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Mathematics
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Fractional Partial Differential Equations: Theory and Applications
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Fractional Partial Differential Equations: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E6: Functional Interpolation".

Deadline for manuscript submissions: closed (31 July 2024) | Viewed by 6578

Special Issue Editor

Dr. Harendra Singh

E-Mail Website
Guest Editor
Department of Mathematics, Post-Graduate College, Ghazipur 233001, India
Interests: fractional partial differential equations; numerical methods

Special Issue Information

Dear Colleagues,

The main objective of the proposal is to collect papers in the area of fractional partial differential equations. This aim of this issue is to encourage scientific and informative exercises by promoting the theory and application of fractional differential equations linked to the hurdles of experimental research. Papers will be invited to distinguish significant research subjects in identified regions in the fields of applied sciences, engineering, and technology. We will mainly focus on application-based papers in different areas of sciences. These results are immensely helpful for engineers, mathematicians, scientists, and researchers working in this area.

Dr. Harendra Singh
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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

  • analytical methods
  • numerical methods
  • partial differential equations
  • fractional models
  • biological models
  • computer applications in mathematics
  • interdisciplinary research

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Published Papers (3 papers)

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Research

23 pages, 9910 KiB  
Article
Image Restoration with Fractional-Order Total Variation Regularization and Group Sparsity
by Jameel Ahmed Bhutto, Asad Khan and Ziaur Rahman
Mathematics 2023, 11(15), 3302; https://doi.org/10.3390/math11153302 - 27 Jul 2023
Cited by 9 | Viewed by 2092
Abstract
In this paper, we present a novel image denoising algorithm, specifically designed to effectively restore both the edges and texture of images. This is achieved through the use of an innovative model known as the overlapping group sparse fractional-order total variation regularization model [...] Read more.
In this paper, we present a novel image denoising algorithm, specifically designed to effectively restore both the edges and texture of images. This is achieved through the use of an innovative model known as the overlapping group sparse fractional-order total variation regularization model (OGS-FOTVR). The OGS-FOTVR model ingeniously combines the benefits of the fractional-order (FO) variation domain with an overlapping group sparsity measure, which acts as its regularization component. This is further enhanced by the inclusion of the well-established L2-norm, which serves as the fidelity term. To simplify the model, we employ the alternating direction method of multipliers (ADMM), which breaks down the model into a series of more manageable sub-problems. Each of these sub-problems can then be addressed individually. However, the sub-problem involving the overlapping group sparse FO regularization presents a high level of complexity. To address this, we construct an alternative function for this sub-problem, utilizing the mean inequality principle. Subsequently, we employ the majorize-minimization (MM) algorithm to solve it. Empirical results strongly support the effectiveness of the OGS-FOTVR model, demonstrating its ability to accurately recover texture and edge information in images. Notably, the model performs better than several advanced variational alternatives, as indicated by superior performance metrics across three image datasets, PSNR, and SSIM. Full article
(This article belongs to the Special Issue Fractional Partial Differential Equations: Theory and Applications)
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13 pages, 531 KiB  
Article
Analytical Solution of the Local Fractional KdV Equation
by Kholoud Saad Albalawi, Ibtehal Alazman, Jyoti Geetesh Prasad and Pranay Goswami
Mathematics 2023, 11(4), 882; https://doi.org/10.3390/math11040882 - 9 Feb 2023
Cited by 2 | Viewed by 1666
Abstract
This research work is dedicated to solving the n-generalized Korteweg–de Vries (KdV) equation in a fractional sense. The method is a combination of the Sumudu transform and the Adomian decomposition method. This method has significant advantages for solving differential equations that are both [...] Read more.
This research work is dedicated to solving the n-generalized Korteweg–de Vries (KdV) equation in a fractional sense. The method is a combination of the Sumudu transform and the Adomian decomposition method. This method has significant advantages for solving differential equations that are both linear and nonlinear. It is easy to find the solutions to fractional-order PDEs with less computing labor. Full article
(This article belongs to the Special Issue Fractional Partial Differential Equations: Theory and Applications)
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17 pages, 497 KiB  
Article
Computational Analysis of the Fractional Riccati Differential Equation with Prabhakar-type Memory
by Jagdev Singh, Arpita Gupta and Devendra Kumar
Mathematics 2023, 11(3), 644; https://doi.org/10.3390/math11030644 - 27 Jan 2023
Cited by 17 | Viewed by 1924
Abstract
The key objective of the current work is to examine the behavior of the nonlinear fractional Riccati differential equation associated with the Caputo–Prabhakar derivative. An efficient computational scheme, that is, a mixture of homotopy analysis technique and sumudu transform, is used to solve [...] Read more.
The key objective of the current work is to examine the behavior of the nonlinear fractional Riccati differential equation associated with the Caputo–Prabhakar derivative. An efficient computational scheme, that is, a mixture of homotopy analysis technique and sumudu transform, is used to solve the nonlinear fractional Riccati differential equation. The convergence and uniqueness analysis for the solution of the implemented technique is shown. In addition, the numerical consequences are demonstrated in the form of graphical representations to verify the reliability of the applied method in obtaining the solution to the mathematical model with Prabhakar-type memory. Full article
(This article belongs to the Special Issue Fractional Partial Differential Equations: Theory and Applications)
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