A second-order numerical method for space–time variable-order diffusion equation

2021 ◽  
Vol 389 ◽  
pp. 113358
Author(s):  
Shujuan Lü ◽  
Tao Xu ◽  
Zhaosheng Feng
Keyword(s):  
Diffusion Equation ◽  
Numerical Method ◽  
Space Time ◽  
Second Order ◽  
Time Variable ◽  
Variable Order

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

Wellposedness and regularity of a variable-order space-time fractional diffusion equation

2020 ◽  
Vol 18 (04) ◽  
pp. 615-638 ◽  
Author(s):  
Xiangcheng Zheng ◽  
Hong Wang
Keyword(s):  
Diffusion Equation ◽  
Linear Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Order Linear ◽  
Integer Order ◽  
Time Fractional Diffusion Equation ◽  
Full Regularity

We prove wellposedness of a variable-order linear space-time fractional diffusion equation in multiple space dimensions. In addition we prove that the regularity of its solutions depends on the behavior of the variable order (and its derivatives) at time [Formula: see text], in addition to the usual smoothness assumptions. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at [Formula: see text] or have certain singularity at [Formula: see text] like its constant-order fractional analogue if the variable order has a non-integer value at time [Formula: see text].

scholarly journals DOUBLE-QUASI-WAVELET NUMERICAL METHOD FOR THE VARIABLE-ORDER TIME FRACTIONAL AND RIESZ SPACE FRACTIONAL REACTION–DIFFUSION EQUATION INVOLVING DERIVATIVES IN CAPUTO–FABRIZIO SENSE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040047
Author(s):  
SACHIN KUMAR ◽  
PRASHANT PANDEY ◽  
J. F. GÓMEZ-AGUILAR ◽  
D. BALEANU
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Numerical Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Variable Order ◽  
Nonlinear Reaction Diffusion Equation ◽  
Spatial Derivatives ◽  
Time Fractional Derivative ◽  
Fractional Type

Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.

Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Zhenhai Liu ◽  
Shengda Zeng ◽  
Yunru Bai
Keyword(s):  
Extreme Points ◽  
Initial Boundary ◽  
Space Time ◽  
Time Variable ◽  
Maximum Principles ◽  
Variable Order ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Initial Boundary Value

AbstractIn this paper, we deal with maximum principles for multi-term space-time variable-order Riesz-Caputo fractional differential equations (MT-STVO-RCFDEs, for short). We firstly derive several important inequalities for variable-order fractional derivatives at extreme points. Based on these inequalities, we obtain the maximum principles. Finally, these principles are employed to show that the uniqueness of solutions of the (MT-STVO-RCFDEs) and continuous dependance of solutions on initial-boundary value conditions.

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