scholarly journals Convergence of approximate solutions of a quasilinear partial differential equation

1994 ◽  
Vol 50 (3) ◽  
pp. 425-433 ◽  
Author(s):  
T.R. Cranny
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Original Problem ◽  
A Priori ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Special Cases ◽  
Convergence Of Approximate Solutions ◽  
Hölder Estimates ◽  
Quasilinear Partial Differential Equation

This article is a sequel to a paper in which a quasilinear partial differential equation with nonlinear boundary condition was approximated using mollifiers, and the existence of solutions to the approximating problem shown under quite general conditions. In this paper we show that standard a priori Hölder estimates ensure the convergence of these solutions to a classical solution of the original problem. Some partial results giving such estimates for special cases are described.

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

Abstract The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

scholarly journals Sinc-Galerkin method for solving hyperbolic partial differential equations

2018 ◽  
Vol 8 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Galerkin Method ◽  
Hyperbolic Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Numerical Examples

In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

scholarly journals On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

SYSTEM CHARACTERISTICS OF DISTRIBUTED PARAMETER SYSTEMS

2020 ◽  
Vol 70 (3) ◽  
pp. 34-44
Author(s):  
Kamen Perev
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Distributed Parameter Systems ◽  
Point Of View ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Special Cases ◽  
The Green Function ◽  
Set Up

The paper considers the problem of distributed parameter systems modeling. The basic model types are presented, depending on the partial differential equation, which determines the physical processes dynamics. The similarities and the differences with the models described in terms of ordinary differential equations are discussed. A special attention is paid to the problem of heat flow in a rod. The problem set up is demonstrated and the methods of its solution are discussed. The main characteristics from a system point of view are presented, namely the Green function and the transfer function. Different special cases for these characteristics are discussed, depending on the specific partial differential equation, as well as the initial conditions and the boundary conditions.

scholarly journals XX.—A Demonstration of Lagrange's Rule for the Solution of a Linear Partial Differential Equation, with some Historical Remarks on Defective Demonstrations hitherto Current

1892 ◽  
Vol 36 (2) ◽  
pp. 551-562 ◽  
Author(s):  
G. Chrystal
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
English Text ◽  
Exact Nature ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Special Cases ◽  
Linear Differential ◽  
Historical Remarks

It seems strange that a principle so fundamental and so widely used as Lagrange's Rule for Solving a Linear Differential Equation should hitherto have been almost invariably provided with an inadequate demonstration. I noticed several years ago that the demonstrations in our current English text-books were apparently insufficient; but, as the method by which I treated Linear Partial Differential Equations in my lectures did not involve the use of them, it did not occur to me to analyse them closely with a view to discovering in what the exact nature of the defect consisted. The consideration of certain special cases recently led me to examine the matter more closely, and I was greatly surprised to find that most of the general demonstrations given are vitiated by a very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the examination of particular cases at all.

scholarly journals A generalized Hankel transform and its use for solving certain partial differential equations

1999 ◽  
Vol 41 (1) ◽  
pp. 105-117 ◽  
Author(s):  
I. Ali ◽  
S. Kalla
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Integral Transform ◽  
Diffusion Theory ◽  
Ground Surface ◽  
Ground Level ◽  
Hankel Transform ◽  
Partial Differential ◽  
Special Cases

AbstractWe introduce a generalized form of the Hankel transform, and study some of its properties. A partial differential equation associated with the problem of transport of a heavy pollutant (dust) from the ground level sources within the framework of the diffusion theory is treated by this integral transform. The pollutant concentration is expressed in terms of a given flux of dust from the ground surface to the atmosphere. Some special cases are derived.

scholarly journals Investigation of dynamics of elastic element of vibration device

2021 ◽  
pp. 67-83
Author(s):  
П.А. Вельмисов ◽  
А.В. Анкилов ◽  
Г.А. Анкилов
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Complex Variable ◽  
Elastic Element ◽  
Numerical Experiments ◽  
Original Problem ◽  
Fourier Method ◽  
Partial Differential ◽  
The Galerkin Method ◽  
Liquid Flows

ва подхода к решению аэрогидродинамической части задачи, основанные на методах теории функций комплексного переменного и методе Фурье. В результате применения каждого подхода решение исходной задачи сведено к исследованию дифференциального уравнения с частными производными для деформации элемента, позволяющего изучать его динамику. На основе метода Галеркина произведены численные эксперименты для конкретных примеров механической системы, подтверждающие идентичность решений, найденных для каждого дифференциального уравнения с частными производными. The dynamics of an elastic element of a vibration device, simulated by a channel, inside which a stream of a liquid flows, is investigated. Two approaches to solving the aerohydrodynamic part of the problem, based on the methods of the theory of functions of a complex variable and the Fourier method, are given. As a result of applying each approach, the solution to the original problem is reduced to the study of a partial differential equation for the deformation of an element, which makes it possible to study its dynamics. Based on the Galerkin method, the numerical experiments were carried out for specific examples of mechanical system, confirming the identity of the solutions found for each partial differential equation.

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