scholarly journals Bicompact schemes for multidimensional hyperbolic equations on Cartesian meshes with solution-based AMR

2019 ◽  
pp. 1-26
Author(s):  
Mikhail Dmitrievich Bragin ◽  
Boris Vadimovich Rogov
Keyword(s):  
Hyperbolic Equations ◽  
Cartesian Meshes ◽  
Multidimensional Hyperbolic Equations ◽  
Bicompact Schemes

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.

scholarly journals The Inverse Problem for a General Class of Multidimensional Hyperbolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Gani Aldashev ◽  
Serik A. Aldashev
Keyword(s):  
Inverse Problem ◽  
General Class ◽  
Hyperbolic Equations ◽  
Existence Of Solution ◽  
Geophysical Prospecting ◽  
Hyperbolic Pde ◽  
Right Hand ◽  
Fourier Series Method ◽  
The Right ◽  
Multidimensional Hyperbolic Equations

Inverse problems for hyperbolic equations are found in geophysical prospecting and seismology, and their multidimensional analogues are especially important for applied work. However, whereas results have been established for the some narrow classes of hyperbolic equations, no results exist for more general classes. This paper proves the solvability of the inverse problem for a general class of multidimensional hyperbolic equations. Our approach consists of properly choosing the shape of the overidentifying condition that is needed to determine the right-hand side of the hyperbolic PDE and then applying the Fourier series method. We are then able to establish the results of the existence of solution for the cases when the unknown right-hand side is time- independent or space independent.

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.

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