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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 31 October 2025 | Viewed by 6285

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Dr. Patricia J. Y. Wong

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School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
Interests: differential equations; difference equations; integral equations; numerical analysis
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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

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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

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Published Papers (6 papers)

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Research

13 pages, 1830 KiB

Open AccessArticle

Maximum Principle-Preserving Computational Algorithm for the 3D High-Order Allen–Cahn Equation

by Junseok Kim and Youngjin Hwang

Mathematics 2025, 13(7), 1085; https://doi.org/10.3390/math13071085 - 26 Mar 2025

Viewed by 216

Abstract

We propose an unconditionally stable computational algorithm that preserves the maximum principle for the three-dimensional (3D) high-order Allen–Cahn (AC) equation. The presented algorithm applies an operator-splitting technique that decomposes the original equation into nonlinear and linear diffusion equations. To guarantee the unconditional stability of the numerical solution, we solve the nonlinear equation using the frozen coefficient technique, which simplifies computations by approximating variable coefficients by constants within small regions. For the linear equation, we use an implicit finite difference scheme under the operator-splitting method. To validate the efficiency of the proposed algorithm, we conducted several computational tests. The numerical results confirm that the scheme achieves unconditional stability even for large time step sizes and high-order polynomial potential. In addition, we analyze motion by mean curvature in three-dimensional space and show that the numerical solutions closely match the analytical solutions. Finally, the robustness of the method is evaluated under noisy data conditions, and its ability to accurately classify complex data structures is demonstrated. These results confirm the efficiency and reliability of the proposed computational algorithm for simulating phase-field models with a high-order polynomial potential. Full article

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

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21 pages, 739 KiB

Open AccessArticle

Effects of Diffusion and Delays on the Dynamic Behavior of a Competition and Cooperation Model

by Hassan Y. Alfifi

Mathematics 2025, 13(7), 1026; https://doi.org/10.3390/math13071026 - 21 Mar 2025

Viewed by 151

Abstract

This study investigates a model of competition and cooperation between two enterprises with reaction, diffusion, and delays. The stability and Hopf bifurcation for variants with two, one, and no delays are considered by examining a system of delay ODE equations analytically and numerically, applying the Galerkin method. A condition is obtained that helps characterize the existence of Hopf bifurcation points. Full maps of stability analysis are discussed in detail. With bifurcation diagrams, three different cases of delay are shown to determine the stable and unstable regions. It is found that when

τ_{i} > 0

, there are two different stability regions, and that without a delay (

τ_{i} = 0

), there is only one stable region. Furthermore, the effects of delays and diffusion parameters on all other free rates in the system are considered; these can significantly affect the stability areas, with important economic consequences for the development of enterprises. Moreover, the relationship between the diffusion and delay parameters is discussed in more detail: it is found that the value of the time delay at the Hopf point increases exponentially with the diffusion coefficient. An increase in the diffusion coefficient can also lead to an increase in the Hopf-point values of the intrinsic growth rates. Finally, bifurcation diagrams are used to identify specific instances of limit cycles, and 2-D phase portraits for both systems are presented to validate all theoretical results discussed in this work. Full article

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

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22 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

Cited by 2 | Viewed by 1309

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)

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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

Cited by 1 | Viewed by 1331

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)

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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 13 | Viewed by 1542

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)

15 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 961

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)

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