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

Deadline for manuscript submissions: closed (23 August 2023) | Viewed by 19643

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Special Issue Editor

Dr. Patricia J. Y. Wong

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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 drawn a large amount of interest from researchers.

This Special Issue aims to collect original and significant contributions on:

The 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.

Equally welcome are investigations in the above topics on partial differential equations involving fractional derivatives with respect to at least one of the independent variables.

Prof. 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

Published Papers (14 papers)

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Research

21 pages, 353 KiB

Open AccessFeature PaperArticle

General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions

by Mi Jin Lee and Jum-Ran Kang

Mathematics 2023, 11(22), 4593; https://0-doi-org.brum.beds.ac.uk/10.3390/math11224593 - 09 Nov 2023

Viewed by 530

Abstract

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition

a_{2} > 0

and without [...] Read more.

a_{2} > 0

and without imposing any restrictive growth assumption on the damping term

f_{1},

using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function

ψ

, namely

ψ^{'} (t) \leq - μ (t) G (ψ (t))

, where G is a convex and increasing function near the origin, and

μ

is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions

μ

and G, as well as the function F defined by

f_{0}

, which characterizes the growth behavior of

f_{1}

at the origin. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

17 pages, 1654 KiB

Open AccessArticle

Lump-Type Solutions, Lump Solutions, and Mixed Rogue Waves for Coupled Nonlinear Generalized Zakharov Equations

by Aly R. Seadawy, Syed T. R. Rizvi and Hanadi Zahed

Mathematics 2023, 11(13), 2856; https://0-doi-org.brum.beds.ac.uk/10.3390/math11132856 - 26 Jun 2023

Cited by 3 | Viewed by 797

Abstract

This article studies diverse forms of lump-type solutions for coupled nonlinear generalized Zakharov equations (CNL-GZEs) in plasma physics through an appropriate transformation approach and bilinear equations. By utilizing the positive quadratic assumption in the bilinear equation, the lump-type solutions are derived. Similarly, by employing a single exponential transformation in the bilinear equation, the lump one-soliton solutions are derived. Furthermore, by choosing the double exponential ansatz in the bilinear equation, the lump two-soliton solutions are found. Interaction behaviors are observed and we also establish a few new solutions in various dimensions (3D and contour). Furthermore, we compute rogue-wave solutions and lump periodic solutions by employing proper hyperbolic and trigonometric functions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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11 pages, 295 KiB

Open AccessArticle

Global Boundedness in a Logarithmic Keller–Segel System

by Jinyang Liu, Boping Tian, Deqi Wang, Jiaxin Tang and Yujin Wu

Mathematics 2023, 11(12), 2743; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122743 - 16 Jun 2023

Viewed by 729

Abstract

In this paper, we propose a user-friendly integral inequality to study the coupled parabolic chemotaxis system with singular sensitivity under the Neumann boundary condition. Under a low diffusion rate, the classical solution of this system is uniformly bounded. Our proof replies on the [...] Read more.

\int_{Ω} \frac{{| v |}^{4}}{v^{2}}

with

v > 0

. It is noteworthy that the inequality used in the paper may be applied to study other chemotaxis systems. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

24 pages, 1031 KiB

Open AccessArticle

Analytical Solutions to the Chavy-Waddy–Kolokolnikov Model of Bacterial Aggregates in Phototaxis by Three Integration Schemes

by Alejandro León-Ramírez, Oswaldo González-Gaxiola and Guillermo Chacón-Acosta

Mathematics 2023, 11(10), 2352; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102352 - 18 May 2023

Cited by 5 | Viewed by 2312

Abstract

In this work, we find analytical solutions to the Chavy-Waddy–Kolokolnikov equation, a continuum approximation for modeling aggregate formation in bacteria moving toward the light, also known as phototaxis. We used three methods to obtain the solutions, the generalized Kudryashov method, the [...] Read more.

e^{- R (ξ)}

-expansion, and exponential function methods, all of them being very efficient for finding traveling wave-like solutions. Findings can be classified into the case where the nonlinear term can be considered a small perturbation of the linear case and the regime of instability and pattern formation. Standing waves and traveling fronts were also found among the physically interesting cases, in addition to recovering stationary spike-like solutions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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17 pages, 1630 KiB

Open AccessArticle

Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source

by R. Soundararajan, V. Subburayan and Patricia J. Y. Wong

Mathematics 2023, 11(9), 2034; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092034 - 25 Apr 2023

Cited by 1 | Viewed by 957

Abstract

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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25 pages, 2783 KiB

Open AccessArticle

Kinetics of a Reaction-Diffusion Mtb/SARS-CoV-2 Coinfection Model with Immunity

by Ali Algarni, Afnan D. Al Agha, Aisha Fayomi and Hakim Al Garalleh

Mathematics 2023, 11(7), 1715; https://0-doi-org.brum.beds.ac.uk/10.3390/math11071715 - 03 Apr 2023

Viewed by 1231

Abstract

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and Mycobacterium tuberculosis (Mtb) coinfection has been observed in a number of nations and it is connected with severe illness and death. The paper studies a reaction–diffusion within-host Mtb/SARS-CoV-2 coinfection model with immunity. This model explores the connections between uninfected epithelial cells, latently Mtb-infected epithelial cells, productively Mtb-infected epithelial cells, SARS-CoV-2-infected epithelial cells, free Mtb particles, free SARS-CoV-2 virions, and CTLs. The basic properties of the model’s solutions are verified. All equilibrium points with the essential conditions for their existence are calculated. The global stability of these equilibria is established by adopting compatible Lyapunov functionals. The theoretical outcomes are enhanced by implementing numerical simulations. It is found that the equilibrium points mirror the single infection and coinfection states of SARS-CoV-2 with Mtb. The threshold conditions that determine the movement from the monoinfection to the coinfection state need to be tested when developing new treatments for coinfected patients. The impact of the diffusion coefficients should be monitored at the beginning of coinfection as it affects the initial distribution of particles in space. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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21 pages, 411 KiB

Open AccessArticle

Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term

by Ben Mansour Dia, Mouhamadou Samsidy Goudiaby and Oliver Dorn

Mathematics 2022, 10(21), 4036; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214036 - 31 Oct 2022

Viewed by 976

Abstract

This paper treats a water flow regularization problem by means of local boundary conditions for the two-dimensional viscous shallow water equations. Using an a-priori energy estimate of the perturbation state and the Faedo–Galerkin method, we build a stabilizing boundary feedback control law for the volumetric flow in a finite time that is prescribed by the solvability of the associated Cauchy problem. We iterate the same approach to build by cascade a stabilizing feedback control law for infinite time. Thanks to a positive arbitrary time-dependent stabilization function, the control law provides an exponential decay of the energy. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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19 pages, 3145 KiB

Open AccessArticle

Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space

by V. Subburayan and S. Natesan

Mathematics 2022, 10(18), 3310; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183310 - 12 Sep 2022

Cited by 2 | Viewed by 1309

Abstract

Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problems. The proposed method is shown to be almost first-order convergent in the spatial direction and first-order convergent in the temporal direction. Theoretical results are illustrated with numerical examples. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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16 pages, 3927 KiB

Open AccessArticle

To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions

by Chein-Shan Liu, Chih-Wen Chang, Yung-Wei Chen and Jian-Hung Shen

Mathematics 2022, 10(17), 3112; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173112 - 30 Aug 2022

Viewed by 947

Abstract

In this paper, the numerical solutions of the backward and forward non-homogeneous wave problems are derived to address the nonlocal boundary conditions. When boundary conditions are not set on the boundaries, numerical instability occurs, and the solution may have a significant boundary error. For this reason, it is challenging to solve such nonlinear problems by conventional numerical methods. First, we derive a nonlocal boundary shape function (NLBSF) from incorporating the Pascal triangle as free functions; hence, the new, two-parameter Pascal bases are created to automatically satisfy the specified conditions for the solution. To satisfy the wave equation in the domain by the collocation method, the solution of the forward nonlocal wave problem can be quickly obtained with high precision. For the backward nonlocal wave problem, we construct the corresponding NLBSF and Pascal bases, which exactly implement two final time conditions, a left-boundary condition and a nonlocal boundary condition; in addition, the numerical method for the backward nonlocal wave problem under two-side, nonlocal boundary conditions is also developed. Nine numerical examples, including forward and backward problems, are tested, demonstrating that this scheme is more effective and stable. Even for boundary conditions with a large noise at final time, the solution recovered in the entire domain for the backward nonlocal wave problem is accurate and stable. The accuracy and efficiency of the method are validated by comparing the estimation results with the existing literature. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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14 pages, 333 KiB

Open AccessArticle

gL1 Scheme for Solving a Class of Generalized Time-Fractional Diffusion Equations

by Xuhao Li and Patricia J. Y. Wong

Mathematics 2022, 10(8), 1219; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081219 - 08 Apr 2022

Cited by 5 | Viewed by 1103

Abstract

In this paper, a numerical scheme based on a general temporal mesh is constructed for a generalized time-fractional diffusion problem of order

α

. The main idea involves the generalized linear interpolation and so we term the numerical scheme the gL1 scheme [...] Read more.

In this paper, a numerical scheme based on a general temporal mesh is constructed for a generalized time-fractional diffusion problem of order

α

. The main idea involves the generalized linear interpolation and so we term the numerical scheme the gL1 scheme. The stability and convergence of the numerical scheme are analyzed using the energy method. It is proven that the temporal convergence order is

(2 - α)

for a general temporal mesh. Simulation is carried out to verify the efficiency of the proposed numerical scheme. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

11 pages, 1771 KiB

Open AccessArticle

Effects of the Wiener Process on the Solutions of the Stochastic Fractional Zakharov System

by Farah M. Al-Askar, Wael W. Mohammed, Mohammad Alshammari and M. El-Morshedy

Mathematics 2022, 10(7), 1194; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071194 - 06 Apr 2022

Cited by 6 | Viewed by 1345

Abstract

We consider in this article the stochastic fractional Zakharov system derived by the multiplicative Wiener process in the Stratonovich sense. We utilize two distinct methods, the Riccati–Bernoulli sub-ODE method and Jacobi elliptic function method, to obtain new rational, trigonometric, hyperbolic, and elliptic stochastic solutions. The acquired solutions are helpful in explaining certain fascinating physical phenomena due to the importance of the Zakharov system in the theory of turbulence for plasma waves. In order to show the influence of the multiplicative Wiener process on the exact solutions of the Zakharov system, we employ the MATLAB tools to plot our figures to introduce a number of 2D and 3D graphs. We establish that the multiplicative Wiener process stabilizes the solutions of the Zakharov system around zero. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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19 pages, 364 KiB

Open AccessArticle

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C^∞ Semigroups

by Bienvenido Barraza Martínez, Jonathan González Ospino, Rogelio Grau Acuña and Jairo Hernández Monzón

Mathematics 2022, 10(5), 751; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050751 - 26 Feb 2022

Viewed by 1102

Abstract

We consider Fourier multiplier systems on

R^{n}

with components belonging to the standard Hörmander class

S_{1, 0}^{m} (R^{n})

, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector

Λ \subset C

(introduced by [...] Read more.

We consider Fourier multiplier systems on

R^{n}

with components belonging to the standard Hörmander class

S_{1, 0}^{m} (R^{n})

, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector

Λ \subset C

(introduced by Denk, Saal, and Seiler) we show the generation of both

C^{\infty}

semigroups and analytic semigroups (in a particular case) on the Sobolev spaces

W_{p}^{k} (R^{n}, C^{q})

with

k \in N_{0}

1 \leq p < \infty

and

q \in N

. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

15 pages, 458 KiB

Open AccessArticle

Analysis of Solutions, Asymptotic and Exact Profiles to an Eyring–Powell Fluid Modell

by José Luis Díaz, Saeed Ur Rahman, Juan Carlos Sánchez Rodríguez, María Antonia Simón Rodríguez, Guillermo Filippone Capllonch and Antonio Herrero Hernández

Mathematics 2022, 10(4), 660; https://0-doi-org.brum.beds.ac.uk/10.3390/math10040660 - 20 Feb 2022

Cited by 1 | Viewed by 1380

Abstract

The aim of this article was to provide analytical and numerical approaches to a one-dimensional Eyring–Powell flow. First of all, the regularity, existence, and uniqueness of the solutions were explored making use of a variational weak formulation. Then, the Eyring–Powell equation was transformed into the travelling wave domain, where analytical solutions were obtained supported by the geometric perturbation theory. Such analytical solutions were validated with a numerical exercise. The main finding reported is the existence of a particular travelling wave speed

a = 1.212

for which the analytical solution is close to the actual numerical solution with an accumulative error of <

10^{- 3}

. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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23 pages, 348 KiB

Open AccessArticle

Optimal Control Theory for a System of Partial Differential Equations Associated with Stratified Fluids

by Edgardo Alvarez, Hernan Cabrales and Tovias Castro

Mathematics 2021, 9(21), 2672; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212672 - 21 Oct 2021

Cited by 2 | Viewed by 1654

Abstract

In this paper, we investigate the existence of an optimal solution of a functional restricted to non-linear partial differential equations, which ruled the dynamics of viscous and incompressible stratified fluids in

R^{3}

. Additionally, we use the first derivative of the considered [...] Read more.

R^{3}

. Additionally, we use the first derivative of the considered functional to establish the necessary condition of the optimality for the optimal solution. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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