Symmetric Methods and Analysis for Differential and Integral Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 5089

Special Issue Editors

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Interests: convergence of numerical methods; diffusion; finite difference methods; iterative methods; numerical stability; partial differential equations
Department of Mathematics, National University of Singapore, Singapore, Singapore
Interests: fractional partial differential equation; machine learning; stochastic dynamical systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential and integral equations are widely used to model many natural phenomena in different scientific fields. The evolution of the solutions can be well learned by certain high-performance numerical methods. Well-designed and high-performance numerical methods can considerably reduce the computational costs of long-term simulations of real-world problems. They can also perform better in the prediction of mathematical models. Therefore, it is very important to develop effective numerical schemes as well as their rigorous numerical analysis.

In light of the aforementioned regarding the significance of numerical schemes and analysis, the potential topics for this issue include but are not limited to the following:

  • The construction of effective numerical methods for solving differential and integral equations;
  • The convergence analysis of symmetric numerical schemes;
  • The stability analysis of symmetric numerical schemes;
  • The dissipativity of symmetric numerical methods;
  • Iterative algorithms and their application.

Symmetric and structure-preserving numerical methods for differential and integral equations.

Prof. Dr. Dongfang Li
Dr. Hongyu Qin
Dr. Xiaoli Chen
Guest Editors

Manuscript Submission Information

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Keywords

  • differential equations
  • Integral equations
  • numerical methods
  • numerical analysis

Published Papers (4 papers)

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Research

17 pages, 725 KiB  
Article
A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems
by Chein-Shan Liu, Chung-Lun Kuo and Chih-Wen Chang
Symmetry 2024, 16(2), 191; https://doi.org/10.3390/sym16020191 - 06 Feb 2024
Viewed by 506
Abstract
In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered [...] Read more.
In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments. Full article
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13 pages, 1560 KiB  
Article
Improved TV Image Denoising over Inverse Gradient
by Minmin Li, Guangcheng Cai, Shaojiu Bi and Xi Zhang
Symmetry 2023, 15(3), 678; https://doi.org/10.3390/sym15030678 - 08 Mar 2023
Cited by 4 | Viewed by 1079
Abstract
Noise in an image can affect one’s extraction of image information, therefore, image denoising is an important image pre-processing process. Many of the existing models have a large number of estimated parameters, which increases the time complexity of the model solution and the [...] Read more.
Noise in an image can affect one’s extraction of image information, therefore, image denoising is an important image pre-processing process. Many of the existing models have a large number of estimated parameters, which increases the time complexity of the model solution and the achieved denoising effect is less than ideal. As a result, in this paper, an improved image-denoising algorithm is proposed based on the TV model, which effectively solves the above problems. The L1 regularization term can make the solution generated by the model sparser, thus facilitating the recovery of high-quality images. Reducing the number of estimated parameters, while using the inverse gradient to estimate the regularization parameters, enables the parameters to achieve global adaption and improves the denoising effect of the model in combination with the TV regularization term. The split Bregman iteration method is used to decouple the model into several related subproblems, and the solutions of the coordinated subproblems are derived as optimal solutions. It is also shown that the solution of the model converges to a Karush–Kuhn–Tucker point. Experimental results show that the algorithm in this paper is more effective in both preserving image texture structure and suppressing image noise. Full article
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18 pages, 951 KiB  
Article
Application of Aboodh Homotopy Perturbation Transform Method for Fractional-Order Convection–Reaction–Diffusion Equation within Caputo and Atangana–Baleanu Operators
by Humaira Yasmin
Symmetry 2023, 15(2), 453; https://doi.org/10.3390/sym15020453 - 08 Feb 2023
Cited by 5 | Viewed by 1395
Abstract
This article is an analysis of fractional nonlinear convection–reaction–diffusion equations involving the fractional Atangana–Baleanu and Caputo derivatives. An efficient Aboodh homotopy perturbation transform method, which combines the homotopy perturbation method with the Aboodh transformation, is applied to investigate this fractional-order proposed model, analytically. [...] Read more.
This article is an analysis of fractional nonlinear convection–reaction–diffusion equations involving the fractional Atangana–Baleanu and Caputo derivatives. An efficient Aboodh homotopy perturbation transform method, which combines the homotopy perturbation method with the Aboodh transformation, is applied to investigate this fractional-order proposed model, analytically. A modified technique known as the Aboodh homotopy perturbation transform method is formulated to approximate these derivatives. The analytical simulation is investigated graphically as well as in tabular form. Full article
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11 pages, 401 KiB  
Article
Evaluation of Fractional-Order Pantograph Delay Differential Equation via Modified Laguerre Wavelet Method
by Aisha Abdullah Alderremy, Rasool Shah, Nehad Ali Shah, Shaban Aly and Kamsing Nonlaopon
Symmetry 2022, 14(11), 2356; https://doi.org/10.3390/sym14112356 - 09 Nov 2022
Cited by 2 | Viewed by 1015
Abstract
Wavelet transforms or wavelet analysis represent a recently created mathematical tool for assistance in resolving various issues. Wavelets can also be used in numerical analysis. In this study, we solve pantograph delay differential equations using the Modified Laguerre Wavelet method (MLWM), an effective [...] Read more.
Wavelet transforms or wavelet analysis represent a recently created mathematical tool for assistance in resolving various issues. Wavelets can also be used in numerical analysis. In this study, we solve pantograph delay differential equations using the Modified Laguerre Wavelet method (MLWM), an effective numerical technique. Fractional derivatives are defined using the Caputo operator. The convergence of the suggested strategy is carefully examined. The suggested strategy is straightforward, effective, and simple in comparison with previous approaches. Specific examples are provided to demonstrate the current scenario’s reliability and accuracy. Compared with other methodologies, our results show a higher accuracy level. With the aid of tables and graphs, we demonstrate the effectiveness of the proposed approach by comparing results of the actual and suggested methods and demonstrating their strong agreement. For better understanding of the proposed method, we show the pointwise solution in the tables provided which confirm the accuracy at each point of the proposed method. Additionally, the results of employing the suggested method to various fractional-orders are compared, which demonstrates that when a value shifts from fractional-order to integer-order, the result approaches the exact solution. Owing to its novelty and scientific significance, the suggested technique can also be used to solve additional nonlinear delay differential equations of fractional-order. Full article
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