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13 pages, 297 KiB

Open AccessArticle

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

by María S. Bruzón, Rafael de la Rosa, María L. Gandarias and Rita Tracinà

Mathematics 2022, 10(6), 954; https://doi.org/10.3390/math10060954 - 17 Mar 2022

Cited by 1 | Viewed by 1597

Abstract

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. [...] Read more.

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

19 pages, 329 KiB

Open AccessEditor’s ChoiceArticle

Applications of Solvable Lie Algebras to a Class of Third Order Equations

by María S. Bruzón, Rafael de la Rosa, María L. Gandarias and Rita Tracinà

Mathematics 2022, 10(2), 254; https://doi.org/10.3390/math10020254 - 14 Jan 2022

Cited by 3 | Viewed by 1860

Abstract

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for [...] Read more.

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

10 pages, 235 KiB

Open AccessArticle

Exact Solutions and Conservation Laws of a Generalized (1 + 1) Dimensional System of Equations via Symbolic Computation

by Sivenathi Oscar Mbusi, Ben Muatjetjeja and Abdullahi Rashid Adem

Mathematics 2021, 9(22), 2916; https://doi.org/10.3390/math9222916 - 16 Nov 2021

Cited by 9 | Viewed by 1357

Abstract

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific [...] Read more.

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

14 pages, 288 KiB

Open AccessArticle

Approximate Noether Symmetries of Perturbed Lagrangians and Approximate Conservation Laws

by Matteo Gorgone and Francesco Oliveri

Mathematics 2021, 9(22), 2900; https://doi.org/10.3390/math9222900 - 15 Nov 2021

Cited by 4 | Viewed by 1622

Abstract

In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate [...] Read more.

In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate conservation laws. Some illustrative applications are presented. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

12 pages, 281 KiB

Open AccessArticle

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

by Nosheen Goshi, Sohail Zafar, Tabasam Rashid and Juan L. G. Guirao

Mathematics 2021, 9(19), 2364; https://doi.org/10.3390/math9192364 - 23 Sep 2021

Cited by 1 | Viewed by 1653

Abstract

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive [...] Read more.

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph

Q_{n}

with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph

F_{k}

and the class of circulant graph

C_{n} (1, 2)

. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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6 pages, 1502 KiB

Open AccessArticle

Lagrangian Formulation, Conservation Laws, Travelling Wave Solutions: A Generalized Benney-Luke Equation

by Sivenathi Oscar Mbusi, Ben Muatjetjeja and Abdullahi Rashid Adem

Mathematics 2021, 9(13), 1480; https://doi.org/10.3390/math9131480 - 24 Jun 2021

Cited by 10 | Viewed by 1793

Abstract

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation [...] Read more.

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation on equations arising from a Lagrangian density function with high order derivatives of the field variables, is also discussed. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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17 pages, 3136 KiB

Open AccessArticle

Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

by Chaudry Masood Khalique and Karabo Plaatjie

Mathematics 2021, 9(12), 1439; https://doi.org/10.3390/math9121439 - 21 Jun 2021

Cited by 10 | Viewed by 2698

Abstract

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained [...] Read more.

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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20 pages, 1028 KiB

Open AccessArticle

Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

by Maria Santos Bruzón, Gaetana Gambino and Maria Luz Gandarias

Mathematics 2021, 9(9), 1009; https://doi.org/10.3390/math9091009 - 29 Apr 2021

Cited by 5 | Viewed by 2670

Abstract

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new [...] Read more.

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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11 pages, 285 KiB

Open AccessArticle

Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV

by Paz Albares and Pilar Garcia Estévez

Mathematics 2021, 9(9), 926; https://doi.org/10.3390/math9090926 - 22 Apr 2021

Cited by 2 | Viewed by 1830

Abstract

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax [...] Read more.

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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31 pages, 422 KiB

Open AccessArticle

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

by Alexander V. Aksenov and Andrei D. Polyanin

Mathematics 2021, 9(4), 345; https://doi.org/10.3390/math9040345 - 9 Feb 2021

Cited by 14 | Viewed by 3352

Abstract

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct [...] Read more.

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time,

u = u (x, t)

, these equations contain the same function at a past time,

w = u (x, t - τ)

, where

τ > 0

is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown

u = u (x, t)

, also contain the same functions with dilated or contracted arguments,

w = u (p x, q t)

, where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

10 pages, 267 KiB

Open AccessArticle

Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions

by Sameerah Jamal

Mathematics 2020, 8(11), 2019; https://doi.org/10.3390/math8112019 - 12 Nov 2020

Cited by 6 | Viewed by 1687

Abstract

In this paper, we discuss travelling wave solutions for image smoothing based on a fourth-order partial differential equation. One of the recurring issues of digital imaging is the amount of noise. One solution to this is to minimise the total variation norm of [...] Read more.

In this paper, we discuss travelling wave solutions for image smoothing based on a fourth-order partial differential equation. One of the recurring issues of digital imaging is the amount of noise. One solution to this is to minimise the total variation norm of the image, thus giving rise to non-linear equations. We investigate the variational properties of the Lagrange functionals associated with these minimisation problems. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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17 pages, 984 KiB

Open AccessArticle

An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials

by Ahmad Sami Bataineh, Osman Rasit Isik, Abedel-Karrem Alomari, Mohammad Shatnawi and Ishak Hashim

Mathematics 2020, 8(9), 1473; https://doi.org/10.3390/math8091473 - 1 Sep 2020

Cited by 3 | Viewed by 2073

Abstract

In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type [...] Read more.

In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing

n, m

for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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1 pages, 166 KiB

Open AccessObituary

In Memory of Prof. Dr. Maria Santos Bruzón Gallego

by Maria Luz Gandarias, Mariano Torrisi and Rita Tracinà

Mathematics 2021, 9(21), 2767; https://doi.org/10.3390/math9212767 - 31 Oct 2021

Viewed by 1673

Abstract

It is with great sadness that we write this memoriam for our beloved friend and colleague Maria de los Santos Bruzón who was an editor of this Special Issue [...] Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

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