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 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 General Method of Exact Solution of the Concentration?Dependent Diffusion Equation

1960 ◽  
Vol 13 (1) ◽  
pp. 1 ◽  
Author(s):  
JR Philip
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
General Method

Only three forms of D(6) have previously been known to yield exact solutions of the equation

SOME PERIODIC AND BLOW-UP SOLUTIONS FOR LANDAU–LIFSHITZ EQUATION

2011 ◽  
Vol 26 (32) ◽  
pp. 2437-2452 ◽  
Author(s):  
PENGHONG ZHONG ◽  
SHU WANG ◽  
SHENGTAO CHEN
Keyword(s):  
Exact Solution ◽  
Periodic Solution ◽  
Exact Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Three Dimensional ◽  
Ferromagnetic Materials ◽  
Lifshitz Equation ◽  
Dimensional Spacetime ◽  
Vortex Solution

In this paper, we construct the exact solution of two- or three-dimensional spacetime Landau–Lifshitz equation raised in the ferromagnetic materials. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite time blow-up solution, vortex solution and periodic solution. In the end, we sketch some solutions and their spatial curvatures.

scholarly journals Fractal Ion Acoustic Waves of the Space-Time Fractional Three Dimensional KP Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
M. A. Abdou ◽  
Saud Owyed ◽  
S. Saha Ray ◽  
Yu-Ming Chu ◽  
Mustafa Inc ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Acoustic Waves ◽  
Three Dimensional ◽  
Fractional Operators ◽  
Trigonometric Functions ◽  
Fractional Partial Differential Equations ◽  
Kp Equation ◽  
Ion Acoustic Waves ◽  
New Exact Solutions ◽  
Conformable Derivatives

Methods known as fractional subequation and sine-Gordon expansion (FSGE) are employed to acquire new exact solutions of some fractional partial differential equations emerging in plasma physics. Fractional operators are employed in the sense of conformable derivatives (CD). New exact solutions are constructed in terms of hyperbolic, rational, and trigonometric functions. Computational results indicate the power of the method.

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 of the reaction–diffusion equation with variable coefficients via the mathematical computation

2018 ◽  
Vol 11 (04) ◽  
pp. 1850051 ◽  
Author(s):  
Jin Hyuk Choi ◽  
Hyunsoo Kim
Keyword(s):  
Diffusion Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Mathematical Computation

In this paper, we construct new exact solutions of the reaction–diffusion equation with time dependent variable coefficients by employing the mathematical computation via the Painlevé test. We describe the behaviors and their interactions of the obtained solutions under certain constraints and various variable coefficients.

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.

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