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12 pages, 295 KiB

Open AccessArticle

Asymptotic Stability for the 2D Navier–Stokes Equations with Multidelays on Lipschitz Domain

by Ling-Rui Zhang, Xin-Guang Yang and Ke-Qin Su

Mathematics 2022, 10(23), 4561; https://0-doi-org.brum.beds.ac.uk/10.3390/math10234561 - 01 Dec 2022

Viewed by 656

This paper is concerned with the asymptotic stability derived for the two-dimensional incompressible Navier–Stokes equations with multidelays on Lipschitz domain, which models the control theory of 2D fluid flow. By a new retarded Gronwall inequality and estimates of stream function for Stokes equations, [...] Read more.

This paper is concerned with the asymptotic stability derived for the two-dimensional incompressible Navier–Stokes equations with multidelays on Lipschitz domain, which models the control theory of 2D fluid flow. By a new retarded Gronwall inequality and estimates of stream function for Stokes equations, the complete trajectories inside pullback attractors are asymptotically stable via the restriction on the generalized Grashof number of fluid flow. The results in this presented paper are some extension of the literature by Yang, Wang, Yan and Miranville in 2021, as well as also the preprint by Su, Yang, Miranville and Yang in 2022 Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

16 pages, 3904 KiB

Open AccessArticle

An Integrable Evolution System and Its Analytical Solutions with the Help of Mixed Spectral AKNS Matrix Problem

by Sheng Zhang, Jiao Gao and Bo Xu

Mathematics 2022, 10(21), 3975; https://0-doi-org.brum.beds.ac.uk/10.3390/math10213975 - 26 Oct 2022

Cited by 1 | Viewed by 727

Abstract

In this work, a novel integrable evolution system in the sense of Lax’s scheme associated with a mixed spectral Ablowitz-Kaup-Newell-Segur (AKNS) matrix problem is first derived. Then, the time dependences of scattering data corresponding to the mixed spectral AKNS matrix problem are given [...] Read more.

In this work, a novel integrable evolution system in the sense of Lax’s scheme associated with a mixed spectral Ablowitz-Kaup-Newell-Segur (AKNS) matrix problem is first derived. Then, the time dependences of scattering data corresponding to the mixed spectral AKNS matrix problem are given in the inverse scattering analysis. Based on the given time dependences of scattering data, the reconstruction of potentials is carried out, and finally analytical solutions with four arbitrary functions of the derived integrable evolution system are formulated. This study shows that some other systems of integrable evolution equations under the resolvable framework of the inverse scattering method with mixed spectral parameters can be constructed by embedding different spectral parameters and time-varying coefficient functions to the known AKNS matrix spectral problem. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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12 pages, 2498 KiB

Open AccessArticle

Application of the Exp

(- φ (ξ))

-Expansion Method to Find the Soliton Solutions in Biomembranes and Nerves

by Attia Rani, Muhammad Shakeel, Mohammed Kbiri Alaoui, Ahmed M. Zidan, Nehad Ali Shah and Prem Junsawang

Mathematics 2022, 10(18), 3372; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183372 - 16 Sep 2022

Cited by 14 | Viewed by 1167

Abstract

Heimburg and Jackson devised a mathematical model known as the Heimburg model to describe the transmission of electromechanical pulses in nerves, which is a significant step forward. The major objective of this paper was to examine the dynamics of the Heimburg model by [...] Read more.

Heimburg and Jackson devised a mathematical model known as the Heimburg model to describe the transmission of electromechanical pulses in nerves, which is a significant step forward. The major objective of this paper was to examine the dynamics of the Heimburg model by extracting closed-form wave solutions. The proposed model was not studied by using analytical techniques. For the first time, innovative analytical solutions were investigated using the exp

(- φ (ξ))

-expansion method to illustrate the dynamic behavior of the electromechanical pulse in a nerve. This approach generates a wide range of general and broad-spectral solutions with unknown parameters. For the definitive value of these constraints, the well-known periodic- and kink-shaped solitons were recovered. By giving different values to the parameters, the 3D, 2D, and contour forms that constantly modulate in the form of an electromechanical pulse traveling through the axon in the nerve were created. The discovered solutions are innovative, distinct, and useful and might be crucial in medicine and biosciences. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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15 pages, 43310 KiB

Open AccessArticle

Analytical Method for Generalized Nonlinear Schrödinger Equation with Time-Varying Coefficients: Lax Representation, Riemann-Hilbert Problem Solutions

by Bo Xu and Sheng Zhang

Mathematics 2022, 10(7), 1043; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071043 - 24 Mar 2022

Cited by 6 | Viewed by 1360

Abstract

In this paper, a generalized nonlinear Schrödinger (gNLS) equation with time-varying coefficients is analytically studied using its Lax representation and the associated Riemann-Hilbert (RH) problem equipped with a symmetric scattering matrix in the Hermitian sense. First, Lax representation and the associated RH problem [...] Read more.

In this paper, a generalized nonlinear Schrödinger (gNLS) equation with time-varying coefficients is analytically studied using its Lax representation and the associated Riemann-Hilbert (RH) problem equipped with a symmetric scattering matrix in the Hermitian sense. First, Lax representation and the associated RH problem of the considered gNLS equation are established so that solution of the gNLS equation can be transformed into the associated RH problem. Secondly, using the solvability of unique solution of the established RH problem, time evolution laws of the scattering data reconstructing potential of the gNLS equation are determined. Finally, based on the determined time evolution laws of scattering data, the long-time asymptotic solution and N-soliton solution of the gNLS equation are obtained. In addition, some local spatial structures of the obtained one-soliton solution and two-soliton solution are shown in the figures. This paper shows that the RH method can be extended to nonlinear evolution models with variable coefficients, and the curve propagation of the obtained N-soliton solution in inhomogeneous media is controlled by the selection of variable–coefficient functions contained in the models. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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10 pages, 2639 KiB

Open AccessArticle

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

by Sheng Zhang and Bo Xu

Mathematics 2022, 10(3), 486; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030486 - 02 Feb 2022

Cited by 3 | Viewed by 1455

Abstract

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund [...] Read more.

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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22 pages, 410 KiB

Open AccessArticle

Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

by Alexander Kazakov and Anna Lempert

Mathematics 2022, 10(2), 232; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020232 - 12 Jan 2022

Cited by 3 | Viewed by 1403

Abstract

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations [...] Read more.

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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28 pages, 1162 KiB

Open AccessArticle

Dufour and Soret Effect on Viscous Fluid Flow between Squeezing Plates under the Influence of Variable Magnetic Field

by Muhammad Kamran Alam, Khadija Bibi, Aamir Khan and Samad Noeiaghdam

Mathematics 2021, 9(19), 2404; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192404 - 27 Sep 2021

Cited by 14 | Viewed by 1854

Abstract

The aim of this article is to investigate the effect of mass and heat transfer on unsteady squeeze flow of viscous fluid under the influence of variable magnetic field. The flow is observed in a rotating channel. The unsteady equations of mass and [...] Read more.

The aim of this article is to investigate the effect of mass and heat transfer on unsteady squeeze flow of viscous fluid under the influence of variable magnetic field. The flow is observed in a rotating channel. The unsteady equations of mass and momentum conservation are coupled with the variable magnetic field and energy equations. By using some appropriate similarity transformations, the partial differential equations obtained are then converted into a system of ordinary differential equations and are solved by Homotopy Analysis Method (HAM). The influence of the natural parameters are investigated for the velocity field components, magnetic field components, heat and mass transfer. A direct effect of the squeeze Reynold number is observed on both concentration and temperature. Moreover, increasing the magnetic Reynold number shows an increase in the fluid temperature, but in the case of concentration, an inverse relation is observed. Furthermore, a decreasing effect of the Dufour number is observed on both concentration and temperature distribution. Besides, in case of the Soret number, a direct effect is observed on concentration, but an inverse effect can be seen on temperature distribution. Different effects are shown through graphs in this study and an error analysis is also presented through tables and graphs. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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11 pages, 355 KiB

Open AccessArticle

Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation

by Almudena P. Márquez and María S. Bruzón

Mathematics 2021, 9(17), 2131; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172131 - 02 Sep 2021

Cited by 8 | Viewed by 1565

Abstract

This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we [...] Read more.

This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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Review

Jump to: Research

9 pages, 263 KiB

Open AccessReview

Optical Solitons with Cubic-Quintic-Septic-Nonic Nonlinearities and Quadrupled Power-Law Nonlinearity: An Observation

by Islam Samir, Ahmed H. Arnous, Yakup Yıldırım, Anjan Biswas, Luminita Moraru and Simona Moldovanu

Mathematics 2022, 10(21), 4085; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214085 - 02 Nov 2022

Cited by 16 | Viewed by 1119

Abstract

The current paper considers the enhanced Kudryashov’s technique to retrieve solitons with a governing model having cubic-quintic-septic-nonic and quadrupled structures of self-phase modulation. The results prove that it is redundant to extend the self-phase modulation beyond cubic-quintic nonlinearity or dual-power law of nonlinearity. [...] Read more.

The current paper considers the enhanced Kudryashov’s technique to retrieve solitons with a governing model having cubic-quintic-septic-nonic and quadrupled structures of self-phase modulation. The results prove that it is redundant to extend the self-phase modulation beyond cubic-quintic nonlinearity or dual-power law of nonlinearity. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

36 pages, 506 KiB

Open AccessReview

Operators and Boundary Problems in Finance, Economics and Insurance: Peculiarities, Efficient Methods and Outstanding Problems

by Sergei Levendorskiĭ

Mathematics 2022, 10(7), 1028; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071028 - 23 Mar 2022

Cited by 2 | Viewed by 1458

Abstract

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique [...] Read more.

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique solution of a boundary problem for a parabolic equation. In the case of PDO with constant symbols, simple probabilistic tools explained in this paper can be used to explicitly calculate expectations under very weak conditions on the process and study the regularity of the solution. Assuming that the Feynman–Kac theorem holds, and a more general boundary problem can be localized, the local results can be used to study the existence and regularity of solutions, and derive efficient numerical methods. In the paper, difficulties for the realization of this program are analyzed, several outstanding problems are listed, and several closely efficient methods are outlined. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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