Boundary problem for the singular heat equation - Makhnei, O.V. (Carpathian Mathematical Publications, 2017)
The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation \[a(x)Tτ= x (λ(x)Tx)\] with coefficient a(x) that is thegeneralized derivative of a function of bounded variation, λ(x)>0, λ^-1(x) is a bounded and measurable function. The boundary conditions have the form { arraylp_11T(0,τ)+p_12T^[1]_x (0,τ)+ q_11T(l,τ)+q_12T^[1]_x (l,τ)= ψ_1(τ), p_21T(0,τ)+p_22T^[1]_x (0,τ)+ q_21T(l,τ)+q_22T^[1]_x (l,τ)= ψ_2(τ),array. where by T^[1]_x (x,τ) we denote the quasiderivative λ(x)Tx. A solution of this problem seek by thereduction method in the form of sum of two functions T(x,τ)=u(x,τ)+v(x,τ). This method allows to reduce solving of proposed problem to solving oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function u(x,τ) and a mixed problem with zero boundaryconditions for some inhomogeneous equation with an unknown function v(x,τ). The first of these problems is solved through the introduction of the quasiderivative. Fourier method andexpansions in eigenfunctions of some boundary value problem forthe second-order quasidifferential equation (λ(x)X'(x))'+ ω a(x)X(x)=0 are used for solving of the second problem. The function v(x,τ) is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.