A Problem for a Factorized Equation with a Pseudoparabolic Differential Operator
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Problem for a Factorized Equation with a Pseudoparabolic Differential Operator L. B. Mironova1* 1
Elabuga Institute of Kazan Federal University, 89 Kazanskaya str., Elabuga, 423600 Russia
Received September 29, 2019; revised September 29, 2019; accepted June 29, 2020
Abstract—We state a new problem for a fourth-order factorized differential equation, whose operator is the product of first- and third-order differential operators, and prove its unique solvability. We construct the solution in terms of the Riemann function of the corresponding third-order pseudoparabolic differential operator, determining one of functions that enter in the formula for the problem solution from resolvents of two integral equations. DOI: 10.3103/S1066369X20080058 Key words: factorized equation, pseudoparabolic equation, equation with a dominating partial derivative, Riemann method, Goursat problem, Volterra equation.
Various problems for factorized equations, whose operator represents the product of a first-order operator and a second-order hyperbolic one, are considered in papers ([1]–[5], [6], Ch. 4, § 1–2, pp. 212–226). In ([6], Ch. 4, § 2, pp. 223–226), V.I. Zhegalov considers the problem stated in a rectangular domain with data on characteristic lines for the factorized equation ∂ ∂ + (uxy + a(x, y)ux + b(x, y)uy + c(x, y)u) = 0. (1) ∂x ∂y Note that Eq. (1) belongs to one of four classes of third-order canonical equations with two independent variables described in the paper [7]. Authors of the paper [8] study problems for factorized equations with Bianchi operators, namely, for the equation ∂ ∂ ∂ + + L(u) = 0, (2) ∂x ∂y ∂z L(u) ≡ uxyz + a(x, y, z)uxy + b(x, y, z)uyz + c(x, y, z)uxz +d(x, y, z)ux + e(x, y, z)uy + f (x, y, z)uz + g(x, y, z)u, in the space of three independent variables and for its analog in the four-dimensional space. A.N. Mironov [9] expands these results to the equation in the space of an arbitrary dimension. In this paper, we consider the following equation with variable coefficients: ∂ ∂ + (uxxy + a20 uxx + a11 uxy + a10 ux + a01 uy + a00 u) = 0, (3) ∂x ∂y whose differential operator represents the product of a first-order operator and a third-order pseudoparabolic one. The equation uxxy + a20 uxx + a11 uxy + a10 ux + a01 uy + a00 u = f (x, y), *
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which belongs to the class of equations with a dominating partial derivative (it is often referred to as a third-order pseudoparabolic equation), was studied by many mathematicians (see, for example, [10]–[15]). It has many applications, in particular, it describes the diffusion processes in fractured media and moisture transfer process in soil [16]–[19]. Let us define the class of functions C (k,l) as follows: a function u belongs to C (k1 ,k2 ) , if there r1 +r2 (r = 0, . . . , ki ). We treat a solution to Eq. (3) that belongs exist continuous derivatives ∂ r1 u ∂x ∂y r2 i to the class C (3,2) as regular. Let D = {0 < x < x1 , 0 < y < y1 }, and let X and Y be parts of ∂D that lie on axes x and y, respectively. D
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