Boundary Value Problem for Loaded Equation of Parabolic-Hyperbolic Type of the Third Order in an Infinite Three-Dimensional Domain

Abstract

In this paper, it is formulated and studied one of the problems for the loaded of parabolic-hyperbolic type equations of the third order in an infinite three-dimensional domain. The main method for study of the formulated problem is the Fourier transform. The uniqueness and existence of a regular solution of the problem are proved.

Boundary-Value Problem for a Loaded Mixed-Type Equation with a Characteristic Line of Type Change

“…The solution of this problem exists and is unique [17]. As in [15], we conclude that Problem 1 is uniquely solvable for 0 < b=a  1:…”

“…If a D 0 and b D 0 or b D 0 and c D 0 and, in addition, ˛i D 0; then we get the equations studied in [13,14]. Further, if a ¤ 0 and b D 0; then we obtain the equation with loaded operator considered in [15]. In view of these facts, we focus our attention on the problems for Eq.…”

Nonlocal <span aria-describedby="tippy-18">Boundary Problem for a Loaded Equation of Mixed Type in a Special Domain

“…However, boundary value problems for loaded equations of mixed type with an integral operator of fractional order in doubly connected domains have still not been sufficiently studied. Note that local and nonlocal boundary value problems for a loaded equation of elliptic -hyperbolic type in a doubly connected domain were studied in [22][23][24][25], in which the loaded part contains a differential operator or a trace of the desired function. The works [26][27] outline a technique for formulating correct boundary value problems with displacement for loaded second-order linear hyperbolic equations in special domains.…”

“…Borodin [8] studied the Dirichlet problem for a loaded equation of elliptic type. In the works of K.U.Khubiev [9], B.Islomov and D.M.Kuryazov [10], M.I.Ramazanov [11], analogues of the Tricomi and Gellerstedt problem for loaded hyperbolic-parabolic type equations were studied, and the Dirichlet problem for loaded equation with the Lavrentiev-Bitsadze operator in a rectangular domain Disclaimer/Publisher's Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.…”

Problem With Shift for a "Pointwise" Loaded Hyperbolic-Parabolic Equation

“…Среди работ, близких по тематике к исследуемой задаче, отметим также работы [11][12][13][14][15][16][17][18][19][20][21], посвященные гиперболо-параболическим и гиперболическим нагруженным уравнениям.…”