A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Jun-Sheng Duan ◽  
Randolph Rach ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Decomposition Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Nonlinear Boundary Value Problems ◽  
Adomian Decomposition ◽  
Nonlinear Boundary Value ◽  
Matching Equation

AbstractIn this paper, we present a reliable algorithm to calculate positive solutions of homogeneous nonlinear boundary value problems (BVPs). The algorithm converts the nonlinear BVP to an equivalent nonlinear Fredholm– Volterra integral equation.We employ the multistage Adomian decomposition method for BVPs on two or more subintervals of the domain of validity, and then solve the matching equation for the flux at the interior point, or interior points, to determine the solution. Several numerical examples are used to highlight the effectiveness of the proposed scheme to interpolate the interior values of the solution between boundary points. Furthermore we demonstrate two novel techniques to accelerate the rate of convergence of our decomposition series solutions by increasing the number of subintervals and adjusting the lengths of subintervals in the multistage Adomian decomposition method for BVPs.

scholarly journals Solution of Two-Point Linear and Nonlinear Boundary Value Problems with Neumann Boundary Conditions Using a New Modified Adomian Decomposition Method

2020 ◽  
pp. 49-60
Author(s):  
Justina Mulenga ◽  
Patrick Azere Phiri
Keyword(s):  
Boundary Value Problems ◽  
Decomposition Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Nonlinear Boundary Value Problems ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Nonlinear Boundary Value ◽  
Linear And Nonlinear

In this paper, we present the New Modified Adomian Decomposition Method which is a modification of the Modified Adomian Decomposition Method. The new method incorporates the inverse linear operator theorem into the modified Adomian decomposition method for the calculation of u0. Six linear and nonlinear boundary value problems with Neumann conditions are solved in order to test the method. The results show that the method is effective.

Dual solutions for nonlinear boundary value problems by the Adomian decomposition method

2016 ◽  
Vol 26 (8) ◽  
pp. 2393-2409 ◽  
Author(s):  
Abdul-Majid Wazwaz ◽  
Randolph Rach ◽  
Lazhar Bougoffa
Keyword(s):  
Boundary Value Problems ◽  
Decomposition Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Dual Solutions ◽  
Scientific Applications ◽  
Content Type ◽  
Adomian Decomposition ◽  
Nonlinear Boundary Value

Purpose The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions. Design/methodology/approach The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific applications. In this work, the authors seek to determine the relative merits of the ADM in the context of several important nonlinear boundary value models characterized by the existence of dual solutions. Findings The ADM is shown to readily solve specific nonlinear BVPs possessing more than one solution. The decomposition series solution of these models requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The authors show that the ADM solves these models for any analytic nonlinearity in a practical and straightforward manner. The conclusions are supported by several numerical examples arising in various scientific applications which admit dual solutions. Originality/value This paper presents an accurate work for solving nonlinear BVPs that possess dual solutions. The authors have demonstrated the widespread applicability of the ADM for solving various forms of these nonlinear equations.

Analytic Approximations to Nonlinear Boundary Value Problems Modeling Beam-Type Nano-Electromechanical Systems

2017 ◽  
Vol 72 (3) ◽  
pp. 201-206
Author(s):  
Li Zou ◽  
Songxin Liang ◽  
Yawei Li ◽  
David J. Jeffrey
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Parameter Method ◽  
Nonlinear Boundary Value Problems ◽  
Electromechanical Systems ◽  
Adomian Decomposition ◽  
Nonlinear Boundary Value ◽  
Beam Type ◽  
Two Parameter

AbstractNonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.

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