Submit to Special Issue Submit Abstract to Special Issue Review for Mathematics Propose a Special Issue

Journal Menu

Journal Browser

Applications of Partial Differential Equations, 2nd Edition

Special Issue Editors
Special Issue Information
Keywords
Benefits of Publishing in a Special Issue
Published Papers

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

Deadline for manuscript submissions: 28 February 2025 | Viewed by 3034

Share This Special Issue

Special Issue Editor

Dr. Patricia J. Y. Wong

E-Mail Website
Guest Editor

School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
Interests: differential equations; difference equations; integral equations; numerical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Partial differential equations are indispensable in modeling various phenomena and processes in many fields, such as physics, biology, finance, and engineering. The study on the solutions of partial differential equations, be it on the qualitative theory or quantitative methods, as well as the applications of such investigations to real-world problems, have garnered significant interest of researchers.

This Special Issue "Applications of Partial Differential Equations, 2nd Edition", aims to collect original and significant contributions on the following:

Applications of partial differential equations in modeling real-world phenomena;
Qualitative theory on the solutions of partial differential equations;
Analytical or numerical methods for solving partial differential equations.

Investigations on partial differential equations involving fractional derivatives with respect to at least one of the independent variables are also welcome.

Dr. Patricia J. Y. Wong
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com 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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

nonlinear partial differential equations
diffusion equations
wave-type equations
partial differential equations with delay
partial functional differential equations
fractional partial differential equations
numerical solution
analytical solution
modeling

Benefits of Publishing in a Special Issue

Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here.

Published Papers (4 papers)

Download All Papers

Order results

Result details

Show export options Show export options

Select all

Export citation of selected articles as:

Research

23 pages, 4312 KiB

Open AccessArticle

Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function

by Afzaal Mubashir Hayat, Muhammad Bilal Riaz, Muhammad Abbas, Moataz Alosaimi, Adil Jhangeer and Tahir Nazir

Mathematics 2024, 12(13), 2137; https://doi.org/10.3390/math12132137 - 7 Jul 2024

Viewed by 536

Abstract

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is [...] Read more.

θ

-weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spline functions, whereas the temporal derivative is discretized by the Atangana–Baleanu operator and finite difference scheme. The proposed approach is stable across each temporal direction as well as second-order convergent. The study investigates the convergence order, error norms, and graphical visualization of the solution for various values of the non-integer parameter. The efficacy of the technique is assessed by implementing it on three test examples and we find that it is more efficient than some existing methods in the literature. To our knowledge, no prior application of this approach has been made for the numerical solution of the given problem, making it a first in this regard. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

► Show Figures

Figure 1

19 pages, 1047 KiB

Open AccessArticle

Stability Analysis and Hopf Bifurcation for the Brusselator Reaction–Diffusion System with Gene Expression Time Delay

by Hassan Y. Alfifi and Saad M. Almuaddi

Mathematics 2024, 12(8), 1170; https://doi.org/10.3390/math12081170 - 13 Apr 2024

Viewed by 716

Abstract

This paper investigates the effect of a gene expression time delay on the Brusselator model with reaction and diffusion terms in one dimension. We obtain ODE systems analytically by using the Galerkin method. We determine a condition that assists in showing the existence of theoretical results. Full maps of the Hopf bifurcation regions of the stability analysis are studied numerically and theoretically. The influences of two different sources of diffusion coefficients and gene expression time delay parameters on the bifurcation diagram are examined and plotted. In addition, the effect of delay and diffusion values on all other free parameters in this system is shown. They can significantly affect the stability regions for both control parameter concentrations through the reaction process. As a result, as the gene expression time delay increases, both control concentration values increase, while the Hopf points for both diffusion coefficient parameters decrease. These values can impact solutions in the bifurcation regions, causing the region of instability to grow. In addition, the Hopf bifurcation points for the diffusive and non-diffusive cases as well as delay and non-delay cases are studied for both control parameter concentrations. Finally, various examples and bifurcation diagrams, periodic oscillations, and 2D phase planes are provided. There is close agreement between the theoretical and numerical solutions in all cases. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

► Show Figures

Figure 1

15 pages, 273 KiB

Open AccessArticle

Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

by Jie Wu and Yujie Huang

Mathematics 2024, 12(8), 1143; https://doi.org/10.3390/math12081143 - 10 Apr 2024

Cited by 8 | Viewed by 815

Abstract

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production [...] Read more.

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production

\{\begin{matrix} 𝜕_{t} u = Δ u - \nabla \cdot (χ_{1} u \nabla v_{1}) + \nabla \cdot (χ_{2} u \nabla v_{2}), & x \in R^{2}, t > 0, \\ 0 = Δ v_{j} - λ_{j} v_{j} + w, & x \in R^{2}, t > 0, (j = 1, 2), \\ 𝜕_{t} w + δ w = u, & x \in R^{2}, t > 0, \\ u (0, x) = u_{0} (x), w (0, x) = w_{0} (x), & x \in R^{2}, \end{matrix}

where the parameters

χ_{i} \geq 0, λ_{i} > 0 (i = 1, 2)

and non-negative initial data

(u_{0} (x), w_{0} (x)) \in L^{1} (R^{2}) \cap L^{\infty} (R^{2})

. We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of

χ_{1} \geq χ_{2}

. At the same time, we propose that when the radial solution satisfies

χ_{1} - χ_{2} \leq \frac{2 π δ}{∥ u_{0} ∥_{L^{1} (R^{2})} + {∥ w_{0} ∥}_{L^{1} (R^{2})}}

, the global solution is bounded. During the proof process, we found that adding indirect signals can constrict the blow-up of the global solution. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

16 pages, 2392 KiB

Open AccessArticle

Distributed Fault Diagnosis via Iterative Learning for Partial Differential Multi-Agent Systems with Actuators

by Cun Wang, Zupeng Zhou and Jingjing Wang

Mathematics 2024, 12(7), 955; https://doi.org/10.3390/math12070955 - 23 Mar 2024

Cited by 1 | Viewed by 587

Abstract

Component failures can lead to performance degradation or even failure in multi-agent systems, thus necessitating the development of fault diagnosis methods. Addressing the distributed fault diagnosis problem in a class of partial differential multi-agent systems with actuators, a fault estimator is designed under the introduction of virtual faults to the agents. A P-type iterative learning control protocol is formulated based on the residual signals, aiming to adjust the introduced virtual faults. Through rigorous mathematical analysis utilizing contraction mapping and the Bellman–Gronwall lemma, sufficient conditions for the convergence of this protocol are derived. The results indicate that the learning protocol ensures the tracking of virtual faults to actual faults, thereby facilitating fault diagnosis for the systems. Finally, the effectiveness of the learning protocol is validated through numerical simulation. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

► Show Figures

Journal Menu

Journal Browser

Applications of Partial Differential Equations, 2nd Edition

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (4 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI