Constructing numerical solutions to a nonlinear heat conduction equation with boundary conditions degenerating at the initial moment of time

2016 ◽  
Author(s):  
L. F. Spevak ◽  
A. L. Kazakov
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Conduction Equation ◽  
Initial Moment ◽  
Nonlinear Heat Conduction ◽  
Nonlinear Heat Conduction Equation

Some Algebraically Explicit Analytical Solutions of Unsteady Nonlinear Heat Conduction

2001 ◽  
Vol 123 (6) ◽  
pp. 1189-1191 ◽  
Author(s):  
Ruixian Cai ◽  
Na Zhang
Keyword(s):  
Heat Conduction ◽  
Analytical Solutions ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Conduction Equation ◽  
Nonlinear Heat Conduction ◽  
One Dimensional ◽  
Nonlinear Heat Conduction Equation ◽  
Conduction Theory ◽  
Unsteady Heat Conduction

The analytical solutions of nonlinear unsteady heat conduction equation are meaningful in theory. In addition, they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes, grid generation methods and so forth. However, very few explicit analytical solutions have been known for the unsteady nonlinear heat conduction. In order to develop the heat conduction theory, some algebraically explicit analytical solutions of nonlinear heat conduction equation have been derived in this paper, which include one-dimensional and two-dimensional unsteady heat conduction solutions with thermal conductivity, density and specific heat being functions of temperature.

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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