scholarly journals Numerical methods for solving the multi-term time-fractional wave-diffusion equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Fawang Liu ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Pinghui Zhuang ◽  
Qingxia Liu
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Time Space ◽  
Methods And Techniques ◽  
Wave Diffusion

AbstractIn this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Maximum principles for a class of generalized time-fractional diffusion equations

2020 ◽  
Vol 23 (3) ◽  
pp. 822-836
Author(s):  
Shengda Zeng ◽  
Stanisław Migórski ◽  
Van Thien Nguyen ◽  
Yunru Bai
Keyword(s):  
Fractional Derivatives ◽  
Extreme Points ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
Fractional Diffusion Equations ◽  
Caputo Derivatives ◽  
Time Space ◽  
Generalized Time

AbstractTwo significant inequalities for generalized time fractional derivatives at extreme points are obtained. Then, we apply the inequalities to establish the maximum principles for multi-term time-space fractional variable-order operators. Finally, we employ the principles to investigate two kinds of diffusion equations involving generalized time-fractional Caputo derivatives and space-fractional Riesz-Caputo derivatives.

On the decomposition of solutions: From fractional diffusion to fractional Laplacian

2021 ◽  
Vol 24 (5) ◽  
pp. 1571-1600
Author(s):  
Yulong Li
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Ordinary Differential Equations ◽  
Laplace Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Deep Insight ◽  
Structure Of Solutions

Abstract This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

scholarly journals Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1975
Author(s):  
M. Luísa Morgado ◽  
Magda Rebelo ◽  
Luís L. Ferrás
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Uniform Mesh ◽  
Numerical Schemes ◽  
The Stability ◽  
Distributed Order

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

scholarly journals Monotone iterative technique for time-space fractional diffusion equations involving delay

2021 ◽  
Vol 26 (2) ◽  
pp. 241-258
Author(s):  
Qiang Li ◽  
Guotao Wang ◽  
Mei Wei
Keyword(s):  
Diffusion Equation ◽  
Fractional Laplacian ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Initial Boundary ◽  
Monotone Iterative Technique ◽  
Laplacian Operator ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Time Space

This paper considers the initial boundary value problem for the time-space fractional delayed diffusion equation with fractional Laplacian. By using the semigroup theory of operators and the monotone iterative technique, the existence and uniqueness of mild solutions for the abstract time-space evolution equation with delay under some quasimonotone conditions are obtained. Finally, the abstract results are applied to the time-space fractional delayed diffusion equation with fractional Laplacian operator, which improve and generalize the recent results of this issue.

scholarly journals High-Order Algorithms for Riesz Derivative and their Applications (III)

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Turbulent Diffusion ◽  
Numerical Experiments ◽  
High Order ◽  
Riesz Space ◽  
Sixth Order ◽  
Convergence Orders

AbstractNumerical methods for fractional calculus attract increasing interest due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz type turbulent diffusion equation (or, Riesz space fractional turbulent diffusion equation). Numerical experiments are displayed which support the theoretical analysis.

scholarly journals Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

2011 ◽  
Vol 52 ◽  
pp. 395 ◽  
Author(s):  
Qianqian Yang ◽  
Tim Moroney ◽  
Kevin Burrage ◽  
Ian Turner ◽  
Fawang Liu
Keyword(s):  
Numerical Methods ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Space ◽  
Fractional Reaction
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