The Method of Fundamental Solutions for the Moving Boundary Problem of the One-Dimension Heat Conduction Equation

2014 ◽  
Vol 1039 ◽  
pp. 59-64 ◽  
Author(s):  
Xin Jiang ◽  
Xiao Gang Wang ◽  
Yue Wei Bai ◽  
Chang Tao Pang
Keyword(s):  
Heat Conduction ◽  
Boundary Problem ◽  
Heat Conduction Equation ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Fundamental Solutions ◽  
Metal Wire ◽  
Moving Boundary Problem ◽  
Conduction Equation ◽  
Temperature Function

The melting of the material is regarded as the moving boundary problem of the heat conduction equation. In this paper, the method of fundamental solution is extended into this kind of problem. The temperature function was expressed as a linear combination of fundamental solutions which satisfied the governing equation and the initial condition. The coefficients were gained by use of boundary condition. When the metal wire was melting, process of the moving boundary was gained through the conversation of energy and the Prediction-Correlation Method. A example was given. The numerical solutions agree well with the exact solutions. In another example, numerical solutions of the temperature distribution of the metal wire were obtained while one end was heated and melting.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

2016 ◽  
Vol 33 (1) ◽  
pp. 65-75 ◽  
Author(s):  
M. R. Talaee ◽  
V. Sarafrazi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Time Dependent ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Modified Quadrature Method for Solving BIEs of Steady-State Anisotropic Heat Conduction Problems

2014 ◽  
Vol 687-691 ◽  
pp. 1354-1358
Author(s):  
Xin Luo ◽  
Jin Huang
Keyword(s):  
Integral Equation ◽  
Heat Conduction ◽  
Steady State ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Trapezoidal Rule ◽  
Conduction Equation ◽  
Quadrature Method ◽  
Singular Kernels ◽  
Anisotropic Heat Conduction

In this paper, steady-state anisotropic heat conduction equation can be converted into the first kind integral equation, then modified quadrature formula based on trapezoidal rule is used to deal the integrals with singular kernels. In addition, Sidi transformation is applied to remove the singularities at concave points in concave polygons. This technique improves the accuracy of numerical solutions of the heat conduction equation. Numerical results show the convergence rate of the proposed method is the order three.

Approximate solutions in the theory of thermal explosions for semi-infinite explosives

1968 ◽  
Vol 305 (1481) ◽  
pp. 205-217 ◽  
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Surface Flux ◽  
Approximate Solutions ◽  
Conduction Equation ◽  
Inert Material ◽  
Zero Order Reaction ◽  
Thermal Explosions

If the solution, of the heat conduction equation θ τ ( 0 ) = θ ξ ξ ( 0 ) , ξ > 0 , τ > 0 of a chemically ‘inert’ material is known, then an approximate formula for the explosion time, ד expl. , of an explosive satisfying the heat conduction equation with zero order reaction, θ ד = θ ξξ +exp(-1/θ), ξ > 0, ד 0, and the same initial and boundary conditions as the ‘inert’, is given by the root of the equation, − ∂ θ ( 0 ) ( ξ , τ expt . ) / ∂ ξ | ξ − 0 = ∫ 0 ∞ exp ⁡ [ − 1 / θ ( 0 ) ( ξ , τ expl . ) ] d ξ provided 1/θ (0) (ξ, ד) is suitably expanded about the surface ξ = 0 such that the integrand vanishes as ξ→∞. Similar results hold for one-dimensional cylindrically and spherically symmetric problems. The derivation of the explosion criterion is based on observation of existing numerical solutions where it is seen that (i) almost to the onset of explosion, the solution θ(ξ, ד )does not differ appreciably from θ (0) (ξ, ד ) (ii) the onset of explosion is indicated by the appearance of a temperature maximum at the surface. Simple formulas for ד expl. readily obtainable for a wide variety of boundary conditions, are given for seven sample problems. Among these are included a semi-infinite explosive with constant surface flux, convective surface heat transfer, and constant surface temperature with and without subsurface melting. The derived values of ד expl. are in satisfactory agreement with those obtained from finite-difference solutions for the problems that can be compared.

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