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Applications of Partial Differential Equations, 2nd Edition

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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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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 371

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

Viewed by 366

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

Viewed by 386

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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