Abstract
On the basis of systems of identical equalities and integral boundary characteristics, a new algorithm of solving a boundary-value problem on the nonstationary heat conduction in a canonical body with boundary condition of the second kind has been developed. The scheme proposed for finding approximate analytical solutions of boundary-value problems on nonstationary heat conduction with boundary conditions of the second kind involves the introduction into consideration of a temperature-disturbance front and separation of the whole heating process into two stages. For the first stage of this process, on the basis of the differentiation of the heat-conduction equation over a space variable and the application of symmetric integral and differential operators to the expressions obtained, two sequences of integral and differential identical equalities have been constructed. Each of these sequences includes integral or differential limiting characteristics for a definite boundary condition of the second kind. For the second stage, by way of introduction of a boundary function, differentiation of the heat-conduction equation with respect to a spatial coordinate, and application of integral operators to the expression obtained, a sequence of integral identical equalities involving integral boundary characteristics for the second-kind boundary condition has been constructed. On the basis of the integral and differential identical equalities obtained, closed systems of equations, allowing one to find polynomial coefficients of the temperature profile for the first and second stages of the heating process, have been constructed. A general scheme of determining approximate eigenvalues of boundary-value problems with boundary conditions of the second kind on the basis of construction of an ordinary differential equation and transformation of it into the characteristic equation is proposed. For each of the two stages of the heating process, special integral operators, reducing the boundary-value heat-conduction problem to the ordinary differential equation, are proposed.
Highlights
The scheme proposed for finding approximate analytical solutions of boundary-value problems on nonstationary heat conduction with boundary conditions of the second kind involves the introduction into consideration of a temperature-disturbance front and separation of the whole heating process into two stages
L. Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions / S
Summary
Введя в рассмотрение интегральный оператор ˆ 0 ≡ ∫ (1 - ξ) m (⋅) dξ, из (36) получим. Теперь применим к уравнению (11) интегральный оператор ˆ ξ ≡ ∫ (1- ξ) m (⋅) dξ :. Далее умножим уравнение (40) на (1–ξ)m и проинтегрируем по области ξ ∈[0,1]. С учетом полученного выше тождественного равенства (35) и введения в рассмотрение граничной функции g(t) = T (1,t),придем к уравнению dξ. В этом случае уравнение (41) предстанет в новой операторной форме: 1(Dt ( ˆ ξT )) ≡ Dt ( ( ˆ ξT )) = Q1 - g(t) / (1 + m). Интегрирование (42) с учетом условия ( ˆ ξT (ξ,t1))) ≡ 1( ˆξT (ξ,t1))),дает уравнение g(t1). Далее умножим левую и правую части уравнения (40) на (1 – ξ)m и проинтегрируем по области ξ ∈[ξ,1]:. Применив к уравнению (49) интегральный оператор ξ , получим:
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