Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation

Povstenko, Yuriy

doi:10.3390/e17064028

Cited by 21 publications

(16 citation statements)

References 56 publications

(78 reference statements)

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“…Entropy was presented in thermodynamics by Clausius and Boltzmann. These advances activated the formulation of novel entropy indices and fractional operators allowing their implementation in complex dynamical systems [18,19]. Machado investigated the entropy analysis of fractional derivatives and their approximation [20].…”

Section: Introductionmentioning

confidence: 99%

Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

Yılmazer¹,

Tchier²,

Băleanu³

2016

Entropy

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Abstract:In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE) by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.

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Section: Introductionmentioning

confidence: 99%

Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

Yılmazer¹,

Tchier²,

Băleanu³

2016

Entropy

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“…In the past decades, fractional calculus has growing interest been paid in modelling applications, including the spread of HIV infection of CD4+ T-cells [1], entropy [2], hydrology [3], soft tissues such as mitral valve in the human heart [4], anomalous diffusion in transport dynamics of complex systems [5], engineering and physics. Many other examples can be found in Refs.…”

Section: Introductionmentioning

confidence: 99%

A Second-Order Accurate Implicit Difference Scheme for Time Fractional Reaction-Diffusion Equation with Variable Coefficients and Time Drift Term

Yong-LiangZhao¹

2019

EAJAM

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An implicit finite difference scheme based on the L2-1 σ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the L 2 -norm is O(τ 2 + h 2 ) with time step τ and mesh size h. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.

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“…Due to the nonlocal feature of fractional derivatives, FDEs are more suitable than the traditional differential equations in the description of the anomalous diffusion phenomena. Nowadays, the applications of FDEs have been recognized in numerous fields such as the physics [3], American options pricing [4], entropy [5] and image processing [6]. It is noticeable that finding the closed-form analytical solutions of most FDEs poses a challenge for researchers.…”

Section: Introductionmentioning

confidence: 99%

An implicit integration factor method for a kind of spatial fractional diffusion equations

Zhao

Zhu

et al. 2019

J. Phys.: Conf. Ser.

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A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an 1 L formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete ordinary differential system by using the implicit integration factor method, which is a class of efficient semi-implicit temporal scheme. Numerical results show that the proposed scheme is accurate even for the discontinuous coefficients.

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Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation

Cited by 21 publications

References 56 publications

Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

A Second-Order Accurate Implicit Difference Scheme for Time Fractional Reaction-Diffusion Equation with Variable Coefficients and Time Drift Term

An implicit integration factor method for a kind of spatial fractional diffusion equations

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