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MCA
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Symmetry Methods for Solving Differential Equations
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Symmetry Methods for Solving Differential Equations

A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (26 July 2024) | Viewed by 6769

Special Issue Editor

Prof. Dr. Mehmet Pakdemirli

E-Mail Website
Guest Editor
Professor Emeritus, Department of Mechanical Engineering, Manisa Celal Bayar University, 45140 Manisa, Turkey
Interests: perturbation methods; perturbation-iteration algorithms; symmetries of differential equations; approximate symmetries; analytical and numerical solutions of differential equations; root-finding algorithms; nonlinear vibrations; non-Newtonian fluid mechanics; nonlinear dynamics; heat transfer; mathematical education

Special Issue Information

Dear Colleagues,

Symmetry Analysis is a systematic method of solving differential equations which has been widely applied to many mathematical models in search of analytical solutions. The results of ad-hoc methods can be combined and classified within the context of symmetries of differential equations.

The aim of this Special Issue is to collect high-quality work and provide a dissemination of recent results on the topic. Lie Group Theory, Noether Symmetries, and the Exterior Calculus approach are widely used symmetry methods. Contributions to the development of these methods are within the scope of this Special Issue. Studies on new theories combining symmetry with perturbation methods, such as the approximate symmetry methods, are welcome. Classical Lie Point Symmetries, Equivalence Transformations, Group Classifications, Non-Classical Symmetries, and Lie–Backlund Symmetries are other techniques that may be considered. Papers employing special group transformations (Scaling, Translational, Spiral), as well as other similarity transformations, are also acceptable. Papers on symmetries should address applications of the method to solving ordinary or partial differential equations. Papers that employ methods for solving applied problems in Physics, Chemistry, Biology, Engineering, Administrative Sciences, and other social sciences in the form of differential equations are highly encouraged. Abstract Lie Group Theory papers without any evidence of application to differential equations are discouraged.

Prof. Dr. Mehmet Pakdemirli
Guest Editor

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Published Papers (7 papers)

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Research

24 pages, 6455 KiB  
Article
Using Artificial Neural Network Analysis to Study Jeffrey Nanofluid Flow in Cone–Disk Systems
by Nasser Nammas Albaqami
Math. Comput. Appl. 2024, 29(6), 98; https://doi.org/10.3390/mca29060098 - 31 Oct 2024
Viewed by 432
Abstract
Artificial intelligence (AI) is employed in fluid flow models to enhance the simulation’s accuracy, to more effectively optimize the fluid flow models, and to realize reliable fluid flow systems with improved performance. Jeffery fluid flow through the interstice of a cone-and-disk system is [...] Read more.
Artificial intelligence (AI) is employed in fluid flow models to enhance the simulation’s accuracy, to more effectively optimize the fluid flow models, and to realize reliable fluid flow systems with improved performance. Jeffery fluid flow through the interstice of a cone-and-disk system is considered in this study. The mathematical description of this flow involves converting a partial differential system into a nonlinear ordinary differential system and solving it using a neurocomputational technique. The fluid streaming through the disk–cone gap is investigated under four contrasting frameworks, i.e., (i) passive cone and spinning disk, (ii) spinning cone and passive disk, (iii) cone and disk rotating in the same direction, and (iv) cone and disk rotating in opposite directions. Employing the recently developed technique of artificial neural networks (ANNs) can be effective for handling and optimizing fluid flow exploits. The proposed approach integrates training, testing and analysis, and authentication based on a locus dataset to address various aspects of fluid problems. The mean square error, regression plots, curve-fitting graphs, and error histograms are used to evaluate the performance of the least mean square neural network algorithm (LMS-NNA). The results show that these equations are consistently aligned, and agreement is, on average, in the order of 10−8. While the resting parameters were kept static, the transverse velocity distribution, in all four cases, exhibited an incremental decreasing behavior in the estimates of magnetic and Jeffery fluid factors. Furthermore, the results obtained were compared with those in the literature, and the close agreement confirms our results. To train the model, 80% of the data were used for LMS-NNA, with 10% used for testing and the remaining 10% for validation. The quantitative and qualitative outputs obtained from the neural network strategy and parameter variation were thoroughly examined and discussed. Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
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17 pages, 293 KiB  
Article
Lie Symmetry Analysis, Closed-Form Solutions, and Conservation Laws for the Camassa–Holm Type Equation
by Jonathan Lebogang Bodibe and Chaudry Masood Khalique
Math. Comput. Appl. 2024, 29(5), 92; https://doi.org/10.3390/mca29050092 - 10 Oct 2024
Viewed by 573
Abstract
In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim [...] Read more.
In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim is to construct closed-form solutions of the equation using a powerful technique, namely the Lie group analysis method. Firstly, we derive the Lie point symmetries of the equation. Thereafter, the equation is reduced to non-linear ordinary differential equations using symmetry reductions. Furthermore, the solutions of the equation are derived using the extended Jacobi elliptic function technique, the simplest equation method, and the power series method. In conclusion, we construct conservation laws for the equation using Noether’s theorem and the multiplier approach, which plays a crucial role in understanding the behavior of non-linear equations, especially in physics and engineering, and these laws are derived from fundamental principles such as the conservation of mass, energy, momentum, and angular momentum. Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
35 pages, 970 KiB  
Article
Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation
by Willy Hereman and Ünal Göktaş
Math. Comput. Appl. 2024, 29(5), 91; https://doi.org/10.3390/mca29050091 - 3 Oct 2024
Viewed by 601
Abstract
In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the [...] Read more.
In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems. Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
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22 pages, 1643 KiB  
Article
Periodic and Axial Perturbations of Chaotic Solitons in the Realm of Complex Structured Quintic Swift-Hohenberg Equation
by Naveed Iqbal, Wael W. Mohammed, Mohammad Alqudah, Amjad E. Hamza and Shah Hussain
Math. Comput. Appl. 2024, 29(5), 86; https://doi.org/10.3390/mca29050086 - 30 Sep 2024
Viewed by 613
Abstract
This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems [...] Read more.
This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it. Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
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10 pages, 244 KiB  
Article
Noether Symmetries of the Triple Degenerate DNLS Equations
by Ugur Camci
Math. Comput. Appl. 2024, 29(4), 60; https://doi.org/10.3390/mca29040060 - 30 Jul 2024
Viewed by 662
Abstract
In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether [...] Read more.
In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether symmetries of the second-order Lagrangian density for these equations using the Noether symmetry approach with a gauge term. For this Lagrangian density, we compute the conserved densities and fluxes corresponding to the Noether symmetries with a gauge term, which differ from the conserved densities obtained using Lie symmetries in Webb et al. (J. Plasma Phys. 1995, 54, 201–244; J. Phys. A Math. Gen. 1996, 29, 5209–5240). Furthermore, we find some new Lie symmetries of the dispersive triple degenerate derivative nonlinear Schrödinger equations for non-vanishing integration functions Ki(t) (i=1,2,3). Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
13 pages, 273 KiB  
Article
Complex Connections between Symmetry and Singularity Analysis
by Asghar Qadir
Math. Comput. Appl. 2024, 29(1), 15; https://doi.org/10.3390/mca29010015 - 19 Feb 2024
Cited by 1 | Viewed by 1472
Abstract
In this paper, it is noted that three apparently disparate areas of mathematics—singularity analysis, complex symmetry analysis and the distributional representation of special functions—have a basic commonality in the underlying methods used. The insights obtained from the first of these provides a much-needed [...] Read more.
In this paper, it is noted that three apparently disparate areas of mathematics—singularity analysis, complex symmetry analysis and the distributional representation of special functions—have a basic commonality in the underlying methods used. The insights obtained from the first of these provides a much-needed explanation for the effectiveness of the latter two. The consequent explanations are provided in the form of two theorems and their corollaries. Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
38 pages, 665 KiB  
Article
Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity
by Fiazuddin D. Zaman, Fazal M. Mahomed and Faiza Arif
Math. Comput. Appl. 2023, 28(5), 96; https://doi.org/10.3390/mca28050096 - 12 Sep 2023
Viewed by 1523
Abstract
We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u) [...] Read more.
We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u). We carried out a complete Lie symmetry classification by finding forms for α(u) and f(u). This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach. Full article
(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)
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