Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 4811

Special Issue Editors

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Interests: convergence of numerical methods; diffusion; finite difference methods; iterative methods; numerical stability; partial differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional differential equations and integral differential equations have attracted a great amount of attention in recent years. They widely appear in applied mathematics, physics, biology, chemistry and other disciplines. The typical models include sub-diffusion equations, diffusion-wave equations, space-fractional differential equations, and so on. It is usually difficult to obtain analytical solutions, due to the integral terms in the models. Fortunately, the evolution of differential equations can be well described by using some well-designed and high-order numerical schemes. Therefore, it has become a hot topic to numerically solve and analyze the equations.

In light of the aforementioned points regarding the significance of theoretical and numerical analysis, the potential topics include, but are not limited to, the following:

  • New theoretical results for fractional differential equations and integral differential equations;
  • New numerical methods for solving fractional differential equations;
  • New numerical methods for solving integral differential equations;
  • New numerical methods for solving non-local problems;
  • Numerical analysis of the numerical methods;
  • Application of fractional differential equations.

Prof. Dr. Dongfang Li
Dr. Hongyu Qin
Guest Editors

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Keywords

  • fractional differential equations
  • integral differential equations
  • theoretical analysis
  • numerical methods
  • numerical analysis

Published Papers (5 papers)

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Research

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23 pages, 349 KiB  
Article
Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory
by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Mathematics 2023, 11(18), 3859; https://doi.org/10.3390/math11183859 - 09 Sep 2023
Viewed by 525
Abstract
In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for [...] Read more.
In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. Full article
21 pages, 9043 KiB  
Article
An Efficient Numerical Approach for Solving Systems of Fractional Problems and Their Applications in Science
by Sondos M. Syam, Z. Siri, Sami H. Altoum and R. Md. Kasmani
Mathematics 2023, 11(14), 3132; https://doi.org/10.3390/math11143132 - 16 Jul 2023
Cited by 2 | Viewed by 879
Abstract
In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic [...] Read more.
In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic problem produced by this approach into a set of 2×2 nonlinear equations, instead of solving a system of 2 m × 2 m equations. We apply our approach to three main applications in science: optimal control problems, Riccati equations, and clock reactions. We compare our results with those of other researchers, considering computational time, cost, and absolute errors. Additionally, we validate our numerical method by comparing our results with the integer model when the fractional order approaches one. We present numerous figures and tables to illustrate our findings. The results demonstrate the effectiveness of the proposed approach. Full article
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10 pages, 274 KiB  
Article
A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions
by Boya Zhou and Xiujun Cheng
Mathematics 2023, 11(12), 2594; https://doi.org/10.3390/math11122594 - 06 Jun 2023
Viewed by 637
Abstract
In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence [...] Read more.
In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order O(N2) in L norm is proved, where N refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method. Full article
17 pages, 4635 KiB  
Article
Mixed Convection Flow of Water Conveying Graphene Oxide Nanoparticles over a Vertical Plate Experiencing the Impacts of Thermal Radiation
by Umair Khan, Aurang Zaib, Anuar Ishak, Iskandar Waini and Ioan Pop
Mathematics 2022, 10(16), 2833; https://doi.org/10.3390/math10162833 - 09 Aug 2022
Cited by 1 | Viewed by 1162
Abstract
Water has drawn a lot of interest as a manufacturing lubricant since it is affordable, eco-friendly, and effective. Due to their exceptional mechanical qualities, water solubility, and variety of application scenarios, graphene oxide (GO)-based materials have the potential to increase the lubricant performance [...] Read more.
Water has drawn a lot of interest as a manufacturing lubricant since it is affordable, eco-friendly, and effective. Due to their exceptional mechanical qualities, water solubility, and variety of application scenarios, graphene oxide (GO)-based materials have the potential to increase the lubricant performance of water. The idea of this research was to quantify the linear 3D radiative stagnation-point flow induced by nanofluid through a vertical plate with a buoyancy or a mixed convection effect. The opposing, as well as the assisting, flows were considered in the model. The leading partial differential equations (PDEs) were transformed into dimensionless similarity equations, which were then solved numerically via a bvp4c solver. The influences of various physical constraints on the fluid flow and thermal properties of the nanofluid were investigated and are discussed. Water-based graphene oxide nanoparticles were considered in this study. The numerical outcomes indicated that multiple solutions were obtained in the case of the opposing flow (λ < 0). The critical values increased as the nanoparticle volume fraction became stronger. Furthermore, as the nanoparticles increased in strength, the friction factor increased and the heat transfer quickened. The radiation factor escalated the heat transfer in both solutions. In addition, a temporal stability analysis was also undertaken to verify the results, and it was observed that the branch of the first outcome became physically reliable (stable) whilst the branch of the second outcome became unstable, as time passed. Full article
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Review

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12 pages, 439 KiB  
Review
A Mini-Review on Recent Fractional Models for Agri-Food Problems
by Stefania Tomasiello and Jorge E. Macías-Díaz
Mathematics 2023, 11(10), 2316; https://doi.org/10.3390/math11102316 - 16 May 2023
Viewed by 836
Abstract
This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result [...] Read more.
This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India. Full article
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