Differential Equations and Inverse Problems

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

Deadline for manuscript submissions: 28 August 2024 | Viewed by 5118

Special Issue Editors

1. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, China
2. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore
Interests: inverse and ill-posed problems; partial differential equations; mathematical physics; financial mathematics; mathematical imaging; artificial intelligence; deep learning; reinforcement learning; multiscale methods; homotopy methods
Prof. Dr. Qiang Ma
E-Mail Website
Guest Editor
Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
Interests: structure-preserving algorithms for differential equations; numerical methods for stochastic differential equation
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, China
Interests: ill-posed problems; regularization method; inverse source problems; backward problems; parabolic equation; elliptic equation; fractional diffusion equation; convergence analysis

Special Issue Information

Dear Colleagues,

Differential equations and inverse problems have become a rapidly growing topic because of the new techniques developed recently 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, biology, etc. 

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, structure-preserving methods), and artificial intelligence (such as deep learning, reinforcement learning). Moreover, we encourage submissions of their applications in various practical areas. 

Dr. Tao Liu
Dr. Qiang Ma
Dr. Songshu Liu
Guest Editors

Manuscript Submission Information

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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
  • artificial intelligence
  • deep learning
  • reinforcement learning

Published Papers (7 papers)

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Research

12 pages, 268 KiB  
Article
On a Backward Problem for the Rayleigh–Stokes Equation with a Fractional Derivative
Axioms 2024, 13(1), 30; https://doi.org/10.3390/axioms13010030 - 30 Dec 2023
Viewed by 674
Abstract
The Rayleigh–Stokes equation with a fractional derivative is widely used in many fields. In this paper, we consider the inverse initial value problem of the Rayleigh–Stokes equation. Since the problem is ill-posed, we adopt the Tikhonov regularization method to solve this problem. In [...] Read more.
The Rayleigh–Stokes equation with a fractional derivative is widely used in many fields. In this paper, we consider the inverse initial value problem of the Rayleigh–Stokes equation. Since the problem is ill-posed, we adopt the Tikhonov regularization method to solve this problem. In addition, this paper not only analyzes the ill-posedness of the problem but also gives the conditional stability estimate. Finally, the convergence estimates are proved under two regularization parameter selection rules. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
22 pages, 538 KiB  
Article
Asymptotical Stability Criteria for Exact Solutions and Numerical Solutions of Nonlinear Impulsive Neutral Delay Differential Equations
Axioms 2023, 12(10), 988; https://doi.org/10.3390/axioms12100988 - 18 Oct 2023
Viewed by 652
Abstract
In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is [...] Read more.
In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is also constructed based on the idea. The numerical methods that can preserve the stability and asymptotical stability of the exact solutions are provided. Two numerical examples are provided to demonstrate the theoretical results. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
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15 pages, 337 KiB  
Article
Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel
Axioms 2023, 12(9), 898; https://doi.org/10.3390/axioms12090898 - 21 Sep 2023
Viewed by 576
Abstract
In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method [...] Read more.
In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
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14 pages, 3289 KiB  
Article
An Efficient Convolutional Neural Network with Supervised Contrastive Learning for Multi-Target DOA Estimation in Low SNR
Axioms 2023, 12(9), 862; https://doi.org/10.3390/axioms12090862 - 07 Sep 2023
Viewed by 744
Abstract
In this paper, a modified high-efficiency Convolutional Neural Network (CNN) with a novel Supervised Contrastive Learning (SCL) approach is introduced to estimate direction-of-arrival (DOA) of multiple targets in low signal-to-noise ratio (SNR) regimes with uniform linear arrays (ULA). The model is trained using [...] Read more.
In this paper, a modified high-efficiency Convolutional Neural Network (CNN) with a novel Supervised Contrastive Learning (SCL) approach is introduced to estimate direction-of-arrival (DOA) of multiple targets in low signal-to-noise ratio (SNR) regimes with uniform linear arrays (ULA). The model is trained using an on-grid setting, and thus the problem is modeled as a multi-label classification task. Simulation results demonstrate the robustness of the proposed approach in scenarios with low SNR and a small number of snapshots. Notably, the method exhibits strong capability in detecting the number of sources while estimating their DOAs. Furthermore, compared to traditional CNN methods, our refined efficient CNN significantly reduces the number of parameters by a factor of sixteen while still achieving comparable results. The effectiveness of the proposed method is analyzed through the visualization of latent space and through the advanced theory of feature learning. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
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13 pages, 304 KiB  
Article
Positive Solutions for Periodic Boundary Value Problems of Fractional Differential Equations with Sign-Changing Nonlinearity and Green’s Function
Axioms 2023, 12(9), 819; https://doi.org/10.3390/axioms12090819 - 26 Aug 2023
Viewed by 560
Abstract
In this paper, a class of nonlinear fractional differential equations with periodic boundary condition is investigated. Although the nonlinearity of the equation and the Green’s function are sign-changing, the results of the existence and nonexistence of positive solutions are obtained by using the [...] Read more.
In this paper, a class of nonlinear fractional differential equations with periodic boundary condition is investigated. Although the nonlinearity of the equation and the Green’s function are sign-changing, the results of the existence and nonexistence of positive solutions are obtained by using the Schaefer’s fixed-point theorem. Finally, two examples are given to illustrate the main results. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
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14 pages, 2367 KiB  
Article
The Laplace Transform Shortcut Solution to a One-Dimensional Heat Conduction Model with Dirichlet Boundary Conditions
Axioms 2023, 12(8), 770; https://doi.org/10.3390/axioms12080770 - 09 Aug 2023
Viewed by 895
Abstract
When using the Laplace transform to solve a one-dimensional heat conduction model with Dirichlet boundary conditions, the integration and transformation processes become complex and cumbersome due to the varying properties of the boundary function f(t). Meanwhile, if f(t [...] Read more.
When using the Laplace transform to solve a one-dimensional heat conduction model with Dirichlet boundary conditions, the integration and transformation processes become complex and cumbersome due to the varying properties of the boundary function f(t). Meanwhile, if f(t) has a complex functional form, e.g., an exponential decay function, the product of the image function of the Laplace transform and the general solution to the model cannot be obtained directly due to the difficulty in solving the inverse. To address this issue, operators are introduced to replace f(t) in the transformation process. Based on the properties of the Laplace transform and the convolution theorem, without the direct involvement of f(t) in the transformation, a general theoretical solution incorporating f(t) is derived, which consists of the product of erfc(t) and f(0), as well as the convolution of erfc(t) and the derivative of f(t). Then, by substituting f(t) into the general theoretical solution, the corresponding analytical solution is formulated. Based on the general theoretical solution, analytical solutions are given for f(t) as a commonly used function. Finally, combined with an exemplifying application demonstration based on the test data of temperature T(x, t) at point x away from the boundary and the characteristics of curve T(x, t) − t and curve 𝜕T(x, t)/𝜕tt, the inflection point and curve fitting methods are established for the inversion of model parameters. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
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14 pages, 327 KiB  
Article
Convergence of Collocation Methods for One Class of Impulsive Delay Differential Equations
Axioms 2023, 12(7), 700; https://doi.org/10.3390/axioms12070700 - 19 Jul 2023
Cited by 2 | Viewed by 521
Abstract
This paper is concerned with collocation methods for one class of impulsive delay differential equations (IDDEs). Some results for the convergence, global superconvergence and local superconvergence of collocation methods are given. We choose a suitable piecewise continuous collocation space to obtain high-order numerical [...] Read more.
This paper is concerned with collocation methods for one class of impulsive delay differential equations (IDDEs). Some results for the convergence, global superconvergence and local superconvergence of collocation methods are given. We choose a suitable piecewise continuous collocation space to obtain high-order numerical methods. Some illustrative examples are given to verify the theoretical results. Full article
(This article belongs to the Special Issue Differential Equations and Inverse Problems)
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