scholarly journals THE WELL-POSEDNESS OF MIXED PROBLEM FOR ONE CLASS OF DEGENERATE MULTI-DIMENSIONAL HYPERBOLIC EQUATIONS

2019 ◽  
pp. 5-14
Author(s):  
S. A. Aldashev
Keyword(s):  
Boundary Value Problems ◽  
Classical Solution ◽  
Explicit Form ◽  
Mixed Problem ◽  
Unique Solvability ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Well Posedness ◽  
Elastic Membranes ◽  
Explicit Representations

Oscillations of elastic membranes in 3D are modelled as degenerate multi-dimensional hyperbolic equations. For applied work, it is important to obtain explicit representations of solution of the studied boundary-value problems. This paper shows the unique solvability and obtains the explicit form of the classical solution of the mixed problem for degenerate multi-dimensional hyperbolic equations.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

scholarly journals The Well-Posedness of the Dirichlet Problem in the Cylindric Domain for the Multidimensional Wave Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Serik A. Aldashev
Keyword(s):  
Wave Equation ◽  
Dirichlet Problem ◽  
Boundary Value Problems ◽  
Unique Solvability ◽  
Boundary Value ◽  
Well Posedness ◽  
Hyperbolic Pdes ◽  
Entire Boundary

In the theory of hyperbolic PDEs, the boundary-value problems with conditions on the entire boundary of the domain serve typically as the examples of the ill-posedness. The paper shows the unique solvability of the Dirichlet problem in the cylindric domain for the multidimensional wave equation. We also establish the criterion for the unique solvability of the equation.

A MIXED PROBLEM FOR A DEGENERATE MULTIDIMENSIONAL ELLIPTIC EQUATION

2021 ◽  
pp. 37-41
Author(s):  
Aisulu K. Tanirbergen
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Classical Solution ◽  
Mixed Problem ◽  
Unique Solvability ◽  
Complex Variable ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Cylindrical Domain

This article shows the unique solvability and obtains an explicit form of the classical solution of the mixed prob-lem in a cylindrical domain for a model degenerate multidimensional elliptic equation. The correctness of boundary value problems in the plane for elliptic equations by the method of the theory of ana-lytic functions of a complex variable has been well studied. The first boundary value problem or the Dirichlet problem for multidimensional elliptic equations with degeneration on the boundary has been sufficiently analyzed. However, as we know, the mixed problem for the indicated equations has been studied very little.

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