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Mathematics
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Partial Differential Equation Theory and Its Applications
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Partial Differential Equation Theory and Its Applications

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

Deadline for manuscript submissions: closed (30 June 2024) | Viewed by 7566

Special Issue Editor

Prof. Dr. Zheng-An Yao

E-Mail Website
Guest Editor
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
Interests: partial differential equation theory and its application; computer and communication; fluid mechanics; complex composite materials; numerical solution of partial differential equations; computer and network; information security; image processing; data analysis

Special Issue Information

Dear Colleagues,

In mathematics, a partial differential equation (PDE) is an equation that imposes relations between the various partial derivatives of a multivariable function. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. In addition, partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise.

The Special Issue aims to publish in as much detail as possible scientific advances in the field of partial differential equation theory, analytical and numerical solutions, and their applications. Both original research and review papers are encouraged.

Prof. Dr. Zheng-An Yao
Guest Editor

Manuscript Submission Information

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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

  • partial defferential equations
  • homogenization
  • porous nedium
  • computer and network

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Published Papers (6 papers)

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Research

14 pages, 263 KiB  
Article
Blow-Up Phenomena for a Non-Newton Filtration Equation with Local Linear Boundary Dissipation
by Xinru Zhou and Dengming Liu
Mathematics 2024, 12(13), 2028; https://doi.org/10.3390/math12132028 - 29 Jun 2024
Viewed by 695
Abstract
In this article, we consider the finite time blow-up phenomenon for a class of non-Newton filtration equations with local linear boundary dissipation. Using a modified concavity method, the sufficient condition for the finite time blow-up is given. Furthermore, we obtain the result of [...] Read more.
In this article, we consider the finite time blow-up phenomenon for a class of non-Newton filtration equations with local linear boundary dissipation. Using a modified concavity method, the sufficient condition for the finite time blow-up is given. Furthermore, we obtain the result of the blow-up solution with arbitrary high initial energy. Meanwhile, the upper and lower estimates of the blow-up time are discussed. Full article
(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)
21 pages, 352 KiB  
Article
Global Solution and Stability of a Haptotaxis Mathematical Model for Complex MAP
by Hongbing Chen and Fengling Jia
Mathematics 2024, 12(7), 1116; https://doi.org/10.3390/math12071116 - 8 Apr 2024
Viewed by 974
Abstract
A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired [...] Read more.
A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system (4) has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system (3) from Lρ to L2ρ (ρ≥1) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N≤3. Lastly, by using various analytical methods, a threshold value ξ00(ξ00<0) is found, such that positive constant steady state (u∗,v∗,w∗) becomes unstable when ξ<ξ00. Our results show that the haptotaxis plays a crucial role in determining the stability to the model (3), that is, it can have a destabilizing effect. Full article
(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)
14 pages, 329 KiB  
Article
The Well-Posed Identification of the Interface Heat Transfer Coefficient Using an Inverse Heat Conduction Model
by Sergey Pyatkov and Alexey Potapkov
Mathematics 2023, 11(23), 4739; https://doi.org/10.3390/math11234739 - 23 Nov 2023
Viewed by 826
Abstract
In this study, the inverse problems of recovering the heat transfer coefficient at the interface of integral measurements are considered. The heat transfer coefficient occurs in the transmission conditions of an imperfect contact type. This is representable as a finite part of the [...] Read more.
In this study, the inverse problems of recovering the heat transfer coefficient at the interface of integral measurements are considered. The heat transfer coefficient occurs in the transmission conditions of an imperfect contact type. This is representable as a finite part of the Fourier series with time-dependent coefficients. The additional measurements are integrals of a solution multiplied by some weights. The existence and uniqueness of solutions in Sobolev classes are proven and the conditions on the data are sharp. These conditions include smoothness and consistency conditions on the data and additional conditions on the kernels of the integral operators used in the additional measurements. The proof relies on a priori bounds and the contraction mapping principle. The existence and uniqueness theorem is local in terms of time. Full article
(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)
10 pages, 260 KiB  
Article
Exact Null Controllability of a Wave Equation with Dirichlet–Neumann Boundary in a Non-Cylindrical Domain
by Lizhi Cui and Jing Lu
Mathematics 2023, 11(15), 3331; https://doi.org/10.3390/math11153331 - 29 Jul 2023
Viewed by 1241
Abstract
In this paper, by applying the Hilbert Uniqueness Method in a non-cylindrical domain, we prove the exact null controllability of one wave equation with a moving boundary. The moving endpoint of this wave equation has a Neumann-type boundary condition, while the fixed endpoint [...] Read more.
In this paper, by applying the Hilbert Uniqueness Method in a non-cylindrical domain, we prove the exact null controllability of one wave equation with a moving boundary. The moving endpoint of this wave equation has a Neumann-type boundary condition, while the fixed endpoint has a Dirichlet boundary condition. We derived the exact null controllability and obtained an exact controllability time of the wave equation. Full article
(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)
14 pages, 337 KiB  
Article
Similarity Solution for a System of Fractional-Order Coupled Nonlinear Hirota Equations with Conservation Laws
by Musrrat Ali, Hemant Gandhi, Amit Tomar and Dimple Singh
Mathematics 2023, 11(11), 2465; https://doi.org/10.3390/math11112465 - 26 May 2023
Cited by 1 | Viewed by 1213
Abstract
The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic framework for the establishment [...] Read more.
The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic framework for the establishment of invariant solutions of initial value and boundary value problems, and for the deduction of laws of conservations. This article is aimed at applying Lie symmetry to the fractional-order coupled nonlinear complex Hirota system of partial differential equations. This system is reduced to nonlinear fractional ordinary differential equations (FODEs) by using symmetries and explicit solutions. The reduced equations are exhibited in the form of an Erdelyi–Kober fractional (E-K) operator. The series solution of the fractional-order system and its convergence is investigated. Noether’s theorem is used to devise conservation laws. Full article
(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)
19 pages, 1052 KiB  
Article
Identifying a Space-Dependent Source Term and the Initial Value in a Time Fractional Diffusion-Wave Equation
by Xianli Lv and Xiufang Feng
Mathematics 2023, 11(6), 1521; https://doi.org/10.3390/math11061521 - 21 Mar 2023
Cited by 2 | Viewed by 1664
Abstract
This paper is focused on the inverse problem of identifying the space-dependent source function and initial value of the time fractional nonhomogeneous diffusion-wave equation from noisy final time measured data in a multi-dimensional case. A mollification regularization method based on a bilateral exponential [...] Read more.
This paper is focused on the inverse problem of identifying the space-dependent source function and initial value of the time fractional nonhomogeneous diffusion-wave equation from noisy final time measured data in a multi-dimensional case. A mollification regularization method based on a bilateral exponential kernel is presented to solve the ill-posedness of the problem for the first time. Error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical experiments of interest show that our proposed method is effective and robust with respect to the perturbation noise in the data. Full article
(This article belongs to the Special Issue Partial Differential Equation Theory and Its Applications)
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