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Axioms
Special Issues
Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods
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Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 25 October 2024 | Viewed by 608

Special Issue Editors

Dr. Renhai Wang

E-Mail Website
Guest Editor
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China
Interests: deterministic and stochastic PDEs; fractional PDEs; lattice systems; dynamical systems; attractors; random attractors; invariant measures; large deviation principle, probability theory; stochastic analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jun Zhou

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Interests: nonlinear PDEs; qualitative analysis of PDEs; blow-up of PDEs; wave equations; diffusion equations; high-order parabolic equations; Boussinesq-type equations

Special Issue Information

Dear Colleagues,

The study of partial differential equations has consistently received attention mainly due to their significant or potential applications in many fields. The aim of this Special Issue is to explore and develop the study of partial differential equations by using qualitative and numerical analysis arguments. In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:  

  • Well-posedness of partial differential equations;
  • Blow-up and long-time behavior of partial differential equations;
  • Numerical analysis and methods of partial differential equations;
  • Fractional calculus for partial differential equations;
  • Attractors, invariant measures, large deviations, traveling wave, stochastic control, invariant manifolds and chaos for differential equations;
  • Dynamical systems and measure theory for differential equations;
  • Other subjects on partial differential equations.

Dr. Renhai Wang
Prof. Dr. Jun Zhou
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at mdpi.longhoe.net by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • partial differential equations
  • stochastic
  • blow-up
  • well-posedness
  • dynamics
  • numerical analysis
  • long-time behavior
  • probability

Published Papers (1 paper)

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Research

27 pages, 368 KiB  
Article
Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations
by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat
Axioms 2024, 13(7), 438; https://doi.org/10.3390/axioms13070438 - 28 Jun 2024
Viewed by 212
Abstract
Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable [...] Read more.
Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end. Full article
(This article belongs to the Special Issue Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods)
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