B-spline method for solving Bratu's problem

Çağlar, Hikmet; Çağlar, Nazan; Özer, Mehmet; Valaristos, Antonios; Anagnostopoulos, A. N.

doi:10.1080/00207160802545882

Cited by 92 publications

(47 citation statements)

References 16 publications

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“…Moreover, the basis functions , , as de�ned in (9), are not differentiable at the endpoints = and = . erefore, if the boundary conditions in (2) are not homogeneous, that is, …”

Section: E Sinc Basismentioning

confidence: 99%

“…For instance, Bratu problem is used in a variety of applications such as the solid fuel ignition in thermal combustion theory, the Chandrasekhar model of the expansion of the universe, chemical reaction theory, and radiative heat transfer, nanotechnology. For more details see [1][2][3][4][5][6][7][8][9]. In addition, omas-Fermi equation is used to calculate the electrostatic potential in the omas-Fermi atom model, see [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

“…Although, each nonlinear problem has its own properties, the numerical schemes used to approximate the solution were almost the same. Bratu problem has been treated numerically via several techniques such as weighted residual method [14], �nite difference method [1,6], Laplace transform decomposition method [7,15], Adomian decomposition method [8,9], homotopy analysis method [3], He's variational method [16], differential transformation method [17], and B-spline method [2]. omas-Fermi problem has also been numerically studied using variational approach [18,19], the -expansion method [20], the decomposition method [21,22], and the homotopy analysis method [11,23].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the many different numerical schemes mentioned above do not treat well the approximation of the derivative of the solution at the endpoints. In this work, we consider the SincGalerkin method with suitable modi�cations to investigate the numerical solution of problems of the form (1) subject to (2). e suggested modi�cations allow for an accurate numerical solution with accurate derivative at the endpoints.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems

Hajji

Al‐Mdallal

2013

Journal of Mathematics

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is paper presents a modi�ed �alerkin method based on sinc basis functions to numerically solve nonlinear boundary value problems. e modi�cations allow for the accurate approximation of the solution with accurate derivatives at the endpoints. e algorithm is applied to well-known problems: Bratu and omas-Fermi problems. Numerical results demonstrate the clear advantage of the suggested modi�cations in obtaining accurate numerical solutions as well as accurate derivatives at the endpoints.

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“…Moreover, the basis functions , , as de�ned in (9), are not differentiable at the endpoints = and = . erefore, if the boundary conditions in (2) are not homogeneous, that is, …”

Section: E Sinc Basismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems

Hajji

Al‐Mdallal

2013

Journal of Mathematics

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“…This method has been implemented with success in a nonlinear BVP [14]. In this work, third degree B-spline functions have also been successfully implemented to Troesch's problem.…”

Section: Introductionmentioning

confidence: 99%

B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

Çağlar¹,

Çağlar²,

Özer

2014

Acta Phys. Pol. A

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A B-spline method is presented for solving the Troesch problem. The numerical approximations to the solution are calculated and then their behavior is studied and commenced. The chaotic dynamics exhibited by the solutions of Troesch's problem as they were derived by the decomposition method approximation are examined and an approximate critical value for the parameter λ is introduced also in this study. For the parameter value slightly less than λ ≈ 2.2, the solutions begin to show successive bifurcations, nally entering chaotic regimes at higher λ values. The eectiveness and accuracy of the B-spline method is veried for dierent values of the parameter, below its critical value, where the rst bifurcation occurs.

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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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B-spline method for solving Bratu's problem

Cited by 92 publications

References 16 publications

Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems

Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems

B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

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

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