On the solvability in a weighted space of an initial–boundary value problem for a third-order operator-differential equation with a parabolic principal part

2016 ◽  
Vol 93 (1) ◽  
pp. 85-88 ◽  
Author(s):  
A. R. Aliev ◽  
F. S. Lachinova
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Weighted Space ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Operator Differential Equation ◽  
Order Operator ◽  
Third Order ◽  
Initial Boundary Value

scholarly journals Solvability of initial-boundary value problem of a multiple characteristic fifth-order operator-differential equation

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Nashat Faried ◽  
Labib Rashed ◽  
Abdel Baset I. Ahmed ◽  
Mohamed A. Labeeb
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Operator Differential Equation ◽  
Order Operator ◽  
Multiple Characteristic ◽  
Fifth Order

Abstract In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

scholarly journals A numerical method for solving the second initial-boundary value problem for a multidimensional third-order pseudoparabolic equation

2021 ◽  
Vol 31 (3) ◽  
pp. 384-408
Author(s):  
M.Kh. Beshtokov
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Pseudoparabolic Equation ◽  
Third Order ◽  
One Dimensional ◽  
Initial Boundary Value

The work is devoted to the study of the second initial-boundary value problem for a general-form third-order differential equation of pseudoparabolic type with variable coefficients in a multidimensional domain with an arbitrary boundary. In this paper, a multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter, and a locally one-dimensional difference scheme by A.A. Samarskii is used. Using the maximum principle, an a priori estimate is obtained for the solution of a locally one-dimensional difference scheme in the uniform metric in the $C$ norm. The stability and convergence of the locally one-dimensional difference scheme are proved.

scholarly journals On a third order initial boundary value problem in a plane domain

2012 ◽  
Vol 17 (3) ◽  
pp. 312-326
Author(s):  
Neringa Klovienė
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plane Domain ◽  
Third Order ◽  
Initial Boundary Value

Third order initial boundary value problem is studied in a bounded plane domain σ with C4 smooth boundary ∂σ. The existence and uniqueness of the solution is proved using Galerkin approximations and a priory estimates. The problem under consideration appear as an auxiliary problem by studying a second grade fluid motion in an infinite three-dimensional pipe with noncircular cross-section.

Existence Theorem for the Initial-Boundary Value Problem for a Singular Parabolic Partial Differential Equation

1972 ◽  
Vol 15 (2) ◽  
pp. 229-234
Author(s):  
Julius A. Krantzberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Image Position ◽  
Initial Boundary Value

We consider the initial-boundary value problem for the parabolic partial differential equation1.1in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).

scholarly journals APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

2020 ◽  
pp. 81-98
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Mixed Type ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Boundary Value

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

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