An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab

Boundary Value Problems, 2015

Begell House, Inc, 2015

This paper studies an analytical method which combines the superposition technique along with the solution structure theorem such that a closed-form solution of the hyperbolic heat conduction equation can be obtained by using the fundamental mathematics. In this paper, the non-Fourier heat conduction in a slab at whose a left boundary there is a constant heat fl ux and at the right boundary, a constant temperature Ts = 15, has been investigated. The complicated problem is split into multiple simpler problems that in turn can be combined to obtain a solution to the original problem. The original problem is divided into fi ve subproblems by sett ing the heat generation term, the initial conditions, and the boundary conditions for diff erent values in each subproblem. All the solutions given in this paper can be easily proven by substituting them into the governing equation. The results show that the temperature will start retreating at approximately t = 2 and for t = 2 the temperature at the left boundary decreases leading to a decrease in the temperature in the domain. Also, the shape of the profi les remains nearly the same aft er t = 4. The solution presented in this study can be used as benchmark problems for validation of future numerical methods.

Journal of Heat Transfer, 2013

2005

Applied Mathematical Modelling, 1981

Steady state temperature fields in domains with temperature dependent heat conductivity and mixed boundary conditions involving a temperature dependent heat transfer coefficient and radiation were considered. The nonlinear heat conduction equation was transformed into Laplace's equation using Kirchhoff's transform. Due to this transform the non-linearity is transferred from the differential equation only to third kind boundary conditions. The remaining boundary conditions of first and second kind, became linear. Applying Green's theorem to transformed problem results in integral equation containing boundary integrals only. Discretization of this integral equation gives a system of algebraic equations with linear matrix and nonlinear right hand sides. Such set of equations can be solved iteratively. Numerical examples are included.

2006

Journal of Siberian Federal University. Mathematics & Physics, 2020

The construction of solutions to the problem with a free boundary for the non-linear heat equation which have the heat wave type is considered in the paper. The feature of such solutions is that the degeneration occurs on the front of the heat wave which separates the domain of positive values of the unknown function and the cold (zero) background. A numerical algorithm based on the boundary element method is proposed. Since it is difficult to prove the convergence of the algorithm due to the non-linearity of the problem and the presence of degeneracy the comparison with exact solutions is used to verify numerical results. The construction of exact solutions is reduced to integrating the Cauchy problem for ODE. A qualitative analysis of the exact solutions is carried out. Several computational experiments were performed to verify the proposed method

Mechanical Engineering Journal, 2017

Boundary Value Problems

In literature, the well-acknowledged initial condition T t x; tj t0 0 is generally used when solving the non-Fourier problems. However, in references, some cases with a suddenly applied heat flux boundary under such an initial condition violate the first law of thermodynamics. This unreasonable situation is first demonstrated in the cases of constant heat flux using the Cattaneo-Vernotte model of a one-dimensional finite medium with the other side being isothermal or insulated. Then it is also demonstrated in the cases of arbitrary heat flux by reduction to absurdity. To adjust the unreasonable situation, an innovative definition of initial condition is proposed to obtain accurate solutions. Besides, the modified cases of a constant heat flux boundary are also discussed in this work with different relaxation time. In those cases, this study, for the first time, discovers that the heat wave can be inversely reflected under the isothermal boundary. Then when τ is extremely big, the heat flux input may result in a cooling response. Nomenclature A n , B n = coefficients defined in Eq. (21) B 0 n = coefficient defined in Eq. (40) = thermal diffusion coefficient E = internal energy Fx; t = inner heat generation function fμ; γ = function defined in Eq. (12a) gμ = function defined in Eq. (39) Ht = Heaviside step function, 0; t 0 1; t > 0 k = thermal conductivity L = length of finite medium Lu; ∂u=∂nj ∂Ω = symbol standing for three kinds of boundary qx; t = heat flux vector qt = function of surface heat flux q = dimensionless heat flux q 0 = constant heat flux sinε n γ = symbol defined in Eq. (22) Tx; t = temperature t = time V CV = velocity of heat wave V CV = dimensionless velocity of heat wave W χ = operator standing for the solution of Eq. (14) Xμ, Yγ = factors of separation of variables defined in Eq. (15a) X n μ, Y n γ = functions defined in Eq. (16) x = spatial variable γ = dimensionless time γ 0 = time when heat flux firstly reaches the end ε n = parameter defined in Eq. (22) ε n = symbol defined in Eq. (26) θ = dimensionless temperature λ = eigenvalues defined in Eq. (17) λ n = set of eigenvalues μ = dimensionless spatial variable ξ, ζ = auxiliary function τ = dimensionless relaxation time τ 0 = relaxation time φμ, ψμ = functions defined in Eq. (12c) χμ = function defined in Eq. (13) Subscript n = number of series