scholarly journals Numerical approximation of stochastic time-fractional diffusion

2019 ◽  
Vol 53 (4) ◽  
pp. 1245-1268 ◽  
Author(s):  
Bangti Jin ◽  
Yubin Yan ◽  
Zhi Zhou
Keyword(s):  
Fractional Derivative ◽  
Gaussian Noise ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Convergence Rates ◽  
Piecewise Linear ◽  
Fractional Diffusion ◽  
Strong And Weak Convergence ◽  
Standard Galerkin Method ◽  
Stochastic Time

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order α ∈ (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ ∈ [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the L2-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.

scholarly journals Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 130
Author(s):  
Suphawat Asawasamrit ◽  
Yasintorn Thadang ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Conditions ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Nonlinear Alternative

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

scholarly journals Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

Meccanica ◽  
2021 ◽  
Author(s):  
Tomasz Blaszczyk ◽  
Krzysztof Bekus ◽  
Krzysztof Szajek ◽  
Wojciech Sumelka
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Rate Of Convergence ◽  
Continuum Mechanics ◽  
Caputo Fractional Derivative ◽  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Variable Order ◽  
Piecewise Constant ◽  
Quadratic Interpolation

AbstractIn this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

scholarly journals A novel series method for fractional diffusion equation within Caputo fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 695-699 ◽  
Author(s):  
Sheng-Ping Yan ◽  
Wei-Ping Zhong ◽  
Xiao-Jun Yang
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Series Solution ◽  
Expansion Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Series Method ◽  
Time Fractional Diffusion Equation ◽  
Series Expansion Method

In this paper, we suggest the series expansion method for finding the series solution for the time-fractional diffusion equation involving Caputo fractional derivative.

scholarly journals Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative

2022 ◽  
Vol 6 (1) ◽  
pp. 35
Author(s):  
Ndolane Sene
Keyword(s):  
Diffusion Equation ◽  
Prandtl Number ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Fractional Diffusion ◽  
Fluid Models ◽  
Fractional Operators ◽  
Fractional Heat Equation

This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators.

Fractional diffusion-wave equations: Hidden regularity for weak solutions

2021 ◽  
Vol 24 (4) ◽  
pp. 1015-1034
Author(s):  
Paola Loreti ◽  
Daniela Sforza
Keyword(s):  
Fractional Derivative ◽  
Weak Solutions ◽  
Caputo Fractional Derivative ◽  
Wave Equations ◽  
Regularity Result ◽  
Fractional Diffusion ◽  
Interpolation Spaces ◽  
Diffusion Wave ◽  
Regularity Properties

Abstract We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

2017 ◽  
Vol 22 (4) ◽  
pp. 1028-1048 ◽  
Author(s):  
Yonggui Yan ◽  
Zhi-Zhong Sun ◽  
Jiwei Zhang
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fast Evaluation ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Order Scheme ◽  
Speed Up

AbstractThe fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on theL2-1σformula proposed in [A. Alikhanov,J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

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