[11]  Hossein Jafari,et al.  Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition , 2006, Appl. Math. Comput..

[12]  O. Agrawal.  Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain , 2002 .

[13]  Nabil T. Shawagfeh,et al.  Analytical approximate solutions for nonlinear fractional differential equations , 2002, Appl. Math. Comput..

[14]  Zhanbing Bai,et al.  Positive solutions for boundary value problem of nonlinear fractional differential equation , 2005 .

[15]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[16]  S. Khuri,et al.  An Algorithm for Solving Boundary Value Problems , 2000 .

[17]  Y. Cherruault,et al.  New ideas for proving convergence of decomposition methods , 1995 .

[18]  George Adomian,et al.  A new approach to boundary value equations and application to a generalization of Airy's equation , 1989 .

[19]  G. Adomian,et al.  Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition , 1993 .

[20]  Abdul-Majid Wazwaz,et al.  Adomian decomposition method for a reliable treatment of the Bratu-type equations , 2005, Appl. Math. Comput..