Differential Equations and Inverse Problems, 2nd Edition

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 October 2025 | Viewed by 2515

Special Issue Editors

College of Sciences, Northeastern University, Shenyang 110819, China
Interests: deep learning; reinforcement learning; multiscale methods (multigrid and wavelet); homotopy method; inverse and Ill-posed problems; parameter reconstruction
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Guest Editor
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Interests: structure-preserving algorithms for differential equations; numerical methods for stochastic differential equation
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Guest Editor
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
Interests: computational finance; iterative methods; computational mathematics; stochastic differential equations
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Guest Editor
School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China
Interests: nonlinear analysis; partial differential equations; exactly solvable systems

Special Issue Information

Dear Colleagues,

This Special Issue builds upon our previous Special Issue, “Differential Equations and Inverse Problems”. Differential equations and inverse problems have become a rapidly growing topic because of recent new techniques and amazing achievements in computational sciences. With the progress of science and technology, differential equations and inverse problems have quickly developed, and new waves have been successively set off in a broad range of disciplines, such as mathematics, physics, engineering, business, economics, earth science, and biology.

The purpose of this Special Issue is to gather contributions from experts on the theory and numerical aspects of differential equations and inverse problems, including, but not limited to, differential equations and fractional differential equations, initial value problems and related inverse problems, boundary value problems and related inverse problems, inverse problems in imaging, image reconstruction in tomography, stability analysis, regularization methods, novel numerical algorithms (such as multigrid methods, wavelet methods, homotopy methods, and structure-preserving methods), and artificial intelligence (such as deep learning and reinforcement learning). Moreover, we encourage submissions of their applications in various practical areas.

Dr. Tao Liu
Dr. Qiang Ma
Dr. Fazlollah Soleymani
Prof. Dr. Lifeng Guo
Guest Editors

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Keywords

  • partial differential equations
  • ordinary differential equations
  • stochastic differential equations
  • fractional differential equations
  • fractional calculus
  • inverse and ill-posed problems
  • imaging
  • image reconstruction
  • tomography
  • tomographic reconstruction
  • regularization methods
  • numerical methods
  • structure-preserving methods
  • minimax methods
  • artificial intelligence
  • deep learning
  • reinforcement learning
  • compact radial basis function approximations
  • approximations based on neural networks
  • node layouts for irregular domains
  • multi-asset option pricing problems
  • numerical methods for matrix functions
  • iterative methods for the solution of nonlinear equations
  • Schulz-type iterative method for generalized matrix inversion
  • biostatistics and pattern recognition
  • combination counting
  • separation theory
  • information security
  • existence of solutions
  • exact solutions
  • fractional Fourier transform and its applications
  • interdisciplinary application of mathematics and other disciplines

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Related Special Issue

Published Papers (3 papers)

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Research

13 pages, 4248 KiB  
Article
Efficient Newton-Type Solvers Based on for Finding the Solution of Nonlinear Algebraic Problems
by Haifa Bin Jebreen, Hongzhou Wang and Yurilev Chalco-Cano
Axioms 2024, 13(12), 880; https://doi.org/10.3390/axioms13120880 - 19 Dec 2024
Viewed by 716
Abstract
The purpose of this study is to improve the computational efficiency of solvers for nonlinear algebraic problems with simple roots. To this end, a multi-step solver based on Newton’s method is utilized. Divided difference operators are applied at two substeps in various forms [...] Read more.
The purpose of this study is to improve the computational efficiency of solvers for nonlinear algebraic problems with simple roots. To this end, a multi-step solver based on Newton’s method is utilized. Divided difference operators are applied at two substeps in various forms to enhance the convergence speed and, consequently, the solver’s efficiency index. Attraction basins for the proposed solvers and their competitors are presented, demonstrating that the proposed solvers exhibit large attraction basins in the scalar case while maintaining high convergence rates. Theoretical findings are supported by numerical experiments. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Edition)
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12 pages, 2211 KiB  
Article
Computing the Set of RBF-FD Weights Within the Integrals of a Kernel-Based Function and Its Applications
by Tao Liu, Bolin Ding and Stanford Shateyi
Axioms 2024, 13(12), 875; https://doi.org/10.3390/axioms13120875 - 17 Dec 2024
Viewed by 600
Abstract
This paper offers an approach to computing Radial Basis Function–Finite Difference (RBF-FD) weights by integrating a kernel-based function. We derive new weight sets that effectively approximate both the first and second differentiations of a function, demonstrating their utility in interpolation and the resolution [...] Read more.
This paper offers an approach to computing Radial Basis Function–Finite Difference (RBF-FD) weights by integrating a kernel-based function. We derive new weight sets that effectively approximate both the first and second differentiations of a function, demonstrating their utility in interpolation and the resolution of Partial Differential Equations (PDEs). Particularly, the paper evaluates the theoretical weights in interpolation tasks, highlighting the observed numerical orders, and further applies these weights to solve two distinct time-dependent PDE problems. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Edition)
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13 pages, 400 KiB  
Article
Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Pulse Vaccination Strategy
by Gui-Lai Zhang, Zhi-Yong Zhu, Lei-Ke Chen and Song-Shu Liu
Axioms 2024, 13(12), 854; https://doi.org/10.3390/axioms13120854 - 4 Dec 2024
Viewed by 679
Abstract
In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free [...] Read more.
In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free periodic solution of the impulsive linearly implicit Euler method are obtained. We further show that the limit of the disease-free periodic solution of the impulsive linearly implicit Euler method is the disease-free periodic solution of the exact solution when the step size tends to 0. Finally, two numerical experiments are given to confirm the conclusions. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Edition)
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