scholarly journals CORRECTNESS OF THE MIXED PROBLEM FOR ONE CLASS OF DEGENERATE MULTIDIMENSIONAL HYPERBOLO-PARABOLIC EQUATIONS

2020 ◽  
Vol 6 (334) ◽  
pp. 27-35
Author(s):  
C.A. Aldashev ◽  
◽  
E. Kazez ◽  
◽  
◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Mixed Problem ◽  
Electromagnetic Fields ◽  
Parabolic Equations ◽  
Hyperbolic Equations ◽  
Heat Propagation ◽  
Explicit Representation ◽  
Electromagnetic Process ◽  
Elastic Membranes ◽  
Multidimensional Hyperbolic Equations

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, we get degenerate multi-dimensional hyperbolic equations. If the medium has a high conductivity, then we go to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to degenerate multidimensional hyperbolic-parabolic equations. Also, it is known that the oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Studying the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Consequently, by studying the mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it is necessary to obtain an explicit representation of the solutions of the studied problems. The mixed problem for degenerate multidimensional hyperbolic equations was previously considered. As far as is known, these questions for degenerate multidimensional hyperbolic-parabolic equations have not been studied. In this paper, unique solvability is shown and an explicit form of the classical solution of the mixed problem for one class of degenerate multidimensional hyperbolic-parabolic equations is obtained.

scholarly journals IXED PROBLEM FOR THREE-DIMENSIONAL HYPERBOLIC EQUATIONS WITH DEGENERATION OF TYPE AND ORDER

2020 ◽  
Vol 6 (334) ◽  
pp. 13-18
Author(s):  
C. A. Aldashev ◽  
◽  
Z. N. Kanapyanova ◽  
◽  
◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Mixed Problem ◽  
Electromagnetic Fields ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
Heat Propagation ◽  
Electromagnetic Process ◽  
Elastic Membranes ◽  
Cylindrical Region ◽  
Multidimensional Hyperbolic Equations

It is known that in space during mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, then we get degenerating multidimensional hyperbolic equations. Therefore, the analysis of electromagnetic fields in complex environments (for example, if the conductivity of the medium changes) is reduced to degenerating multidimensional hyperbolic equations. It is also known that oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Therefore, by studying mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerating multidimensional hyperbolic equations. When studying these applications, it becomes necessary to obtain a clear representation of the solutions to the investigated problems. The mixed problem for degenerating multidimensional hyperbolic equations in generalized spaces is well researched. This task is also studied in the works of S. A. Aldashev, where it is shown that its correctness significantly depends on the height of the cylindrical region under consideration. A.V. Bitsadze drew attention to the importance of studies of multidimensional hyperbolic equations with degeneration of type and order. Mixed problems for these equations have not previously been studied. In this work, the solvability of a mixed problem is shown and a clear form of a classical solution for three-dimensional hyperbolic equations with degeneration of type and order is obtained.

scholarly journals TRICOMI PROBLEM FOR MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATION

2021 ◽  
Vol 26 (4) ◽  
pp. 7-14
Author(s):  
S. A. Aldashev
Keyword(s):  
Parabolic Equation ◽  
Electromagnetic Fields ◽  
Parabolic Equations ◽  
Hyperbolic Equations ◽  
Explicit Representation ◽  
Tricomi Problem ◽  
Electromagnetic Process ◽  
Representation Of Solutions ◽  
The Media ◽  
Multidimensional Hyperbolic Equations

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the media. If the medium is non-conducting, then we obtain multidimensional hyperbolic equations. If the mediums conductivity is higher, then we arrive at multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to multidimensional hyperbolic-parabolic equations. When studying these applications, one needs to obtain an explicit representation of solutions to the problems under study. Boundary-value problems for hyperbolic-parabolic equations on a plane are well studied; however, their multidimensional analogs have been analyzed very little. The Tricomi problem for the above equations has been previously investigated, but this problem in space has not been studied earlier. This article shows that the Tricomi problem is not uniquely solvable for a multidimensional mixed hyperbolic-parabolic equation. An explicit form of these solutions is given.

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.

scholarly journals Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Necmettin Aggez
Keyword(s):  
Space Variable ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Integral Condition ◽  
Variable Coefficients ◽  
Difference Schemes ◽  
Order Difference ◽  
One Dimensional ◽  
Nonlocal Integral Condition ◽  
Multidimensional Hyperbolic Equations

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-dimensional hyperbolic equations are supported by numerical examples.

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