Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

Gia, Quôc Thông Lê; McLean, William

doi:10.1007/s10444-013-9311-6

Cited by 13 publications

(2 citation statements)

References 18 publications

(28 reference statements)

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“…In this regard the researchers have done a large amount of valuable work. For example one can find the coupling of Laplace transform with boundary particle method in [46], with finite element method in [47,48], finite difference [36], with boundary element [49], with RBFs method on unit sphere in [50], with Kansa method in [51] and references therein. In this work we propose a local meshless method which is based on LT for the approximation of TFADE with ABC derivative given as…”

Section: Introductionmentioning

confidence: 99%

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

Kamran¹,

Ahmadian²,

Salahshour³

et al. 2021

Phys. Scr.

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In this article we aim to approximate linear time fractional advection diffusion equations (TFADE) with Atangana-Baleanu- Caputo(ABC) derivative using local meshless method and Laplace transformation(LT). The method comprises of three steps. In the first step the the time variable is eliminated using LT. In the second step the reduced problem is solved using local meshless method. In the third step the solution of TFADE with ABC derivative is retrieved from local meshless methods solution by representing it as Bromwich integral. We then approximate the integral using some suitable quadrature rule. The stability and convergence of the method are discussed. The local meshless method is utilized to overcome the ill-conditioning issue of the interpolation matrices in global meshless methods and to over come the shape parameters sensitivity. Also in comparison with time stepping methods the LT is employed and contour integration technique is utilized to deal with the ABC derivative, which circumvent the calculation of costly convolution integrals in the approximation of ABC derivative, and also avoids the effect of time step on the stability and accuracy. Some test problems are considered in one and two dimensions to validate the proposed numerical method. The two dimensional problem is solved in regular and irregular domains. The computational experiments confirms that this method is computationally efficient and highly accurate for such type of problems.

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Section: Introductionmentioning

confidence: 99%

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

Kamran¹,

Ahmadian²,

Salahshour³

et al. 2021

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…In Sheen et al, 12,13 when the Laplace transform method was used for time discretization, the finite element method (FEM) was also used for the discretization of space operators. The simplicity of the implementation of the radial basis function (RBF) method instead of the FEM caused Laplace transform to be combined with the RBFs for the parabolic equations on the sphere 17,18 . The Laplace transform and Fourier transforms were used to the analytical study of the distributed‐order time‐fractional (DOTF) subdiffusion 6 and DOTF diffusion‐wave equations 19 …”

Section: Introductionmentioning

confidence: 99%

Local discontinuous Galerkin method for distributed‐order time‐fractional diffusion‐wave equation: Application of Laplace transform

Mohammadi-Firouzjaei

Adibi

Dehghan

2020

Math Methods in App Sciences

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In this paper, the Laplace transform combined with the local discontinuous Galerkin method is used for distributed‐order time‐fractional diffusion‐wave equation. In this method, at first, we convert the equation to some time‐independent problems by Laplace transform. Then, we solve these stationary equations by the local discontinuous Galerkin method to discretize diffusion operators at the same time. Next, by using a numerical inversion of the Laplace transform, we find the solution of the original equation. One of the advantages of this procedure is its capability to be implemented in a parallel environment. It's another advantage is that the number of stationary problems that should be solved is much less than that is needed in time‐marching methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the method.

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Exploring oversampling in RBF least-squares collocation method of lines for surface diffusion

Chen,

Ling

2024

Numer Algor

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Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

Cited by 13 publications

References 18 publications

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

Local discontinuous Galerkin method for distributed‐order time‐fractional diffusion‐wave equation: Application of Laplace transform

Exploring oversampling in RBF least-squares collocation method of lines for surface diffusion

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