A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems

Roul, Pradip; Thula, Kiran

doi:10.1080/00207160.2017.1417592

Cited by 50 publications

(48 citation statements)

References 23 publications

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“…In this section, we consider four nonlinear numerical examples to demonstrate the applicability, efficiency, and accuracy of the proposed method and to check its rate of convergence. The results obtained by the proposed method are compared with those obtained by the SQBCM and FDM . If the exact solution is known, the maximum absolute error E n can be calculated as

E_{n} = \max_{0 \leq j \leq n} | y (x_{j}) - Y (x_{j}) |,

where y ( x j ) and Y ( x j ) represent the exact and OQBCM solutions at the j th mesh point, respectively.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…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%

“…Yousef used Legendre wavelets method to solve singular initial value problems of Lane‐Emden type. Roul and Thula have considered the application of the SQBCM for the numerical solution of problems and .…”

Section: Introductionmentioning

confidence: 99%

“…To demonstrate the applicabilty, efficieny, and accuracy of the new method, we consider four nonlinear examples, including two Bratu‐type problems and two Lane‐Emden–type problems. Comparison is made between the proposed OQBCM and SQBCM . It is shown that the suggested method yields a sixth‐order convergent approximation to the solution of the problem, which is two orders of magnitude higher compared with SQBCM and leads to optimal approximation to the second‐order BVP.…”

Section: Introductionmentioning

confidence: 99%

“…Comparison is made between the proposed OQBCM and SQBCM . It is shown that the suggested method yields a sixth‐order convergent approximation to the solution of the problem, which is two orders of magnitude higher compared with SQBCM and leads to optimal approximation to the second‐order BVP. Furthermore, our method has been compared with the FDM …”

Section: Introductionmentioning

confidence: 99%

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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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E_{n} = \max_{0 \leq j \leq n} | y (x_{j}) - Y (x_{j}) |,

where y ( x j ) and Y ( x j ) represent the exact and OQBCM solutions at the j th mesh point, respectively.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…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%

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

Self Cite

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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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A high‐order efficient technique and its convergence analysis for Bratu‐type and Lane‐Emden‐type problems

Roul

Goura

2022

Math Methods in App Sciences

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The present work deals with the construction of a high‐order efficient technique for solving one‐dimensional nonlinear Bratu‐type and Lane‐Emden‐type problems describing various physical, chemical and biological phenomena in applied science and engineering. Convergence analysis of the method is studied. Numerical experiments are carried out to demonstrate the efficiency, accuracy and performance of proposed method. The method is shown to have sixth order convergence. The computed results are compared with those obtained by other numerical techniques to justify the advantage of present scheme. It is shown that the proposed method is fast convergent and provides accurate solutions for Bratu‐type and Lane‐Emden‐type problems with low computational cost.

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A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems

Cited by 50 publications

References 23 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

A high‐order efficient technique and its convergence analysis for Bratu‐type and Lane‐Emden‐type problems

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