Solution of partial differential equations (3) – the heat conduction equation

2019 ◽  
pp. 414-415
Author(s):  
John Bird
Keyword(s):  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Partial Differential

Systematic Construction of Neural Forms for Solving Partial Differential Equations Inside Rectangular Domains, Subject to Initial, Boundary and Interface Conditions

2020 ◽  
Vol 29 (05) ◽  
pp. 2050009
Author(s):  
Pola Lydia Lagari ◽  
Lefteri H. Tsoukalas ◽  
Salar Safarkhani ◽  
Isaac E. Lagaris
Keyword(s):  
Neural Network ◽  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Systematic Approach ◽  
Thermal Contact ◽  
Initial Boundary ◽  
Spatial Domain ◽  
Interface Conditions ◽  
Partial Differential

A systematic approach is developed for constructing proper trial solutions to Partial Differential Equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. The spatial domain considered is of the rectangular hyper-box type. On each face either Dirichlet or Neumann conditions may apply. Robin conditions may be accommodated as well. Interface conditions that induce discontinuities, have not been treated to date in the relevant neural network literature. As an illustration a common problem of heat conduction through a system of two rods in thermal contact is considered.

The numerical solution of the heat conduction equation in one dimension

1964 ◽  
Vol 60 (4) ◽  
pp. 897-907 ◽  
Author(s):  
M. Wadsworth ◽  
A. Wragg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Heat Conduction Equation ◽  
Linear Equations ◽  
Conduction Equation ◽  
Partial Differential ◽  
Direct Numerical Method

AbstractThe replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.

scholarly journals On the solution of reaction-diffusion equations with double diffusivity

1987 ◽  
Vol 10 (1) ◽  
pp. 163-172
Author(s):  
B. D. Aggarwala ◽  
C. Nasim
Keyword(s):  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Special Cases ◽  
Two Temperatures

In this paper, solution of a pair of Coupled Partial Differential equations is derived. These equations arise in the solution of problems of flow of homogeneous liquids in fissured rocks and heat conduction involving two temperatures. These equations have been considered by Hill and Aifantis, but the technique we use appears to be simpler and more direct, and some new results are derived. Also, discussion about the propagation of initial discontinuities is given and illustrated with graphs of some special cases.

Heat Conduction Equation Finite Volume Method to Achieve on MATLAB

2012 ◽  
Vol 195-196 ◽  
pp. 712-717
Author(s):  
Qiong Xue ◽  
Xiao Feng Xiao ◽  
Niang Zhi Fan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Volume Method

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated. Transient heat conduction analysises of infinite plate with uniform thickness and two dimensional rectangle region have been realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

scholarly journals Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations

2018 ◽  
Vol 7 (4.36) ◽  
pp. 379
Author(s):  
Erwin Sulaeman ◽  
S. M. Afzal Hoq ◽  
Abdurahim Okhunov ◽  
Marwan Badran
Keyword(s):  
Finite Element ◽  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Heat Conduction Problem ◽  
Promising Result ◽  
Three Dimensional ◽  
Integration Scheme ◽  
Partial Differential

Solution of partial differential equations (PDEs) of three dimensional steady state heat conduction and its error analysis are elaborated in the present paper by using a Trilinear Galerkin Finite Element method (TGFEM). An eight-node hexahedron element model is developed for the TGFEM based on a trilinear basis function where physical domain is meshed by structured grid.  The stiffness matrix of the hexahedron element is formulated by using a direct integration scheme without the necessity to use the Jacobian matrix.  To check the accuracy of the established scheme, comparisons of the results using error analysis between the present TGFEM and exact solution is conducted for various number of the elements.  For this purpose, analytical solution is derived in detailed for a particular heat conduction problem.  The comparison shows promising result where its convergence is approximately O(h²) for matrix norms L1, L2 and L¥.  

Heat Conduction Equation Finite Volume Method to Achieve on MATLAB

2012 ◽  
Vol 510 ◽  
pp. 205-210
Author(s):  
Xiao Feng Xiao ◽  
Qiong Xue
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Volume Method

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated.. Transient heat conduction analysis of infinite plate with uniform thickness and two dimensional rectangle region are realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

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