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.

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 Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Karel Van Bockstal
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Operators ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Unique Weak Solution ◽  
Bounded Lipschitz Domain

AbstractIn this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( k ∗ v ) ( t ) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

scholarly journals Locally one dimensional scheme of the Dirichlet boundary value problem for fractional diffusion equation with space caputo fractional derivative

2016 ◽  
Author(s):  
А.К. Баззаев
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Dirichlet Boundary Value Problem ◽  
One Dimensional

В работе рассматриваются локально-одномерные схемы для уравнения диффузии дробного порядка с дробной производной в младших членах с граничными условиями первого рода. С помощью принципа максимума получена априорная оценка для решения разностной задачи в равномерной метрике. Доказаны устойчивость и сходимость построенных локально-одномерных схем.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

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