Mixed problem for the singular partial differential equation of parabolic type
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
Abstract
The scheme for solving of a mixed problem is proposed for a differential equation a(x)∂T∂τ=∂∂x(c(x)∂T∂x)−g(x)T with coefficients a(x), g(x) that are the generalized derivatives of functions of bounded variation, c(x)>0, c−1(x) is a bounded and measurable function. The boundary and initial conditions have the form p1T(0,τ)+p2T[1]x(0,τ)=ψ1(τ),q1T(l,τ)+q2T[1]x(l,τ)=ψ2(τ), T(x,0)=φ(x), where p1p2≤0, q1q2≥0 and by T[1]x(x,τ) we denote the quasiderivative c(x)∂T∂x. A solution of this problem seek by the reduction 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 of two problems: a quasistationary boundary problem with initial and boundary conditions for the search of the function u(x,τ) and a mixed problem with zero boundary conditions 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 and expansions in eigenfunctions of some boundary value problem for the second-order quasidifferential equation (c(x)X′(x))′−g(x)X(x)+ωa(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.
Authors and Affiliations
O. Makhnei
Keywords
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- EP ID EP532827
- DOI 10.15330/cmp.10.1.165-171
- Views 47
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