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Symmetry
Special Issues
Various Approaches for Generalized Integral Transforms
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Various Approaches for Generalized Integral Transforms

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

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 14248

Special Issue Editor

Prof. Dr. Hwa joon Kim

E-Mail Website
Guest Editor
Department of IT Engineering, Kyungdong University, Yangju 11458, Korea
Interests: mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The method of integral transforms is a tool to solve ordinary differential equations—a way to find analytical solutions of partial differential equations. The solutions of partial differential equations can be classified into analytical and numerical solutions. The analytical solutions can be found by the technique of separation of variables, the method of characteristics, integral transforms, change of variables, methods for non-linear equations such as the Cauchy–Kowalevski theorem, the split-step method, perturbation analysis, Lie group method, and semi-analytical methods such as homotopy analysis. The numerical solutions can be found by finite element methods, finite volume methods, finite difference methods, and meshfree methods. The method of integral transforms also gives a reasonable tool for solving engineering problems such as electrical networks, springs, signal processing, heat flow, and electrostatics.

Therefore, we would like to tackle new and interesting engineering problems in this Special Issue. We welcome research papers on generalized integral transforms including review articles describing the latest research trends.

Potential topics include but are not limited to the following:

  • New results on generalized integral transforms;
  • Laplace transforms, Fourier transforms, fractional Fourier transforms, linear canonical transforms, Riesz transforms, etc.;
  • Applications in artificial intelligence such as convolution theories and update of weights;
  • Wavelet and curvelet transforms;
  • Hypercomplex generalizations of integral transforms to complex vector spaces, quaternions, octonions, and Clifford algebra;
  • Applications in engineering;
  • Integral transforms involving hypergeometric functions;
  • Numerical methods and implementations involving integral transforms.

Please note that all submitted papers must be within the general scope of the Symmetry journal.

Prof. Dr. Hwa joon Kim
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. Symmetry is an international peer-reviewed open access monthly 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 2400 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.

Published Papers (10 papers)

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Research

15 pages, 298 KiB  
Article
The Solution of Coupled Burgers’ Equation by G-Laplace Transform
by Reem K. Alhefthi and Hassan Eltayeb
Symmetry 2023, 15(9), 1764; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15091764 - 15 Sep 2023
Viewed by 646
Abstract
The coupled Burgers’ equation is a fundamental partial differential equation with applications in various scientific fields. Finding accurate solutions to this equation is crucial for understanding physical phenomena and mathematical models. While different methods have been explored, this work highlights the importance of [...] Read more.
The coupled Burgers’ equation is a fundamental partial differential equation with applications in various scientific fields. Finding accurate solutions to this equation is crucial for understanding physical phenomena and mathematical models. While different methods have been explored, this work highlights the importance of the G-Laplace transform. The G-transform is effective in solving a wide range of non-constant coefficient differential equations, setting it apart from the Laplace, Sumudu, and Elzaki transforms. Consequently, it stands as a powerful tool for addressing differential equations characterized by variable coefficients. By applying this transformative approach, the study provides reliable and exact solutions for both homogeneous and non-homogeneous coupled Burgers’ equations. This innovative technique offers a valuable tool for gaining deeper insights into this equation’s behavior and significance in diverse disciplines. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
20 pages, 317 KiB  
Article
Application of the Double Sumudu-Generalized Laplace Transform Decomposition Method to Solve Singular Pseudo-Hyperbolic Equations
by Hassan Eltayeb
Symmetry 2023, 15(9), 1706; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15091706 - 06 Sep 2023
Cited by 1 | Viewed by 696
Abstract
In this study, the technique established by the double Sumudu transform in combination with a new generalized Laplace transform decomposition method, which is called the double Sumudu-generalized Laplace transform decomposition method, is applied to solve general two-dimensional singular pseudo-hyperbolic equations subject to the [...] Read more.
In this study, the technique established by the double Sumudu transform in combination with a new generalized Laplace transform decomposition method, which is called the double Sumudu-generalized Laplace transform decomposition method, is applied to solve general two-dimensional singular pseudo-hyperbolic equations subject to the initial conditions. The applicability of the proposed method is analyzed through demonstrative examples. The results obtained show that the procedure is easy to carry out and precise when employed for different linear and non-linear partial differential equations. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
13 pages, 276 KiB  
Article
Solution of Fractional Third-Order Dispersive Partial Differential Equations and Symmetric KdV via Sumudu–Generalized Laplace Transform Decomposition
by Hassan Eltayeb and Reem K. Alhefthi
Symmetry 2023, 15(8), 1540; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15081540 - 04 Aug 2023
Cited by 4 | Viewed by 727
Abstract
This research work introduces a novel method called the Sumudu–generalized Laplace transform decomposition method (SGLDM) for solving linear and nonlinear non-homogeneous dispersive Korteweg–de Vries (KdV)-type equations. The SGLDM combines the Sumudu–generalized Laplace transform with the Adomian decomposition method, providing a powerful approach to [...] Read more.
This research work introduces a novel method called the Sumudu–generalized Laplace transform decomposition method (SGLDM) for solving linear and nonlinear non-homogeneous dispersive Korteweg–de Vries (KdV)-type equations. The SGLDM combines the Sumudu–generalized Laplace transform with the Adomian decomposition method, providing a powerful approach to tackle complex equations. To validate the efficacy of the method, several model problems of dispersive KdV-type equations are solved, and the resulting approximate solutions are expressed in series form. The findings demonstrate that the SGLDM is a reliable and robust method for addressing significant physical problems in various applications. Finally, we conclude that this transform is a symmetry to other symmetric transforms. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
16 pages, 428 KiB  
Article
On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method
by Kamran, Sharif Ullah Khan, Salma Haque and Nabil Mlaiki
Symmetry 2023, 15(6), 1214; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15061214 - 07 Jun 2023
Cited by 4 | Viewed by 1212
Abstract
Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional [...] Read more.
Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional order is developed. The analytic inversion can be very difficult for complex forms of the transform function. Therefore, numerical methods are used for the inversion of the Laplace transform. In general, the numerical inverse Laplace transform is an ill-posed problem. This difficulty has led to various numerical methods for the inversion of the Laplace transform. In this work, the Weeks method is utilized for the numerical inversion of the Laplace transform. In our proposed numerical method, first, the fractional-order differential equation is converted to an algebraic equation using Laplace transform. Then, the transformed equation is solved in Laplace space using algebraic techniques. Finally, the Weeks method is utilized for the inversion of the Laplace transform. Weeks method is one of the most efficient numerical methods for the computation of the inverse Laplace transform. We have considered five test problems for validation of the proposed numerical method. Based on the comparison between analytical results and the Weeks method results, the reliability and effectiveness of the Weeks method for fractional-order differential equations was confirmed. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
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11 pages, 288 KiB  
Article
More on the Unified Mittag–Leffler Function
by Chahnyong Jung, Ghulam Farid, Hafsa Yasmeen and Kamsing Nonlaopon
Symmetry 2022, 14(3), 523; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030523 - 03 Mar 2022
Cited by 1 | Viewed by 1416
Abstract
Symmetry is a fascinating property of numerous mathematical notions. In mathematical analysis a function f:[a,b]→R symmetric about a+b2 satisfies the equation [...] Read more.
Symmetry is a fascinating property of numerous mathematical notions. In mathematical analysis a function f:[a,b]→R symmetric about a+b2 satisfies the equation f(a+b−x)=f(x). In this paper, we investigate the relationship of unified Mittag–Leffler function with some known special functions. We have obtained some integral transforms of unified Mittag–Leffler function in terms of Wright generalized function. We also established a recurrence relation along with another important result. Furthermore, we give formulas of Riemann–Liouville fractional integrals and fractional integrals containing unified Mittag–Leffler function for symmetric functions. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
9 pages, 261 KiB  
Article
Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform
by Daniela Inoan and Daniela Marian
Symmetry 2021, 13(11), 2181; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112181 - 15 Nov 2021
Cited by 10 | Viewed by 1417
Abstract
In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: [...] Read more.
In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
21 pages, 370 KiB  
Article
Certain Applications of Generalized Kummer’s Summation Formulas for 2F1
by Junesang Choi
Symmetry 2021, 13(8), 1538; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081538 - 21 Aug 2021
Cited by 13 | Viewed by 1821
Abstract
We present generalizations of three classical summation formulas 2F1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number [...] Read more.
We present generalizations of three classical summation formulas 2F1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number of interesting formulas for double-finite series involving quotients of Gamma functions. We also consider several other applications of these formulas. Certain symmetries occur often in mathematical formulae and identities, both explicitly and implicitly. As an example, as mentioned in Remark 1, evident symmetries are naturally implicated in the treatment of generalized hypergeometric series. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
14 pages, 1864 KiB  
Article
The Rheological Analytical Solution and Parameter Inversion of Soft Soil Foundation
by Heng Zhang, Chao Su, Jiawei Bai, Rongyao Yuan, Yujun Ma and Wenjun Wang
Symmetry 2021, 13(7), 1228; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071228 - 08 Jul 2021
Cited by 2 | Viewed by 1451
Abstract
In soft soil engineering projects, the building loads are always required to be symmetrically distributed on the surface of the foundation to prevent uneven settlement. Even if the buildings and soft clay are controlled by engineers, it can still lead to the rheology [...] Read more.
In soft soil engineering projects, the building loads are always required to be symmetrically distributed on the surface of the foundation to prevent uneven settlement. Even if the buildings and soft clay are controlled by engineers, it can still lead to the rheology of the foundation. The analytical solution based on the Laplace integral transformation method has positive significance for providing a simple and highly efficient way to solve engineering problems, especially in the long-term uneven settlement deformation prediction of buildings on soft soil foundations. This paper proposes an analytical solution to analyze the deformation of soft soil foundations. The methodology is based on calculus theory, Laplace integral transformation, and viscoelastic theory. It combines an analytical solution with finite theory to solve the construction sequences and loading processes. In addition, an improved quantum genetic algorithm is put forward to inverse the parameters of soft soil foundations. The analytical solution based on Laplace integral transformation is validated through an engineering case. The results clearly illustrate the accuracy of the method. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
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17 pages, 285 KiB  
Article
Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities
by Michael Th. Rassias, Bicheng Yang and Andrei Raigorodskii
Symmetry 2021, 13(6), 1006; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061006 - 04 Jun 2021
Cited by 4 | Viewed by 1715
Abstract
In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of [...] Read more.
In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
22 pages, 374 KiB  
Article
On (p,q)-Analogues of Laplace-Typed Integral Transforms and Applications
by Sansumpan Jirakulchaiwong, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas and Hwajoon Kim
Symmetry 2021, 13(4), 631; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13040631 - 09 Apr 2021
Cited by 3 | Viewed by 1508
Abstract
In this paper, we establish (p,q)-analogues of Laplace-type integral transforms by using the concept of (p,q)-calculus. Moreover, we study some properties of (p,q)-analogues of Laplace-type integral transforms and [...] Read more.
In this paper, we establish (p,q)-analogues of Laplace-type integral transforms by using the concept of (p,q)-calculus. Moreover, we study some properties of (p,q)-analogues of Laplace-type integral transforms and apply them to solve some (p,q)-differential equations. Full article
(This article belongs to the Special Issue Various Approaches for Generalized Integral Transforms)
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