scholarly journals A new fractional derivative model for the anomalous diffusion problem

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 1005-1011
Author(s):  
Zhanqing Chen ◽  
Peitao Qiu ◽  
Xiao-Jun Yang ◽  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Heat Conduction Problem ◽  
Diffusion Problem ◽  
The Laplace Transform ◽  
Complex Phenomena ◽  
First Time

In this paper, a new fractional derivative within the exponential decay kernel is addressed for the first time. A new anomalous diffusion model is proposed to describe the heat-conduction problem. With the use of the Laplace transform, the analytical solution is discussed in detail. The presented result is as an accurate and efficient approach proposed for the heat-conduction problem in the complex phenomena.

Anomalous diffusion equation using a new general fractional derivative within the Miller–Ross kernel

2020 ◽  
Vol 34 (27) ◽  
pp. 2050289
Author(s):  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Fractional Integral ◽  
Liouville Type

In view of the generalization of Miller–Ross kernel in the sense of Riemann–Liouville type, we propose the new definitions of the general fractional integral (GFI) and general fractional derivative (GFD) to discuss the anomalous diffusion equation, which is distinct from those classic calculus operators. The obtained analytical solution of the application described in the graph is effective and accurate making the use of Laplace transform.

scholarly journals General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 11-18 ◽  
Author(s):  
Feng Gao
Keyword(s):  
Heat Transfer ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Power Law ◽  
Anomalous Diffusion ◽  
Fourier Transforms ◽  
Diffusion Phenomena ◽  
Complex Phenomena ◽  
Transfer Problems ◽  
First Time

In this paper we address the general fractional calculus of Liouville-Weyl and Liouville-Caputo general fractional derivative types with non-singular power-law kernel for the first time. The Fourier transforms and the anomalous diffusions are discussed in detail. The formulations are adopted to describe complex phenomena of the heat transfer problems.

scholarly journals The local fractional iteration solution for the diffusion problem in fractal media

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 743-746 ◽  
Author(s):  
Ya-Juan Hao ◽  
Ai-Min Yang
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Diffusion Problem ◽  
Coupling Method ◽  
Iteration Algorithm ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
Fractal Media ◽  
First Time

In this paper, we address the coupling method for the local fractional variational iteration algorithm III and local fractional Laplace transform for the first time, which is called as the local fractional Laplace transform variational iteration algorithm III. The proposed technology is used to find the local fractional iteration solution for the diffusion problem in fractal media via local fractional derivative.

scholarly journals A new computational method for fractal heat-diffusion via local fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 773-776
Author(s):  
Geng-Yuan Liu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Heat Conduction Problem ◽  
Computational Method ◽  
Heat Diffusion ◽  
Heat Diffusion Equation ◽  
Local Fractional Derivative

The fractal heat-conduction problem via local fractional derivative is investigated in this paper. The solution of the fractal heat-diffusion equation is obtained. The characteristic equation method is proposed to find the analytical solution of the partial differential equation in fractal heat-conduction problem.

scholarly journals Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem

2013 ◽  
Vol 17 (3) ◽  
pp. 715-721 ◽  
Author(s):  
Chun-Feng Liu ◽  
Shan-Shan Kong ◽  
Shu-Juan Yuan
Keyword(s):  
Heat Conduction ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Heat Conduction Equation ◽  
Heat Conduction Problem ◽  
Variational Iteration Method ◽  
High Accuracy ◽  
Variational Iteration

A reconstructive scheme for variational iteration method using the Yang-Laplace transform is proposed and developed with the Yang-Laplace transform. The identification of fractal Lagrange multiplier is investigated by the Yang-Laplace transform. The method is exemplified by a fractal heat conduction equation with local fractional derivative. The results developed are valid for a compact solution domain with high accuracy.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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