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17 pages, 377 KiB

Open AccessArticle

On the Generalized Fractional Convection–Diffusion Equation with an Initial Condition in

R^{n}

by Chenkuan Li, Reza Saadati, Safoura Rezaei Aderyani and Min-Jie Luo

Fractal Fract. 2025, 9(6), 347; https://doi.org/10.3390/fractalfract9060347 - 27 May 2025

Cited by 1 | Viewed by 409

Abstract

Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional [...] Read more.

Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional convection–diffusion equation with a non-homogenous source term, based on an inverse operator, a newly-constructed space, and the multivariate Mittag–Leffler function. Several illustrative examples are provided to show the power and simplicity of our main theorems in solving certain fractional convection–diffusions equations. Additionally, we compare these results with solutions obtained using the AI model DeepSeek-R1, highlighting the effectiveness and validity of our proposed methods and main theorems. Full article

(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)

39 pages, 391 KiB

Open AccessArticle

Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations

by Chenkuan Li and Wenyuan Liao

Fractal Fract. 2025, 9(4), 200; https://doi.org/10.3390/fractalfract9040200 - 25 Mar 2025

Cited by 3 | Viewed by 390

Abstract

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a [...] Read more.

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a generalized Mittag-Leffler function, as well as a few popular fixed-point theorems. These studies have good applications in general since uniqueness, existence and stability are key and important topics in many fields. Several examples are presented to demonstrate applications of results obtained by computing approximate values of the generalized Mittag-Leffler functions. (ii) We use the inverse operator method and newly established spaces to find analytic solutions to a number of notable partial differential equations, such as a multi-term time-fractional convection problem and a generalized time-fractional diffusion-wave equation in

R^{n}

with initial conditions only, which have never been previously considered according to the best of our knowledge. In particular, we deduce the uniform solution to the non-homogeneous wave equation in n dimensions for all

n \geq 1

, which coincides with classical results such as d’Alembert and Kirchoff’s formulas but is much easier in the computation of finding solutions without any complicated integrals on balls or spheres. Full article

(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)

19 pages, 1052 KiB

Open AccessArticle

Identifying a Space-Dependent Source Term and the Initial Value in a Time Fractional Diffusion-Wave Equation

by Xianli Lv and Xiufang Feng

Mathematics 2023, 11(6), 1521; https://doi.org/10.3390/math11061521 - 21 Mar 2023

Cited by 3 | Viewed by 1775

Abstract

This paper is focused on the inverse problem of identifying the space-dependent source function and initial value of the time fractional nonhomogeneous diffusion-wave equation from noisy final time measured data in a multi-dimensional case. A mollification regularization method based on a bilateral exponential [...] Read more.

This paper is focused on the inverse problem of identifying the space-dependent source function and initial value of the time fractional nonhomogeneous diffusion-wave equation from noisy final time measured data in a multi-dimensional case. A mollification regularization method based on a bilateral exponential kernel is presented to solve the ill-posedness of the problem for the first time. Error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical experiments of interest show that our proposed method is effective and robust with respect to the perturbation noise in the data. Full article

(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)

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15 pages, 399 KiB

Open AccessArticle

The Fractional Tikhonov Regularization Method to Identify the Initial Value of the Nonhomogeneous Time-Fractional Diffusion Equation on a Columnar Symmetrical Domain

by Yong-Gang Chen, Fan Yang, Xiao-Xiao Li and Dun-Gang Li

Symmetry 2022, 14(8), 1633; https://doi.org/10.3390/sym14081633 - 8 Aug 2022

Cited by 1 | Viewed by 2020

Abstract

In this paper, the inverse problem for identifying the initial value of a time fractional nonhomogeneous diffusion equation in a columnar symmetric region is studied. This is an ill-posed problem, i.e., the solution does not depend continuously on the data. The fractional Tikhonov [...] Read more.

In this paper, the inverse problem for identifying the initial value of a time fractional nonhomogeneous diffusion equation in a columnar symmetric region is studied. This is an ill-posed problem, i.e., the solution does not depend continuously on the data. The fractional Tikhonov regularization method is applied to solve this problem and obtain the regularization solution. The error estimations between the regularization solution and the exact solution are also obtained under the priori and the posteriori regularization parameter choice rules, respectively. Some examples are given to show this method’s effectiveness. Full article

(This article belongs to the Special Issue Inverse Problems and Differential Geometry: Theory and Applications)

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22 pages, 345 KiB

Open AccessArticle

Regularization for a Sideways Problem of the Non-Homogeneous Fractional Diffusion Equation

by Yonggang Chen, Yu Qiao and Xiangtuan Xiong

Fractal Fract. 2022, 6(6), 312; https://doi.org/10.3390/fractalfract6060312 - 2 Jun 2022

Viewed by 2173

Abstract

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve [...] Read more.

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve the result of conditional stability under an a priori assumption. Two regularization strategies, based on the truncation of high frequency components, are constructed for solving the inverse problem in the presence of noisy data, and the corresponding error estimates are proved. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

14 pages, 274 KiB

Open AccessArticle

Regularization Error Analysis for a Sideways Problem of the 2D Nonhomogeneous Time-Fractional Diffusion Equation

by Yonggang Chen, Yu Qiao and Xiangtuan Xiong

Mathematics 2022, 10(10), 1742; https://doi.org/10.3390/math10101742 - 19 May 2022

Viewed by 1630

Abstract

The inverse and ill-posed problem of determining a solute concentration for the two-dimensional nonhomogeneous fractional diffusion equation is investigated. This model is much worse than its homogeneous counterpart as the source term appears. We propose a modified kernel regularization technique for the stable [...] Read more.

The inverse and ill-posed problem of determining a solute concentration for the two-dimensional nonhomogeneous fractional diffusion equation is investigated. This model is much worse than its homogeneous counterpart as the source term appears. We propose a modified kernel regularization technique for the stable numerical reconstruction of the solution. The convergence estimates under both a priori and a posteriori parameter choice rules are proven. Full article

21 pages, 13086 KiB

Open AccessArticle

Fractional Modeling Applied to the Dynamics of the Action Potential in Cardiac Tissue

by Sergio Adriani David, Carlos Alberto Valentim and Amar Debbouche

Fractal Fract. 2022, 6(3), 149; https://doi.org/10.3390/fractalfract6030149 - 10 Mar 2022

Cited by 15 | Viewed by 3051

Abstract

We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some [...] Read more.

We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some of the proposed fractional solutions to provide a different modeling perspective on distinct phases of cardiac membrane potential. Results indicate that the fractional diffusion-wave equation may be employed to model membrane potential dynamics with the fractional order working as an extra asset to modulate electricity conduction, particularly for lower fractional order values. Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications (DSTA) — in Memory of Prof. Dr. José A. Tenreiro Machado)

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26 pages, 1316 KiB

Open AccessArticle

Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

by Saadoune Brahimi, Ahcene Merad and Adem Kılıçman

Mathematics 2021, 9(16), 1987; https://doi.org/10.3390/math9161987 - 19 Aug 2021

Cited by 1 | Viewed by 2422

Abstract

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types

\int_{0}^{1} v (x, t) d x

and [...] Read more.

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types

\int_{0}^{1} v (x, t) d x

and

\int_{0}^{1} x^{n} v (x, t) d x

. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results. Full article

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21 pages, 427 KiB

Open AccessArticle

The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations

by Fan Yang, Qu Pu, Xiao-Xiao Li and Dun-Gang Li

Mathematics 2019, 7(11), 1007; https://doi.org/10.3390/math7111007 - 23 Oct 2019

Cited by 17 | Viewed by 2700

Abstract

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value [...] Read more.

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value functions. The convergence estimations are given in a priori regularization parameter choice regulations and a posteriori regularization parameter choice regulations. Numerical examples are given to demonstrate this is effective and practicable. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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