scholarly journals XXIII.—On Systems of Solutions of Homogeneous and Central Equations of the nth Degree and of two or more Variables; with a Discussion of the Loci of such Equations

1890 ◽  
Vol 35 (4) ◽  
pp. 1043-1098
Author(s):  
M'Laren
Keyword(s):  
Exact Solutions ◽  
Plane Curves ◽  
Algebraic Equations ◽  
Linear Variation ◽  
Approximate Methods ◽  
Contour Lines

The purpose of the present paper is to ascertain how far it is possible to find exact solutions or values of x, y, &c., in equations between variables, so that the forms of plane curves and contour-lines of surfaces may be exactly determined. No approximate methods have been admitted, and only those methods have been used which are applicable to algebraic equations of every degree and any number of variables. In the examples given I have generally selected equations of even degree and even powers of the variables. But every such solution evidently includes the solution of the non-central equation of half the degree having corresponding terms and equal coefficients. The methods of solution employed are founded on the following introductory theorem or principle, which may be described as that of Homogeneous or Linear Variation of the quantities.

EXACT SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH SPHERICALLY SYMMETRIC OCTIC POTENTIAL

2012 ◽  
Vol 27 (20) ◽  
pp. 1250112 ◽  
Author(s):  
DAVIDS AGBOOLA ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Spherically Symmetric ◽  
Algebraic Equations ◽  
Potential Parameters

We present exact solutions of the Schrödinger equation with spherically symmetric octic potential. We give closed-form expressions for the energies and the wave functions as well as the allowed values of the potential parameters in terms of a set of algebraic equations.

Brief Review of Exact Solutions for Slip-Flow in Ducts and Channels

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Exact Solutions ◽  
Slip Flow ◽  
Microchannel Flow ◽  
Approximate Methods

Slip flow in ducts is important in numerous contemporary applications, especially microchannel flow. This Note reviews the existing exact solutions for slip flow. These solutions serve as accuracy standards for approximate methods including numerical or semi-numerical means. Some new solutions are also introduced.

scholarly journals Symmetry reduction a promising method for heat conduction equations

2019 ◽  
Vol 23 (4) ◽  
pp. 2219-2227
Author(s):  
Yi Tian
Keyword(s):  
Heat Conduction ◽  
Exact Solutions ◽  
Wave Equations ◽  
Reduction Method ◽  
Variational Iteration Method ◽  
Mathematical Tool ◽  
Approximate Methods ◽  
Similarity Reduction ◽  
Homotopy Perturbation ◽  
Powerful Mathematical Tool

Though there are many approximate methods, e. g., the variational iteration method and the homotopy perturbation, for non-linear heat conduction equations, exact solutions are needed in optimizing the heat problems. Here we show that the Lie symmetry and the similarity reduction provide a powerful mathematical tool to searching for the needed exact solutions. Lie algorithm is used to obtain the symmetry of the heat conduction equations and wave equations, then the studied equations are reduced by the similarity reduction method.

scholarly journals Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

2018 ◽  
Author(s):  
Adem Kilicman ◽  
Rathinavel Silambarasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Exact Solutions ◽  
Exact Analytical Solution ◽  
Nonlinear Partial Differential Equation ◽  
Complex Structures ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
Sivashinsky Equation

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudrayshov method for the exact analytical solution. The modified Kudrayshov method converts the nonlinear partial differential equation to algebraic equations, as a result of various steps, which on solving the so obtained equation systems yields the analytical solution. By this way various exact solutions including complex structures are found and drawn their behaviour in complex plane by Maple to compare the uniqueness of solutions.

scholarly journals Approximate and Exact Solutions to the Singular Nonlinear Heat Equation with a Common Type of Nonlinearity

2020 ◽  
Vol 34 ◽  
pp. 18-34
Author(s):  
A. L. Kazakov ◽  
◽  
L. F. Spevak ◽  
Keyword(s):  
Heat Equation ◽  
Exact Solutions ◽  
Wave Front ◽  
Heat Wave ◽  
Small Neighborhood ◽  
Parabolic Type ◽  
Nonlinear Heat Equation ◽  
Time Interval ◽  
Qualitative Properties ◽  
Approximate Methods

The paper deals with the problem of the motion of a heat wave with a specified front for a general nonlinear parabolic heat equation. An unknown function depends on two variables. Along the heat wave front, the coefficient of thermal conductivity and the source function vanish, which leads to a degeneration of the parabolic type of the equation. This circumstance is the mathematical reason for the appearance of the considered solutions, which describe perturbations propagating along the zero background with a finite velocity. Such effects are generally atypical for parabolic equations. Previously, we proved the existence and uniqueness theorem for the problem considered in this paper. Still, it is local and does not allow us to study the properties of the solution beyond the small neighborhood of the heat wave front. To overcome this problem, the article proposes an iterative method for constructing an approximate solution for a given time interval, based on the boundary element approach. Since it is usually not possible to prove strict convergence theorems of approximate methods for nonlinear equations of mathematical physics with a singularity, verification of the calculation results is relevant. One of the traditional ways is to compare them with exact solutions. In this article, we obtain and study an exact solution of the required type, the construction of which is reduced to integrating the Cauchy problem for an ODE. We obtained some qualitative properties, including an interval estimation of the wave amplitude in one particular case. The performed calculations show the effectiveness of the developed computational algorithm, as well as the compliance of the results of calculations with qualitative analysis.

Approximate methods of solving amplitude-phase problem for continuous signals

2021 ◽  
pp. 37-46
Author(s):  
Ilia V. Boikov ◽  
Yana V. Zelina
Keyword(s):  
Integral Equations ◽  
Collocation Methods ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Phase Problem ◽  
Spline Collocation ◽  
Approximate Methods ◽  
Nonlinear Operator Equations ◽  
One Dimensional

Amplitude and phase problems in physical research are considered. The construction of methods and algorithms for solving phase and amplitude problems is analyzed without involving additional information about the signal and its spectrum. Mathematical models of the amplitude and phase problems in the case of one-dimensional and two-dimensional continuous signals are proposed and approximate methods for their solution are constructed. The models are based on the use of nonlinear singular and bisingular integral equations. The amplitude and phase problems are modeled by corresponding nonlinear singular and bisingular integral equations defined on the numerical axis (in the one-dimensional case) and on the plane (in the two-dimensional case). To solve the constructed nonlinear singular and bisingular integral equations, spline-collocation methods and the method of mechanical quadratures are used. Systems of nonlinear algebraic equations that arise during the application of these methods are solved by the continuous method of solving nonlinear operator equations. A model example shows the effectiveness of the proposed method for solving the phase problem in the two-dimensional case.

Torsion and Flexure of Slender Solid Sections

1958 ◽  
Vol 25 (1) ◽  
pp. 115-121
Author(s):  
W. J. Carter
Keyword(s):  
Exact Solutions ◽  
Turbine Blades ◽  
Torsion Problem ◽  
Rectangular Section ◽  
Approximate Methods ◽  
Elastic Plastic ◽  
Shear Problem

Abstract The solution of the torsion problem for a slender rectangular section has been made previously by approximate methods based on the Prandtl membrane analogy. In this paper approximate methods are employed in the solution of both the torsion and flexural shear problem for slender sections having a variety of shapes, most of them being doubly symmetric. Solutions obtained in this manner are compared with exact solutions, when these are available, and otherwise with solutions obtained by relaxation. It is shown that approximate methods provide an adequate solution for elements such as compressor-turbine blades when pretwist and taper can be neglected. Some attention is given to the problem of elastic-plastic torsion and elastic-plastic flexural shear of slender sections.

scholarly journals On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Index-3

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Melike Karta ◽  
Ercan Çelik
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Variational Iteration

Numerical solution of differential-algebraic equations with Hessenberg index-3 is considered by variational iteration method. We applied this method to two examples, and solutions have been compared with those obtained by exact solutions.

A FIRST STUDY OF GENERIC CUBIC TERMS IN GRAVITATION THEORIES

1993 ◽  
Vol 02 (03) ◽  
pp. 257-278
Author(s):  
H. CAPRASSE ◽  
J. DEMARET ◽  
P. HOUBA
Keyword(s):  
Exact Solutions ◽  
Computer Algebra ◽  
Space Time ◽  
Cosmological Models ◽  
Gravitation Theory ◽  
Algebraic Equations ◽  
Field Equations ◽  
Time Dimension ◽  
Bianchi I ◽  
The Way

The generic cubic contributions to the Lagrangian of gravitation theory are considered. Field equations are determined and put in their simplest form. In the framework of Bianchi I cosmological models with a metric which is power-like in time, algebraic equations are obtained and their exact solutions are derived exploiting computer algebra techniques. These solutions are fully discussed. The analysis is, presently, essentially restricted to a space-time dimension equal to four but results obtained here open the way to an analysis in any dimension.

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