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Research Papers

Quenching Phenomenon of a Time-Fractional Kawarada Equation

[+] Author and Article Information
Yufeng Xu

Department of Applied Mathematics,
Central South University,
Changsha 410083, Hunan, China
e-mail: xuyufeng@csu.edu.cn

Zhibo Wang

School of Applied Mathematics,
Guangdong University of Technology,
Guangzhou 510006, Guangdong, China
e-mail: wzbmath@gdut.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 23, 2018; final manuscript received July 31, 2018; published online August 22, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(10), 101010 (Aug 22, 2018) (7 pages) Paper No: CND-18-1183; doi: 10.1115/1.4041085 History: Received April 23, 2018; Revised July 31, 2018

In this paper, we introduce a class of time-fractional diffusion model with singular source term. The derivative employed in this model is defined in the Caputo sense to fit the conventional initial condition. With assistance of corresponding linear fractional differential equation, we verify that the solution of such model may not be globally well-defined, and the dynamics of this model depends on the order of fractional derivative and the volume of spatial domain. In simulation, a finite difference scheme is implemented and interesting numerical solutions of model are illustrated graphically. Meanwhile, the positivity, monotonicity, and stability of the proposed scheme are proved. Numerical analysis and simulation coincide the theoretical studies of this new model.

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Figures

Grahic Jump Location
Fig. 1

The numerical solution for u(x, t) (left) and ut(x, T) (right), when α = 0.4, T = 1, L = 1

Grahic Jump Location
Fig. 2

The numerical solution for u(x, t) (left) and ut(x, T) (right), when α = 0.8, T = 1, L = 1

Grahic Jump Location
Fig. 3

The numerical solution for u(x, t) (left) and ut(x, Tq) (right), when α = 0.4, Tq ≈ 0.2860, L = 2

Grahic Jump Location
Fig. 4

The numerical solution for u(x, t) (left) and ut(x, Tq) (right), when α = 0.8, Tq ≈ 0.6460, L = 2

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