scholarly journals Exact solution of optimization task generated by simplest heat conduction equation

2012 ◽  
pp. 141-156
Author(s):  
V.I. Rodionov ◽  
◽  
N.V. Rodionova ◽  
Keyword(s):  
Exact Solution ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Optimization Task

scholarly journals Determining An Unknown Boundary Condition by An Iteration Method

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 409
Author(s):  
Dejian Huang ◽  
Yanqing Li ◽  
Donghe Pei
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Heat Conduction ◽  
Iteration Method ◽  
Heat Conduction Equation ◽  
Heat Conduction Problem ◽  
Variational Iteration Method ◽  
Conduction Equation ◽  
Unknown Boundary ◽  
Initial Condition

This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research.

A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

2015 ◽  
Vol 7 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Jingjun Zhao ◽  
Songshu Liu ◽  
Tao Liu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Kernel Method ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Ill Posed ◽  
Well Posed ◽  
Regularized Solution

AbstractIn this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.

Exact Solution of Two-Dimensional Hyperbolic Heat Conduction Equation With Combined Boundary Conditions and Arbitrary Initial Conditions

2009 ◽  
Author(s):  
Seyed Reza Mahmoudi ◽  
Nikola Toljic
Keyword(s):  
Exact Solution ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Initial Conditions ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
Application Range

Analytical solution of hyperbolic heat conduction equation has so far been limited only to one-dimensional frameworks. With the expanding of the application range for the fast heat sources in microelectronic industries, the two-dimensional solution of non-Fourier heat conduction becomes increasingly important. This paper presents an exact solution of hyperbolic heat conduction equation for a finite plane with sides subjected to combined boundary conditions. In the mathematical model, the heating on boundaries is treated as an apparent heat source whereas sides of the plane with the second kind boundaries are assumed to be insulated. The important characteristic of the proposed solution is its simplicity.

scholarly journals FORMULATION OF THE HEAT CONDUCTION EQUATION FOR HETEROGENEOUS MEDIA WITH MULTIPLE SPATIAL SCALES USING REITERATED HOMOGENIZATION

2016 ◽  
Vol 15 (1) ◽  
pp. 96
Author(s):  
E. Iglesias-Rodríguez ◽  
M. E. Cruz ◽  
J. Bravo-Castillero ◽  
R. Guinovart-Díaz ◽  
R. Rodríguez-Ramos ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Small Parameter ◽  
Heat Conduction Equation ◽  
Heterogeneous Media ◽  
Spatial Scales ◽  
Conduction Equation ◽  
Small Scale ◽  
Multiple Spatial Scales ◽  
Effective Coefficients ◽  
Reiterated Homogenization

Heterogeneous media with multiple spatial scales are finding increased importance in engineering. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. The objective in this paper is to formulate the strong-form Fourier heat conduction equation for such media using the method of reiterated homogenization. The phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter ε. The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter ε . The technique leads to two pairs of local and homogenized equations, linked by effective coefficients. In this manner the medium behavior at the smallest scales is seen to affect the macroscale behavior, which is the main interest in engineering. To facilitate the physical understanding of the formulation, an analytical solution is obtained for the heat conduction equation in a functionally graded material (FGM). The approach presented here may serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.

scholarly journals A Monte Carlo Method of Solving Heat Conduction Problems

1980 ◽  
Vol 102 (1) ◽  
pp. 121-125 ◽  
Author(s):  
S. K. Fraley ◽  
T. J. Hoffman ◽  
P. N. Stevens
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Monte Carlo Method ◽  
Transport Equation ◽  
Heat Conduction Equation ◽  
Analytical Techniques ◽  
Conduction Equation ◽  
Geometric Complexity ◽  
New Approach ◽  
Temperature Distributions

A new approach in the use of Monte Carlo to solve heat conduction problems is developed using a transport equation approximation to the heat conduction equation. A variety of problems is analyzed with this method and their solutions are compared to those obtained with analytical techniques. This Monte Carlo approach appears to be limited to the calculation of temperatures at specific points rather than temperature distributions. The method is applicable to the solution of multimedia problems with no inherent limitations as to the geometric complexity of the problem.

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