GEOMETRIC INTEGRATION BY SOLUTION INTERPOLATION

2009 ◽  
Vol 20 (02) ◽  
pp. 313-322
Author(s):  
PILWON KIM
Keyword(s):  
Differential Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Standard Method ◽  
Interpolation Method ◽  
Nonlinear Heat Equation ◽  
Geometric Integration ◽  
Numerical Schemes ◽  
New Class ◽  
Geometric Integrators

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

scholarly journals On exact solutions of the nonlinear heat equation

2019 ◽  
Vol 5 ◽  
pp. 11-17
Author(s):  
A.F. Barannyk ◽  
◽  
T.A. Barannyk ◽  
I.I. Yuryk ◽  
◽  
...  
Keyword(s):  
Heat Equation ◽  
Exact Solutions ◽  
Nonlinear Heat Equation

scholarly journals Solution Interpolation Method for Highly Oscillating Hyperbolic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Pilwon Kim ◽  
Chang Hyeong Lee
Keyword(s):  
Wave Equation ◽  
Standard Method ◽  
Stability Condition ◽  
Numerical Scheme ◽  
Hyperbolic Equations ◽  
Interpolation Method ◽  
Quasilinear Equation ◽  
Numerical Data ◽  
Discontinuous Coefficients ◽  
Numerical Schemes

This paper deals with a novel numerical scheme for hyperbolic equations with rapidly changing terms. We are especially interested in the quasilinear equationut+aux=f(x)u+g(x)unand the wave equationutt=f(x)uxxthat have a highly oscillating term likef(x)=sin(x/ε),  ε≪1. It also applies to the equations involving rapidly changing or even discontinuous coefficients. The method is based on the solution interpolation and the underlying idea is to establish a numerical scheme by interpolating numerical data with a parameterized solution of the equation. While the constructed numerical schemes retain the same stability condition, they carry both quantitatively and qualitatively better performances than the standard method.

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

2018 ◽  
Vol 59 (3) ◽  
pp. 427-441 ◽  
Author(s):  
A. L. Kazakov ◽  
Sv. S. Orlov ◽  
S. S. Orlov
Keyword(s):  
Heat Equation ◽  
Exact Solutions ◽  
Nonlinear Heat Equation

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.

Influence of Some Parameters on the Solution of a Thermal Model of Vulcanization

1993 ◽  
Vol 66 (1) ◽  
pp. 19-29 ◽  
Author(s):  
Jacques Burger ◽  
Nicole Burger ◽  
Marc Pogu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Heat Equation ◽  
Thermal Behavior ◽  
Thermal Model ◽  
Nonlinear Heat Equation ◽  
Reaction Equation ◽  
Finite Element Procedure ◽  
Vulcanization Process

Abstract The thermal behavior of a piece of rubber during a vulcanization process is modelized by a set of two coupled differential equations. The first one is a classical nonlinear heat equation while the second is a reaction equation. The aim of this article is first to solve numerically the two equations using a finite element procedure for the partial differential equation and a Runge-Kutta like scheme for the reaction equation, and then to study the sensitivity of the solution to variations in some parameters.

scholarly journals The Extended Tanh Method and the Exp-Function Method to Solve a Kind of Nonlinear Heat Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Thermal Properties ◽  
Heat Equation ◽  
Exact Solutions ◽  
Function Method ◽  
Nonlinear Heat Equation ◽  
Temperature Dependent ◽  
Extended Tanh Method ◽  
New Exact Solutions ◽  
Tanh Method

We study a kind of nonlinear heat equation with temperature-dependent thermal properties by the aid of the extended Tanh method and the Exp-function method. We obtain abundant new exact solutions of the equation. By comparing both of the methods, we find that the Exp-function method gives more solutions in this problem.

A NONLINEAR INVERSE PHASE TRANSITION PROBLEM FOR THE HEAT EQUATION

1991 ◽  
Vol 01 (02) ◽  
pp. 167-182 ◽  
Author(s):  
L. PREZIOSI ◽  
L.M. DE SOCIO
Keyword(s):  
Phase Transition ◽  
Heat Equation ◽  
Space Dimension ◽  
Interpolation Method ◽  
Solution Procedure ◽  
Nonlinear Heat Equation ◽  
Moving Interface ◽  
First Case ◽  
Phase Transition Problem ◽  
Quantitative Results

This paper proposes a method for the solution of two inverse problems which are governed by the nonlinear heat equation in one space dimension. In the first case phase transition occurs at the moving interface which divides two media. In the second one a random heat source is placed at a moving point. In both cases the temperature is assigned, as a function of time and within a random error, at a given fixed point. The solution procedure leads to quantitative results and is based on the Stochastic Adaptative Interpolation method.

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