The Evolution of Travelling Waves in Fractional Order Autocatalysis with Decay. II. The Initial Boundary Value Problem

2000 ◽  
Vol 60 (5) ◽  
pp. 1707-1748 ◽  
Author(s):  
D. J. Needham ◽  
J. A. Leach ◽  
P. M. McCabe
Keyword(s):  
Boundary Value Problem ◽  
Fractional Order ◽  
Travelling Waves ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value

Travelling waves in an initial-boundary value problem

1981 ◽  
Vol 90 (1-2) ◽  
pp. 41-61 ◽  
Author(s):  
E. J. M. Veling
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Expression ◽  
Travelling Waves ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Problem ◽  
Travelling Wave ◽  
Boundary Values ◽  
Initial Boundary Value

SynopsisIn this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.

scholarly journals A grid method for solving the first initial boundary value problem for a loaded differential equation of fractional order convection diffusion

2020 ◽  
pp. 27-40
Author(s):  
Мурат Хамидбиевич Бештоков
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniform Grid ◽  
Convection Diffusion ◽  
Loaded Differential Equation ◽  
Initial Boundary Value

Рассмотрена первая начально-краевая задача для нагруженного дифференциального уравнения конвекции диффузии дробного порядка. На равномерной сетке построена разностная схема, аппроксимирующая эту задачу. Для решения поставленной задачи в предположении существования регулярного решения получены априорные оценки в дифференциальной и разностной формах. Из этих оценок следуют единственность и непрерывная зависимость решения от входных данных задачи, а также сходимость со скоростью $O(h^2+\\tau^2)$. The first initial boundary value problem for a loaded differential equation of fractional order convection diffusion is considered. A difference scheme approximating this problem is constructed on a uniform grid. To solve the problem, assuming the existence of a regular solution, a priori estimates in differential and difference forms are obtained. From these estimates follow the uniqueness and continuous dependence of the solution on the input data of the problem, as well as the convergence with the rate $O(h^2+\\tau^2)$.

The effects of variable diffusivity on the development of travelling waves in a class of reaction-diffusion equations

1994 ◽  
Vol 348 (1687) ◽  
pp. 229-260 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Travelling Waves ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

In this paper we examine the effects of concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics and ecology. We consider piecewise classical solutions to an initial boundary-value problem. The existence of a family of permanent form travelling wave solutions is established and the development of the solution of the initial boundary-value problem to the travelling wave of minimum propagation speed is considered. For certain types of initial data, ‘waiting time’ phenomena are encountered.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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