scholarly journals APPLICATION OF FAST EXPANSIONS TO OBTAIN EXACT SOLUTIONS TO A PROBLEM ON RECTANGULAR MEMBRANE DEFLECTION UNDER ALTERNATING LOAD

2021 ◽  
pp. 127-142
Author(s):  
A.D. Chernyshov ◽  
◽  
V.V. Goryainov ◽  
S.F. Kuznetsov ◽  
O.Yu. Nikiforova ◽  
...  
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Boundary Function ◽  
Maximum Deflection ◽  
New Exact Solutions ◽  
General Terms ◽  
Fast Expansions ◽  
Fixed Membrane ◽  
Fourier Sine Series ◽  
Membrane Deflection

The problem of rectangular membrane deflection under alternating loads is solved in general terms by means of the method of fast expansions. The exact solution is represented by the finite expression borrowed from the theory of fast expansions as a sum of the boundary function and Fourier sine series with two Fourier coefficients taken into account. The obtained exact solution includes free parameters. Changing the values of these parameters, one can derive many new exact solutions. Obtaining of exact solutions to a problem of the rigidly fixed membrane under two types of loads (dome-shaped and sinusoidal) is shown as an example. Graphs of the dome-shaped and sinusoidal loads on the membrane and the curves of the corresponding deflections and stress components are presented in the paper. From the analysis of the exact solutions, it is obvious that only when a symmetrical alternating load is used, the membrane maximum deflection is attained in the center of the membrane, and the stresses reach the highest values in the middle of both long sides. In the case of a non-symmetrical load, the maximum stress occurs in the middle of either one of two long sides of the rectangular membrane, and the maximum deflection is found in the central region.

scholarly journals Three-dimensional exact solutions of the diffusion equation

2021 ◽  
pp. 48-62
Author(s):  
Александр Данилович Чернышов ◽  
Виталий Валерьевич Горяйнов ◽  
Сергей Федорович Кузнецов ◽  
Ольга Юрьевна Никифорова
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
Three Dimensional ◽  
Arithmetic Mean ◽  
Internal Source ◽  
New Exact Solutions ◽  
Fast Expansions ◽  
Free Parameters ◽  
Diffusion Flows

При помощи метода быстрых разложений решается задача диффузии в параллелепипеде с граничными условиями 1-го рода и внутренним источником вещества, зависящим от координат точек параллелепипеда. Получено в общем виде решение, содержащее свободные параметры, с помощью которых можно получить множество новых точных решений с различными свойствами. Показан пример построения точного решения для случая внутреннего источника переменного только по оси OZ . Приведен анализ особенностей диффузионных потоков в параллелепипеде с указанным внутреннем источником. Получено, что концентрация вещества в центре параллелепипеда равна сумме среднеарифметического значения концентраций вещества в его вершинах и амплитуды внутреннего источника умноженного на величину The authors solve the problem of diffusion in a parallelepiped-shaped body with boundary conditions of the 1st kind and an internal source of substance, depending on the parallelepiped points coordinates with the fast expansions method. The proposed exact solution in general form contains free parameters, which can be used to obtain many new exact solutions with different properties. An example of constructing an exact solution with a variable internal source depending on one coordinate z is shown in the work. An analysis of the features of diffusion flows in a parallelepiped with the indicated internal source is given. It was found that the concentration of a substance in the center of a parallelepiped is equal to the sum of the arithmetic mean of the concentration of a substance at its vertices and the amplitude of the internal source multiplied by the value

scholarly journals New Classes of Solutions of Dynamical Problems of Plasticity

2020 ◽  
pp. 792-796
Author(s):  
Sergei I. Senashov ◽  
Olga V. Gomonova ◽  
Irina L. Savostyanova ◽  
Olga N. Cherepanova
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Plastic Layer ◽  
Two Dimensional ◽  
Science And Engineering ◽  
Spatial Variables ◽  
One Dimensional ◽  
New Exact Solutions

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

Constructing exact solutions to anisotropic heat conduction equations by means of conservation laws

2012 ◽  
Vol 9 (1) ◽  
pp. 88-90
Author(s):  
N.H. Ibragimov ◽  
E.D. Avdonina
Keyword(s):  
Exact Solution ◽  
Heat Conduction ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Heat Equations ◽  
New Exact Solutions ◽  
Anisotropic Heat Conduction

Recently the new theorem about conservation laws was proved by N. H. Ibragimov related to the nonlinear self-adjointness definition, and a method of exact solution construction was proposed. In the next paper by the authors, the conservation laws for anisotropic heat conduction equations were constructed. In this work the proposed method is applied to anisotropic heat equations with a source and new exact solutions are constructed.

New exact solutions for the Wick-type stochastic Zakharov–Kuznetsov equation for modelling waves on shallow water surfaces

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
White Noise ◽  
Shallow Water ◽  
Hermite Transform ◽  
New Exact Solutions ◽  
Kudryashov Method ◽  
Stochastic Solution

AbstractIn this article, an exact solution of the Wick-type stochastic Zakharov–Kuznetsov equation has been obtained by using the Kudryashov method. We have used the Hermite transform for transforming the Wick-type stochastic Zakharov–Kuznetsov equation into a deterministic partial differential equation. Also we have applied the inverse Hermite transform for obtaining a set of stochastic solution in the white noise space.

scholarly journals Exact Solutions for(4+1)-Dimensional Nonlinear Fokas Equation Using ExtendedF-Expansion Method and Its Variant

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yinghui He
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Elliptic Function ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Weierstrass Elliptic Function ◽  
New Exact Solutions ◽  
Higher Dimensional

The construction of exact solution for higher-dimensional nonlinear equation plays an important role in knowing some facts that are not simply understood through common observations. In our work,(4+1)-dimensional nonlinear Fokas equation, which is an important physical model, is discussed by using the extendedF-expansion method and its variant. And some new exact solutions expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

scholarly journals Comment on “New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method”

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Hong-Zhun Liu
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Function Method ◽  
New Exact Solutions

We point out in this paper that the claims made by Kim et al. in the commented paper are incorrect and no new exact solution was obtained.

scholarly journals New Exact Solutions for MHD Transient Rotating Flow of a Second-Grade Fluid in a Porous Medium

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Faisal Salah ◽  
Zainal Abdul Aziz ◽  
Dennis Ling Chuan Ching
Keyword(s):  
Exact Solution ◽  
Porous Medium ◽  
Exact Solutions ◽  
Flat Plate ◽  
Laplace Transforms ◽  
Second Grade ◽  
Rotating Flow ◽  
Second Grade Fluid ◽  
New Exact Solutions ◽  
Grade Fluid

The magnetohydrodynamic (MHD) and rotating flow of second-grade fluid over a suddenly moved flat plate is investigated, where the second-grade fluid saturates the porous medium. The new exact solution is derived by using the Fourier sine and Laplace transforms. Many interesting available results in the literature are obtained as limiting cases of our solution. Finally, some graphical results are presented for different values of the material constants.

A NOTE ON THE EXACT SOLUTION OF THE SPECIAL MODIFIED KdV EQUATION

2008 ◽  
Vol 22 (04) ◽  
pp. 289-293
Author(s):  
HONGLEI WANG ◽  
CHUNHUAN XIANG
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
General Equation ◽  
Vries Equation ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
New Exact Solutions ◽  
Modified Kdv Equation ◽  
De Vries ◽  
Physical Fields

The modified KdV (Korteweg–de Vries) equation with two different variable coefficients can be employed in many different physical fields with time changing. In the present work, by using the truncated expansion, some new exact solutions of the equation are obtained. The general equation may change into lots of other forms KdV equation if we select different parameters.

New Exact Solutions of Steady Plane Flows of an In Compressible Fluid of Variable Viscosity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Sobia Younus
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Compressible Fluid ◽  
Variable Viscosity ◽  
Plane Motion ◽  
Transformation Technique ◽  
Vorticity Distribution ◽  
New Exact Solutions ◽  
Steady Plane ◽  
Uniform Stream

<span>Some new exact solutions to the equations governing the steady plane motion of an in compressible<span> fluid of variable viscosity for the chosen form of the vorticity distribution are determined by using<span> transformation technique. In this case the vorticity distribution is proportional to the stream function<span> perturbed by the product of a uniform stream and an exponential stream<br /><br class="Apple-interchange-newline" /></span></span></span></span>

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