An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab (original) (raw)
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Boundary Value Problems, 2015
A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material x > 0 with a zero temperature boundary condition at the face x = 0 is studied with the aim of finding explicit solutions. It is not a standard heat conduction problem because a heat source −Φ(x)F (V (t), t) is considered, where Φ and F are real functions and V represents the heat flux at the face x = 0. Explicit solutions independents of the space or temporal variables are given. Solutions with separated variables when the data functions are defined from the solution X = X(x) of a linear initial value problem of second order and the solution T = T (t) of a non-linear (in general) initial value problem of first order which involves the function F , are also given and explicit solutions corresponding to different definitions of the function F are obtained. A solution by an integral representation depending on the heat flux at the boundary x = 0 for the case in which F = F (V, t) = νV , for some ν > 0, is obtained and explicit expressions for the heat flux at the boundary x = 0 and for its corresponding solution are calculated when h = h(x) is a potential function and Φ = Φ(x) is given by Φ(x) = λx, Φ(x) = −µ sinh (λx) or Φ(x) = −µ sin (λx), for some λ > 0 and µ > 0. The limit when the temporal variable t tends to +∞ of each explicit solution obtained in this paper is studied and the "controlling" effects of the source term −ΦF are analysed by comparing the asymptotic behaviour of each solution with the asymptotic behaviour of the solution to the same problem but in absence of source term. Finally, a relationship between this problem with another non-classical initial and boundary value problem for the heat equation is established and explicit solutions for this second problem are also obtained. As a consequence of our study, several problems which can be used as benchmark problems for testing new numerical methods for solving partial differential equations are obtained.
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.
A Revised Approach for One-Dimensional Time-Dependent Heat Conduction in a Slab
Journal of Heat Transfer, 2013
Classical Green's and Duhamel's integral formulas are enforced for the solution of one dimensional heat conduction in a slab, under general boundary conditions of the first kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.
Existence and uniqueness of solutions of a nonlinear heat equation
2005
A nonlinear partial differential equation of the following form is considered: u ′ − div a(u)∇u + b(u) |∇u| 2 = 0, which arises from the heat conduction problems with strong temperature-dependent material parameters, such as mass density, specific heat and heat conductivity. Existence, uniqueness and asymptotic behavior of initial boundary value problems under appropriate assumptions on the material parameters are established.
Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions
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.
A nonlinear heat equation with temperature-dependent parameters
2006
A nonlinear partial differential equation of the following form is considered: u − div a(u)∇u + b(u) |∇u| 2 = 0, which arises from the heat conduction problems with strong temperature-dependent material parameters, such as mass density, specific heat and heat conductivity. Existence, uniqueness and asymptotic behavior of initial boundary value problems under appropriate assumptions on the material parameters are established for onedimensional case. Existence and asymptotic behavior for two-dimensional case are also proved.
On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation
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
A new simple analytical method for solving the problem of one-dimensional transient heat conduction in a slab of finite thickness is proposed, in which the initial temperature is assumed zero or constant and the boundary surfaces are assumed to be at constant temperature, constant heat flux, or insulated. In this method, the solution is expressed by an infinite series representation, each term of which is the temperature solution of the corresponding initial-boundary value problem for the semi-infinite solid. Each semi-infinite solid extends to infinity in the positive or negative direction of the x axis and the surface is located at various positions along the x axis. Each term and the partial sum in the infinite series automatically satisfy the heat conduction equation and the initial condition. The solution is easily constructed so that the boundary values of the partial sum converge to those of the heat conduction problem as the number of terms N increases to infinity. The basic concept of the solution method for the problem of one-dimensional transient heat conduction in a slab is described. The solution method is applied to various initial-boundary value problems. The formulas of the typical solutions by this method are found to be the same as those of the solutions obtained by other literature using the method of Laplace transformation, which supports the validity of the new solution method proposed in this paper. The usefulness of this method is also examined.
Non-classical heat conduction problem with nonlocal source
Boundary Value Problems
We consider the non-classical heat conduction equation, in the domain D = R n-1 × R + , for which the internal energy supply depends on an integral function in the time variable of the heat flux on the boundary S = ∂D, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable t with a parameter in R n-1. The solution to this Volterra equation is the heat flux (y, s) → V(y, t) = u x (0, y, t) on S, which is an additional unknown of the considered problem. We show that a unique local solution, which can be extended globally in time, exists. Finally a one-dimensional case is studied with some simplifications. We obtain the solution explicitly by using the Adomian method, and we derive its properties.
Analysis of Non-Fourier Heat Conduction Problem with Suddenly Applied Surface Heat Flux
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