A new approach to boundary value equations and application to a generalization of Airy's equation

Adomian, G.; Elrod, M. K.; Rach, Randolph

doi:10.1016/0022-247x(89)90083-8

Cited by 13 publications

(4 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Adomian and his coworkers in many papers have solved nonlinear differential equations for a wide range of nonlinearities, including polynomial nonlinearities (Adomian and Rach, 1985a), negative power nonlinearities (Adomian and Rach, 1985b), composite nonlinearities (Adomian and Rach, 1986a, b), and even decimal power nonlinearities (Adomian and Rach, 1986c) as exhibited in the Lane‐Emden model of astrophysics (Adomian et al , 1995; Wazwaz, 2001), among other classes of strong nonlinearities. This methodology of Adomian essentially unifies the subjects of linear and nonlinear, deterministic and stochastic, ordinary and partial differential equations, initial value problems and boundary value problems (Adomian et al , 1989a, b; Adomian and Rach, 1993a, b), as well as systems of coupled equations into a single fundamental method. The Adomian decomposition method is elegant, powerful, and accurate.…”

Section: Discussionmentioning

confidence: 99%

A new definition of the Adomian polynomials

Rach¹

2008

Kybernetes

135

View full text Add to dashboard Cite

PurposeTo provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.Design/methodology/approachDevelops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.FindingsThe four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.Originality/valueThis paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

show abstract

Section: Discussionmentioning

confidence: 99%

A new definition of the Adomian polynomials

Rach¹

2008

Kybernetes

135

View full text Add to dashboard Cite

show abstract

“…Proof. Substituting (7) and (8) into (5) and equating the terms with the identical powers of , we have…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…These methods have been implemented in several boundary value problems with all types of boundary conditions. In the literature, nonlinear boundary value problems (BVP) have been studied extensively [4][5][6][7][8][9]. In the present paper, we obtain positive solutions of nonlinear fractional order BVP using the HPM.…”

Section: Introductionmentioning

confidence: 99%

Homotopy Perturbation Method to Obtain Positive Solutions of Nonlinear Boundary Value Problems of Fractional Order

Jafari

Bagherian

Moshokoa

2014

Abstract and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…Adomian et al solved a generalization of Airy's equation by decomposition method [9]. In the present communication we apply nonpolynomial spline functions to develop numerical method for obtaining the approximations to the solution of second order two point boundary value problem of the form…”

Section: Introductionmentioning

confidence: 99%

Solution of Boundary Value Problems by Approaching Spline Techniques

Kalyani

Rao

2013

International Journal of Engineering Mathematics

View full text Add to dashboard Cite

In the present work a nonpolynomial spline function is used to approximate the solution of the second order two point boundary value problems. The classes of numerical methods of second order, for a specific choice of parameters involved in nonpolynomial spline, have been developed. Numerical examples are presented to illustrate the applications of this method. The solutions of these examples are found at the nodal points with various step sizes and with various parameters (α, β). The absolute errors in each example are estimated, and the comparison of approximate values, exact values, and absolute errors of at the nodal points are shown graphically. Further, shown that nonpolynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.

show abstract

A new approach to boundary value equations and application to a generalization of Airy's equation

Cited by 13 publications

References 2 publications

A new definition of the Adomian polynomials

A new definition of the Adomian polynomials

Homotopy Perturbation Method to Obtain Positive Solutions of Nonlinear Boundary Value Problems of Fractional Order

Solution of Boundary Value Problems by Approaching Spline Techniques

Contact Info

Product

Resources

About