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

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 254
Author(s):  
María S. Bruzón ◽  
Rafael de la Rosa ◽  
María L. Gandarias ◽  
Rita Tracinà
Keyword(s):  
Three Dimensional ◽  
Nonlinear Ordinary Differential Equation ◽  
Hydrodynamic Models ◽  
Third Order ◽  
Nonlinear Odes ◽  
Gardner Equation ◽  
Macroscopic Models ◽  
First Order ◽  
Group Invariant Solutions ◽  
The Given

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.

Vibroacoustic Comparisons and Acoustic Power Insulation Mechanisms of Multidirectional Stiffened Laminated Plates

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Xiongtao Cao ◽  
Hongxing Hua
Keyword(s):  
Fourier Transform ◽  
High Efficiency ◽  
Three Dimensional ◽  
Acoustic Power ◽  
Higher Order ◽  
Laminated Plates ◽  
Transverse Displacement ◽  
Third Order ◽  
First Order ◽  
Implicit Equations

Vibroacoustic characteristics of multidirectional stiffened laminated plates with or without compliant layers are explored in the wavenumber and spatial domains with the help of the two-dimensional continuous Fourier transform and discrete inverse fast Fourier transform. Implicit equations of motion for the arbitrary angle ply laminated plates are derived from the three-dimensional higher order and Reddy third order shear deformation plate theories. The expressions of acoustic power of the stiffened laminated plates with or without complaint layers are formulated in the wavenumber domain, which is a significant method to calculate acoustic power of the stiffened plates with multiple sets of cross stiffeners. Vibroacoustic comparisons of the stiffened laminated plates are made in terms of the transverse displacement spectra, forced responses, acoustic power, and input power according to the first order, Reddy third order, and three-dimensional higher order plate theories. Sound reduction profiles of compliant layers are further examined by the theoretical deductions. This study shows the feasibility and high efficiency of the first order and Reddy third order plate theories in the broad frequency range and allows a better understanding the principal mechanisms of acoustic power radiated from multidirectional stiffened laminated composite plates with compliant layers, which has not been adequately addressed in its companion paper. (Cao and Hua, 2012, “Sound Radiation From Shear Deformable Stiffened Laminated Plates With Multiple Compliant Layers,” ASME J. Vib. Acoust., 134(5), p. 051001.)

Effect of Dissipative Terms on the Quality of Two and Three-Dimensional Euler Flow Solutions

2008 ◽  
Author(s):  
Seyed Saied Bahrainian
Keyword(s):  
Euler Equations ◽  
Hyperbolic Conservation Laws ◽  
Three Dimensional ◽  
Strong Shock ◽  
Computational Domain ◽  
Third Order ◽  
First Order ◽  
Proper Values ◽  
Dissipative Terms

The Euler equations are a set of non-dissipative hyperbolic conservation laws that can become unstable near regions of severe pressure variation. To prevent oscillations near shockwaves, these equations require artificial dissipation terms to be added to the discretized equations. A combination of first-order and third-order dissipative terms control the stability of the flow solutions. The assigned magnitude of these dissipative terms can have a direct effect on the quality of the flow solution. To examine these effects, subsonic and transonic solutions of the Euler equations for a flow passed a circular cylinder has been investigated. Triangular and tetrahedral unstructured grids were employed to discretize the computational domain. Unsteady Euler equations are then marched through time to reach a steady solution using a modified Runge-Kutta scheme. Optimal values of the dissipative terms were investigated for several flow conditions. For example, at a free stream Mach number of 0.45 strong shock waves were captured on the cylinder by using values of 0.25 and 0.0039 for the first-order and third-order dissipative terms. In addition to the shock capturing effect, it has been shown that smooth pressure coefficients can be obtained with the proper values for the dissipative terms.

scholarly journals Mathematical model of falling of a viscous jet onto a moving surface

2007 ◽  
Vol 18 (6) ◽  
pp. 659-677 ◽  
Author(s):  
A. HLOD ◽  
A. C. T. AARTS ◽  
A. A. F. van de VEN ◽  
M. A. PELETIER
Keyword(s):  
Three Dimensional ◽  
Integral Condition ◽  
Stationary Flow ◽  
Third Order ◽  
Convex Shape ◽  
Dimensional Parameter ◽  
Moving Surface ◽  
First Order ◽  
Existence Region ◽  
Unknown Length

The stationary flow of a jet of a Newtonian fluid that is drawn by gravity onto a moving surface is analyzed. It is assumed that the jet has a convex shape and hits the moving surface tangentially. The flow is modelled by a third-order ODE on a domain of unknown length and with an additional integral condition. By solving part of the equation explicitly, the problem is reformulated as a first-order ODE with an integral constraint. The corresponding existence region in the three-dimensional parameter space is characterized in terms of an easily calculable quantity. In a qualitative sense, the results from the model are found to correspond with experimental observations.

scholarly journals On conditions of solvability of three-dimensional integralsystem in quadratures

2017 ◽  
Vol 21 (10) ◽  
pp. 40-46
Author(s):  
E.A. Sozontova
Keyword(s):  
Differential Equations ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Original System ◽  
Goursat Problem ◽  
System Of Differential Equations ◽  
Third Order ◽  
First Order ◽  
Three Dimensional Space

In this paper we consider the system of equations with partial integrals in three-dimensional space. The purpose is to find sufficient conditions of solvability of this system in quadratures. The proposed method is based on the reduction of the original system, first, to the Goursat problem for a system of differential equations of the first order, and after that to the three Goursat problems for differential equations of the third order. As a result, the sufficient conditions of solvability of the considering system in explicit form were obtained. The total number of cases discussing solvability is 16.

Time Fractional Third-Order Evolution Equation: Symmetry Analysis, Explicit Solutions, and Conservation Laws

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Dumitru Baleanu ◽  
Mustafa Inc ◽  
Abdullahi Yusuf ◽  
Aliyu Isa Aliyu
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Series Solution ◽  
Three Dimensional ◽  
Nonlinear Ordinary Differential Equation ◽  
Symmetry Analysis ◽  
Power Series Solution ◽  
Lie Symmetry Analysis ◽  
Third Order ◽  
Contour Plots

In this work, Lie symmetry analysis for the time fractional third-order evolution (TOE) equation with Riemann–Liouville (RL) derivative is analyzed. We transform the time fractional TOE equation to nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in Erdelyi–Kober (EK) sense. We obtain a kind of an explicit power series solution for the governing equation based on the power series theory. Using Ibragimov's nonlocal conservation method to time fractional partial differential equations (FPDEs), we compute conservation laws (CLs) for the TOE equation. Two dimensional (2D), three-dimensional (3D), and contour plots for the explicit power series solution are presented.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 585-608 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Odes ◽  
First Order ◽  
Open Questions ◽  
Motion Time

Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle–Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as time-dependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka–Volterra system and three-dimensional Rössler system as well as other examples including non-autonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

scholarly journals An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Ali K. Joohy ◽  
Ghassan A. Al-Juaifri ◽  
Mohammed S. Mechee
Keyword(s):  
Differential Equation ◽  
Real Line ◽  
Theoretical Work ◽  
Nonlinear Ordinary Differential Equation ◽  
Complex Domain ◽  
Third Order ◽  
Nonlinear Odes ◽  
The Real ◽  
Extended Technique ◽  
Complex Differential Equations

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

scholarly journals Zero-Hopf Bifurcations of 3D Quadratic Jerk System

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1454
Author(s):  
Bo Sang ◽  
Bo Huang
Keyword(s):  
Three Dimensional ◽  
Canonical System ◽  
Averaging Theory ◽  
Hopf Bifurcations ◽  
Third Order ◽  
Local Behaviour ◽  
First Order ◽  
The Core ◽  
Local Bifurcations ◽  
Jerk System

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity.

scholarly journals Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields

2010 ◽  
Vol 2010 ◽  
pp. 1-23
Author(s):  
M. P. Markakis
Keyword(s):  
Vector Fields ◽  
Ad Hoc ◽  
Three Dimensional ◽  
Solid Surfaces ◽  
Two Dimensional ◽  
Nonlinear Odes ◽  
First Order ◽  
Parabolic Geometry ◽  
Abel Differential Equation ◽  
Analytical Tools

Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.

Evidence for general base catalysis by protic solvents in a kinetic study of alcoholyses and hydrolyses of 1-(phenoxycarbonyl)pyridinium ions under both solvolytic and non-solvolytic conditions

2009 ◽  
Vol 74 (1) ◽  
pp. 43-55 ◽  
Author(s):  
Dennis N. Kevill ◽  
Byoung-Chun Park ◽  
Jin Burm Kyong
Keyword(s):  
Second Order ◽  
Base Catalysis ◽  
Substitution Reactions ◽  
Third Order ◽  
Methoxy Derivative ◽  
First Order ◽  
General Base ◽  
Protic Solvents ◽  
Kinetics Of ◽  
Pyridinium Ions

The kinetics of nucleophilic substitution reactions of 1-(phenoxycarbonyl)pyridinium ions, prepared with the essentially non-nucleophilic/non-basic fluoroborate as the counterion, have been studied using up to 1.60 M methanol in acetonitrile as solvent and under solvolytic conditions in 2,2,2-trifluoroethan-1-ol (TFE) and its mixtures with water. Under the non- solvolytic conditions, the parent and three pyridine-ring-substituted derivatives were studied. Both second-order (first-order in methanol) and third-order (second-order in methanol) kinetic contributions were observed. In the solvolysis studies, since solvent ionizing power values were almost constant over the range of aqueous TFE studied, a Grunwald–Winstein equation treatment of the specific rates of solvolysis for the parent and the 4-methoxy derivative could be carried out in terms of variations in solvent nucleophilicity, and an appreciable sensitivity to changes in solvent nucleophilicity was found.

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