The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

Singh, Randhir; Kumar, Jitendra

doi:10.1007/s12190-013-0699-4

Cited by 44 publications

(31 citation statements)

References 39 publications

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“…In the study of nonlinear phenomena in engineering and applied sciences, many mathematical models lead to singular boundary value problems (SBVP), see previous studies. [5][6][7][8][9][10][11][12][13][14][15][16] Several methods have been developed for solving (1) with q(x) = 1, g(x) or x and f(x, y, gy ′ ) = f(x, y). For example, finite difference methods, 17,18 spline methods, 19,20 finite element method, 21 collocation method, 22 variational iteration method, 23 homotopy perturbation method, [24][25][26][27] Adomian decomposition method (ADM), 28 and B-spline collocation method.…”

Section: Introductionmentioning

confidence: 99%

“…For example, finite difference methods, 17,18 spline methods, 19,20 finite element method, 21 collocation method, 22 variational iteration method, 23 homotopy perturbation method, [24][25][26][27] Adomian decomposition method (ADM), 28 and B-spline collocation method. 29,30 However, numerical analysis literature does not contain much on numerical solutions of DDSBVP described by (1) to (3). To date, there has been only one semianalytical technique developed for DDSBVP considered, the MADM developed very recently in previous studies.…”

Section: Introductionmentioning

confidence: 99%

“…To date, there has been only one semianalytical technique developed for DDSBVP considered, the MADM developed very recently in previous studies. 1,31 It is well known that the ADM yields the solution in the form of series. But, the convergence of series solution is not always guaranteed.…”

Section: Introductionmentioning

confidence: 99%

“…The effectiveness and accuracy of the proposed approach are evaluated by solving three SBVPs, out of which one is a real-life problem, which describes stress distribution on a rotationally shallow membrane cap. 1,32,33 Numerical results obtained by the present scheme are compared with that obtained by the MADM. 1 Comparison suggests that the proposed method requires lesser number of solution components in the solution series than the method used in the study of Kumar and Singh 1 to produce similar results.…”

Section: Introductionmentioning

confidence: 99%

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Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

Roul

2018

Math Methods in App Sciences

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In this paper, a fast-converging recursive scheme is presented to approximate the solution of a class of derivative dependent doubly singular boundary value problems (DDSBVP). First, the original problem is reformulated as an equivalent integral equation. The resulting integral equation is then efficiently tackled by an improved homotopy analysis method (IHAM). This method contains a parameter, which greatly accelerates the convergence of the series solution. The convergence of the method is carried out. To illustrate the efficiency and accuracy of the proposed recursive approach, we consider three nonlinear examples, including one physical model problem, which describes stress distribution on a rotationally shallow membrane cap. Results show that our method excels over the existing methods.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

Roul

2018

Math Methods in App Sciences

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“…It is well known that the ADM allows us to solve nonlinear BVPs without restrictive assumptions such as linearization, discretization and perturbation. Many researchers [14][15][16][17][18][19][20][21][22][23] have shown interest to study the ADM for different scientific models. According to the ADM, we rewrite the problem (1) in an operator form…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of fourth-order Volterra integro-differential equations by the Green’s function and decomposition method

Singh

Wazwaz

2016

Math Sci

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We propose a reliable technique based on Adomian decomposition method (ADM) for the numerical solution of fourth-order boundary value problems for Volterra integro-differential equations. We use Green's function technique to convert boundary value problem into the integral equation before establishing the recursive scheme for the solution components of a specific solution. The advantage of the proposed technique over the standard ADM or modified ADM is that it provides not only better numerical results but also avoids solving a sequence of transcendental equations for unknown constant. Approximations of the solutions are obtained in the form of series. Convergence and error analysis is also discussed. The accuracy and generality of the proposed scheme are demonstrated by solving some numerical examples.

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Existence and nonexistence results for a class of fourth‐order coupled singular boundary value problems arising in the theory of epitaxial growth

Verma

Pandit

Agarwal

2020

Math Methods in App Sciences

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In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear, non‐self‐adjoint, and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled system. To prove the existence of the coupled solutions, we apply the monotone iterative technique on equivalent coupled integral equations in the presence of upper and lower solutions. We also compute the bounds for the parameters, which indicates the existence of coupled solutions for small values of the parameters, and nonexistence for large positive values of these parameters. To illustrate the theory developed, we consider test cases and compute the sequence of upper and lower solutions. To verify the bounds on the parameters, we have used a type of Adomian decomposition method.

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The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

Cited by 44 publications

References 39 publications

Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

Numerical solutions of fourth-order Volterra integro-differential equations by the Green’s function and decomposition method

Existence and nonexistence results for a class of fourth‐order coupled singular boundary value problems arising in the theory of epitaxial growth

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