An algorithm based on the variational iteration technique for the Bratu-type and the Lane–Emden problems

Das, Nilima; Singh, Randhir; Wazwaz, Abdul‐Majid; Kumar, Jitendra

doi:10.1007/s10910-015-0575-6

Cited by 62 publications

(35 citation statements)

References 27 publications

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“…It is well known that Bratu's boundary value problem (BVP) is of the following form:

y^{''} false(x false) + λ N false(y false) = 0, 0 < x < 1, λ > 0,

y false(0 false) = 0, y false(1 false) = 0,

where N ( y ) = e − y or e y . This problem arises in the study of various physical and chemical models in applied science and engineering such as the fuel ignition model of the thermal combustion theory, chemical reaction theory, thermal reaction process, Chandrasekhar model of the expansion of the universe, thermo‐electro‐hydrodynamic model for electrospinning process for the manufacturing of the nanofibers, radiative heat transfer, and nanotechnology. The exact solution of with N ( y ) = e y is given by

y (x) = - 2 \ln (\frac{\cosh ((x - \frac{1}{2}) \frac{θ}{2})}{\cosh (\frac{θ}{4})}),

where θ is a solution of

θ - \sqrt{2 λ} \cosh (\frac{θ}{4}) = 0 .

The Lane‐Emden BVP is of the form

y^{''} + \frac{α}{x} y^{'} + f false(x, y false) = 0, α > 0,

y^{'} false(0 false) = 0, y false(1 false) = c,

where c is a finite constant.…”

Section: Introductionmentioning

confidence: 99%

“…where N(y) = e −y or e y . This problem arises in the study of various physical and chemical models in applied science and engineering [1][2][3][4][5][6][7][8] such as the fuel ignition model of the thermal combustion theory, chemical reaction theory, thermal reaction process, Chandrasekhar model of the expansion of the universe, thermo-electro-hydrodynamic model for electrospinning process for the manufacturing of the nanofibers, radiative heat transfer, and nanotechnology. The exact solution of (1) with N(y) = e y is given by 9 (x) = −2 ln…”

Section: Introductionmentioning

confidence: 99%

“…Wazwaz 8 considered the application of Adomian decomposition method (ADM) for solving the Bratu-type problems (1) and (2). Das et al 4 proposed a modified version of the variational iteration method to obtain series solution of problems (1) and (2). Moreover, Roul and Madduri 13 used an optimal homotopy analysis method (OHAM) to solve Bratu-type problem.…”

Section: Introductionmentioning

confidence: 99%

“…Kumar and Singh 17 presented a modified ADM for solving problems (3) and (4). Chawla et al 18 (3) and (4). The convergence order of this method is four only, which is a nonoptimal approximation to the solution of second-order BVP.…”

Section: Introductionmentioning

confidence: 99%

“…18 The rest of this paper is structured as follows. In Section 3, the OQBCM is developed for solving Bratu problems (1) and (2) or Lane-Emden problems (3) and (4). Section 3 deals with four nonlinear numerical examples that demonstrate the applicability and accuracy of the proposed method.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

Roul

Thula

Goura

2019

Math Methods in App Sciences

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This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.

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“…It is well known that Bratu's boundary value problem (BVP) is of the following form:

y^{''} false(x false) + λ N false(y false) = 0, 0 < x < 1, λ > 0,

y false(0 false) = 0, y false(1 false) = 0,

y (x) = - 2 \ln (\frac{\cosh ((x - \frac{1}{2}) \frac{θ}{2})}{\cosh (\frac{θ}{4})}),

where θ is a solution of

θ - \sqrt{2 λ} \cosh (\frac{θ}{4}) = 0 .

The Lane‐Emden BVP is of the form

y^{''} + \frac{α}{x} y^{'} + f false(x, y false) = 0, α > 0,

y^{'} false(0 false) = 0, y false(1 false) = c,

where c is a finite constant.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

Roul

Thula

Goura

2019

Math Methods in App Sciences

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Solving a class of local and nonlocal elliptic boundary value problems arising in heat transfer

Singh¹

2021

Heat Trans

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This paper deals with a class of Bratu's type, Troesch's, and nonlocal elliptic boundary value problems arising in the heat transfer process. Due to the strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems analytically as well as numerically. By using the Jacobi spectral collocation method, these problems are solved fruitfully. We have shown the numerical as well as theoretical convergence of the suggested scheme. Numerical results are presented through figures and tables, demonstrating the accuracy of the scheme. Results are compared with some known methods to highlight its neglectable error.

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Approximate Analytical Solution of the Bratu Boundary-Value Problem

Kot

2024

J Eng Phys Thermophy

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An algorithm based on the variational iteration technique for the Bratu-type and the Lane–Emden problems

Cited by 62 publications

References 27 publications

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

Solving a class of local and nonlocal elliptic boundary value problems arising in heat transfer

Approximate Analytical Solution of the Bratu Boundary-Value Problem

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