Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions

A new straightforward approach for solving ordinary and partial second-order boundary value problems with Neumann boundary conditions is introduced in this research. This approach depends mainly on the Adomian decomposition method with a new definition of the differential operator and its inverse, which has been modified for Neumann boundary conditions. The effectiveness of the proposed approach is verified by several linear and nonlinear examples.

[1]  Jafar Biazar,et al.  A new approach to the solution of the prey and predator problem and comparison of the results with the Adomian method , 2005, Appl. Math. Comput..

[2]  G. Adomian,et al.  Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations , 1990 .

[3]  A. Wazwaz.  The numerical solution of fifth-order boundary value problemsby the decomposition method , 2001 .

[4]  L. Casasús,et al.  The Adomian decomposition method in turning point problems , 2005 .

[5]  I. Hashim.  Adomian decomposition method for solving BVPs for fourth-order integro-differential equations , 2006 .

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

[7]  Using the Adomian method for solving a class of boundary differential systems , 2004 .

[8]  Mehdi Dehghan.  Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications , 2004, Appl. Math. Comput..

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

[10]  Daniel Lesnic,et al.  Blow-up solutions obtained using the decomposition method , 2006 .

[11]  Nabil Shawagfeh,et al.  Analytic approximate solution for a nonlinear oscillator equation , 1996 .

[12]  Y. Cherruault,et al.  Decomposition methods: A new proof of convergence , 1993 .

[13]  Shaher Momani,et al.  Variational iteration method for solving nonlinear boundary value problems , 2006, Appl. Math. Comput..

[14]  Mohamed El-Gamel.  Comparison of the solutions obtained by Adomian decomposition and wavelet-Galerkin methods of boundary-value problems , 2007, Appl. Math. Comput..

[15]  Yves Cherruault,et al.  Application of the Adomian method for solving a class of boundary problems , 2004 .

[16]  Salah M. El-Sayed,et al.  A numerical method for solving Jaulent-Miodek equation , 2003 .

[17]  G. Adomian.  Solution of the Thomas-Fermi equation , 1998 .

[18]  Giuseppe Saccomandi,et al.  New results for convergence of Adomian's method applied to integral equations , 1992 .

[19]  Y. Cherruault,et al.  Solving a class of linear partial differential equations with Dirichlet‐boundary conditions by the Adomian method , 2004 .

[20]  G. Adomian,et al.  SOLUTIONS OF NONLINEAR P.D.E. , 1998 .

[21]  G. Adomian,et al.  A nonperturbative analytical solution of immune response with time-delays and possible generalization , 1996 .

[22]  I. Hashim,et al.  Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method , 2007 .

[23]  Y. Cherruault,et al.  On the solution of the non‐linear Korteweg–de Vries equation by the decomposition method , 2002 .

[24]  G. Adomian.  Explicit solutions of nonlinear partial differential equations , 1997 .

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

[26]  Waleed Al-Hayani,et al.  Adomian decomposition method with Green's function for sixth-order boundary value problems , 2011, Comput. Math. Appl..

[27]  Nasser Hassan Sweilam,et al.  Approximate solutions to the nonlinear vibrations of multiwalled carbon nanotubes using Adomian decomposition method , 2010, Appl. Math. Comput..

[28]  Y Cherruault,et al.  Further remarks on convergence of decomposition method. , 1995, International journal of bio-medical computing.

[29]  A. Aminataei,et al.  The comparison of the stability of Adomian decomposition method with numerical methods of equation solution , 2007, Appl. Math. Comput..

[30]  A. Wazwaz.  A NOTE ON USING ADOMIAN DECOMPOSITION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS , 2000 .

[31]  M. Hadizadeh,et al.  On the decomposition method to the heat equation with non‐linear and non‐local boundary conditions , 1998 .

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

[33]  Dogan Kaya,et al.  A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations , 2002, Math. Comput. Simul..

[34]  Damian Slota,et al.  Comparison of the Adomian decomposition method and the variational iteration method in solving the moving boundary problem , 2011, Comput. Math. Appl..

[35]  E. S. Ismail,et al.  Accuracy of the Adomian decomposition method applied to the Lorenz system , 2006 .

[36]  Abdul-Majid Wazwaz.  Equality of partial solutions in the decomposition method for partial differential equations , 1997, Int. J. Comput. Math..

[37]  Y. Cherruault.  Convergence of Adomian's method , 1989 .