A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

Duan, Jun‐Sheng; Rach, Randolph

doi:10.1016/j.amc.2011.09.037

Cited by 163 publications

(112 citation statements)

References 52 publications

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“…Rach and Duan [46] presented the combined solution of the near-field and far-field approximations by the Adomian and asymptotic decomposition methods, where the Padé approximants technique was only used as necessary to calculate the mid-field approximation in order to bridge the near-and far-field approximations. In the ADM, the parametrized recursion scheme [22,36,37] was proposed in order to obtain decomposition solutions with large effective regions of convergence.…”

Section: Convergence Acceleration Multistage Decomposition and Numermentioning

confidence: 99%

“…Ghosh and Roy [57] considered the numeric-analytic form of the ADM for two-point boundary value problems (BVPs). The multistage ADM for BVPs including even Robin boundary conditions has been considered in [58], where a multistage ADM for BVPs has been developed through partitioning the domain into two, or more, subdomains, where a separate series was computed in each subdomain using the Duan-Rach modified recursion scheme for nonlinear BVPs [36]. The sub-solutions are combined to form the decomposition spline by applying the conditions of continuity at the interior boundary points in analogy to the multistage ADM for IVPs [49].…”

Section: Convergence Acceleration Multistage Decomposition and Numermentioning

confidence: 99%

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The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

Duan¹,

Rach²

2015

JMSS

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Abstract:In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Padé approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

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Section: Convergence Acceleration Multistage Decomposition and Numermentioning

confidence: 99%

Section: Convergence Acceleration Multistage Decomposition and Numermentioning

confidence: 99%

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

Duan¹,

Rach²

2015

JMSS

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“…(16) and (17) In the recent years, many authors have been applied the ADM [40][41][42] to various problems and demonstrated the efficiency of the ADM for handling non-linear and solving various chemistry and engineering problems. Using ADM (refer Appendix A in Supplementary information), we can obtain the concentration of methanol and α-pinene in the biofilm phase (See Appendix B in Supplementary information) as follows:…”

Section: Dimensionless Mass Balance Equation In the Biofilm Phasementioning

confidence: 99%

“…To the best of our knowledge, this is the first study reporting an exact analytical expression of concentrations of substrate in the biofilm phase and air phase. The purpose of this communication is to derive approximate analytical expressions for the concentrations in both the phases using the Adomian decomposition method [38][39][40][41][42][43]. This analytical result will be useful for optimize the parameters on the performance of the biofilters in removing hydrophobic compounds such as α-pinene.…”

Section: Introductionmentioning

confidence: 99%

Analytical Expression of Concentration Profiles of Methanol and Pinene in Air Stream and Bio-Film Phase

L¹

2015

J Anal Bioanal Tech

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In this paper, mathematical models of biofilteration of mixtures of hydrophilic (methanol) and hydrophobic (α-pinene) volatile organic compounds (VOC's) are discussed. This model is based on the system of diffusion equations containing a nonlinear term related to the Michaelis-Menten kinetic of the enzymatic reaction. The model describes the mass transfer in air-biofilm interface and chemical oxidation in the air stream phase. An approximate analytical expression of concentration profiles of methanol and α-pinene in air stream and bio-film phase have been derived using the Adomian decomposition method for all possible values of parameters. Furthermore, in this work, the numerical simulation of the problem is also reported using the Matlab program to investigate the dynamics of the system. Graphical results are presented and discussed quantitatively to illustrate the solution. Good agreement between the analytical and numerical data is noted.

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“…To achieve this objective, flow governing equations are formulated, nondimensionalised and approximate solution of the dimensionless coupled nonlinear boundary-value problem are obtained by using a semi-analytical Adomian decomposition method [22,23]. The choice of the method is due to the fact that it does not require any linearisation, discretisation, use of initial guess or pertubation.…”

Section: Introductionmentioning

confidence: 99%

Entropy Generation Rate in Unsteady Buoyancy-Driven Hydromagnetic Couple Stress Fluid Flow Through a Porous Channel

Kareem¹,

Adesanya²,

Falade³

et al. 2017

Int. J. of Pure and Appl. Math.

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In this paper, the entropy generation rate in unsteady buoyancy-driven hydromagnetic couple stress fluid flow through a porous channel has been investigated. The partial differential equations are converted into their corresponding dimensionless equivalence, including the prescribed initial boundary conditions. These equations were solved using the Adomian decomposition method and the behaviour of some pertinent fluid variables, such as velocity, temperature, entropy generation rate and the irreversibility ratio were examined and discussed for different parameters of interest, which include, the Grashof number, the Hartmann's number, the Reynolds number, the Brinkman number and the couple stress pa- Graphs are shown to illustrate the findings.

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A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

Cited by 163 publications

References 52 publications

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

Analytical Expression of Concentration Profiles of Methanol and Pinene in Air Stream and Bio-Film Phase

Entropy Generation Rate in Unsteady Buoyancy-Driven Hydromagnetic Couple Stress Fluid Flow Through a Porous Channel

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