Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs

Topic Information

Dear Colleagues,

This topic aims to delve into the frontier research of fractional calculus, symmetry phenomenon, probability theory, and fractional differential equations. Fractional calculus, as an essential branch of modern mathematics, provides powerful tools for modeling and analyzing complex systems. Fractional differential equations have widespread applications in physics, engineering, and other fields. This topic will gather the latest research achievements in this area, promote academic exchange and cooperation, and drive the development of fractional calculus and fractional differential equations in both theory and application. Potential topics include but are not limited to the following:

• Stability analysis of fractional partial differential equations in infinite dimensional dynamical systems;
• Bifurcation phenomena in fractional partial differential equations within infinite dimensional dynamical systems;
• Fractional stochastic partial differential equations for complex real-world problems;
• Applications of fractional calculus in physical modeling;
• Recent advances in complex system analysis using fractional differential equations;
• Cross-research between fractional calculus and nonlinear science;
• Explorations in the application of fractional calculus in signal and image processing.

Prof. Dr. Renhai Wang
Prof. Dr. Junesang Choi
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
AppliedMath appliedmath	-	-	2021	25.3 Days	CHF 1000	Submit
Axioms axioms	1.9	-	2012	22.8 Days	CHF 2400	Submit
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700	Submit
Mathematical and Computational Applications mca	1.9	-	1996	25.4 Days	CHF 1400	Submit
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600	Submit
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400	Submit

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Published Papers (5 papers)

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All AppliedMath Axioms Fractal Fract MCA Mathematics Symmetry

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22 pages, 341 KiB  

Open AccessArticle

Mild Solutions of the (ρ₁,ρ₂,k₁,k₂,φ)-Proportional Hilfer–Cauchy Problem

by Haihua Wang and Jie Zhao

Symmetry 2024, 16(10), 1349; https://doi.org/10.3390/sym16101349 - 11 Oct 2024

Viewed by 776

Abstract

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and the [...] Read more.

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results. Full article

(This article belongs to the Topic Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs)

9 pages, 255 KiB  

Open AccessArticle

Elephant Random Walk with a Random Step Size and Gradually Increasing Memory and Delays

by Rafik Aguech

Axioms 2024, 13(9), 629; https://doi.org/10.3390/axioms13090629 - 14 Sep 2024

Viewed by 877

Abstract

The ERW model was introduced twenty years ago to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past throughout a parameter p between zero and one. Several variations of the ERW model have recently been introduced. [...] Read more.

The ERW model was introduced twenty years ago to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past throughout a parameter p between zero and one. Several variations of the ERW model have recently been introduced. In this work, we investigate the asymptotic normality of the ERW model with a random step size and gradually increasing memory and delays. In particular, we extend some recent results in this subject. Full article

(This article belongs to the Topic Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs)

87 pages, 15297 KiB  

Open AccessReview

Analytic Theory of Seven Classes of Fractional Vibrations Based on Elementary Functions: A Tutorial Review

by Ming Li

Symmetry 2024, 16(9), 1202; https://doi.org/10.3390/sym16091202 - 12 Sep 2024

Viewed by 720

Abstract

This paper conducts a tutorial review of the analytic theory of seven classes of fractional vibrations based on elementary functions. We discuss the classification of seven classes of fractional vibrations and introduce the problem statements. Then, the analytic theory of class VI fractional [...] Read more.

This paper conducts a tutorial review of the analytic theory of seven classes of fractional vibrations based on elementary functions. We discuss the classification of seven classes of fractional vibrations and introduce the problem statements. Then, the analytic theory of class VI fractional vibrators is given. The analytic theories of fractional vibrators from class I to class V and class VII are, respectively, represented. Furthermore, seven analytic expressions of frequency bandwidth of seven classes of fractional vibrators are newly introduced in this paper. Four analytic expressions of sinusoidal responses to fractional vibrators from class IV to VII by using elementary functions are also newly reported in this paper. The analytical expressions of responses (free, impulse, step, and sinusoidal) are first reported in this research. We dissert three applications of the analytic theory of fractional vibrations: (1) analytical expression of the forced response to a damped multi-fractional Euler–Bernoulli beam; (2) analytical expressions of power spectrum density (PSD) and cross-PSD responses to seven classes of fractional vibrators under the excitation with the Pierson and Moskowitz spectrum, which are newly introduced in this paper; and (3) a mathematical explanation of the Rayleigh damping assumption. Full article

(This article belongs to the Topic Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs)

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18 pages, 315 KiB  

Open AccessArticle

Existence of Heteroclinic Solutions in Nonlinear Differential Equations of the Second-Order Incorporating Generalized Impulse Effects with the Possibility of Application to Bird Population Growth

by Robert de Sousa and Marco António de Sales Monteiro Fernandes

AppliedMath 2024, 4(3), 1047-1064; https://doi.org/10.3390/appliedmath4030056 - 27 Aug 2024

Viewed by 1826

Abstract

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with $\varphi$-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that $\varphi$ be a homeomorphism, using [...] Read more.

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with $\varphi$-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that $\varphi$ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of $L^1$-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth. Full article

(This article belongs to the Topic Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs)

25 pages, 1991 KiB  

Open AccessArticle

Chebyshev Pseudospectral Method for Fractional Differential Equations in Non-Overlapping Partitioned Domains

by Shina Daniel Oloniiju, Nancy Mukwevho, Yusuf Olatunji Tijani and Olumuyiwa Otegbeye

AppliedMath 2024, 4(3), 950-974; https://doi.org/10.3390/appliedmath4030051 - 2 Aug 2024

Viewed by 1465

Abstract

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when [...] Read more.

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when dealing with such problems. This study proposes a novel multi-domain pseudospectral method based on the first kind of Chebyshev polynomials and the Gauss–Lobatto quadrature for fractional initial value problems.The proposed technique involves partitioning the problem’s domain into non-overlapping sub-domains, calculating the fractional differential operator in each sub-domain as the sum of the ‘local’ and ‘memory’ parts and deriving the corresponding differentiation matrices to develop the numerical schemes. The linear stability analysis indicates that the numerical scheme is absolutely stable for certain values of arbitrary non-integer order and conditionally stable for others. Numerical examples, ranging from single linear equations to systems of non-linear equations, demonstrate that the multi-domain approach is more appropriate, efficient and accurate than the single-domain scheme, particularly for problems with long-term dynamics. Full article

(This article belongs to the Topic Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs)

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