scholarly journals General Euler-Poisson-Darboux Equation and Hyperbolic B-Potentials

2019 ◽  
Vol 65 (2) ◽  
pp. 157-338
Author(s):  
E L Shishkina
Keyword(s):  
Wave Operator ◽  
Hyperbolic Equations ◽  
Inverse Operator ◽  
Weight Functions ◽  
Cauchy Problems ◽  
Bessel Operators ◽  
Hyperbolic Operators ◽  
Generalized Translation ◽  
Darboux Equation ◽  
Singular Wave

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz B-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler-Poisson-Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on $$\mathbb {R}^n$$Rn

2020 ◽  
Vol 192 (1) ◽  
pp. 1-38
Author(s):  
Ahmed Abdeljawad ◽  
Alessia Ascanelli ◽  
Sandro Coriasco
Keyword(s):  
Cauchy Problems ◽  
Hyperbolic Operators

scholarly journals A parametrix for quantum gravity?

2016 ◽  
Vol 13 (05) ◽  
pp. 1650060
Author(s):  
Giampiero Esposito
Keyword(s):  
Quantum Gravity ◽  
Hyperbolic System ◽  
Wave Operator ◽  
Wave Equations ◽  
Inverse Operator ◽  
Einstein Equations ◽  
Poisson Brackets ◽  
Maxwell Theory ◽  
Kirchhoff Type ◽  
Nonlinear Hyperbolic System

In the 60s, DeWitt discovered that the advanced and retarded Green functions of the wave operator on metric perturbations in the de Donder gauge make it possible to define classical Poisson brackets on the space of functionals that are invariant under the action of the full diffeomorphism group of spacetime. He therefore tried to exploit this property to define invariant commutators for the quantized gravitational field, but the operator counterpart of such classical Poisson brackets turned out to be a hard task. On the other hand, in the mathematical literature, it is by now clear that, rather than inverting exactly an hyperbolic (or elliptic) operator, it is more convenient to build a quasi-inverse, i.e. an inverse operator up to an operator of lower order which plays the role of regularizing operator. This approximate inverse, the parametrix, which is, strictly, a distribution, makes it possible to solve inhomogeneous hyperbolic (or elliptic) equations. We here suggest that such a construction might be exploited in canonical quantum gravity provided one understands what is the counterpart of classical smoothing operators in the quantization procedure. We begin with the simplest case, i.e. fundamental solution and parametrix for the linear, scalar wave operator; the next step are tensor wave equations, again for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear Einstein equations are studied, relying upon the well-established Choquet-Bruhat construction, according to which the fifth derivatives of solutions of a nonlinear hyperbolic system solve a linear hyperbolic system. The latter is solved by means of Kirchhoff-type formulas, while the former fifth-order equations can be solved by means of well-established parametrix techniques for elliptic operators. But then the metric components that solve the vacuum Einstein equations can be obtained by convolution of such a parametrix with Kirchhoff-type formulas. Some basic functional equations for the parametrix are also obtained, that help in studying classical and quantum version of the Jacobi identity.

scholarly journals On the stability of a laminated Timoshenko problem with interfacial slip in the whole space under frictional dampings or infinite memories

2020 ◽  
Vol 7 (1) ◽  
pp. 194-218
Author(s):  
Aissa Guesmia
Keyword(s):  
Initial Data ◽  
Hyperbolic Equations ◽  
Weight Functions ◽  
Stability Estimates ◽  
Interfacial Slip ◽  
Infinite Memory ◽  
Bounded Interval ◽  
Frictional Damping ◽  
Stability And Instability ◽  
The Stability

Abstract The author of the present paper considered in [16] a model describing a vibrating strucure of an interfacial slip and consists of three coupled hyperbolic equations in one-dimensional bounded interval, where the dissipation is generated by either a frictional damping or an infinite memory, and it is acting only on one component. Some strong, polynomial, exponential and non exponential stability results were proved in [16] depending on the values of the parameters and the regularity of the initial data. The objective of the present paper is to compelete the study of [16] by considering this model in the whole line ℝ and under only one control given by a frictional damping or an infinite memory. When the system is controled via its second or third component (rotation angle displacement or dynamic of the slip), we show that this control alone is sufficient to stabilize our system and get different polynomial stability estimates in the L 2-norm of the solution and its higher order derivatives with respect to the space variable. The decay rate depends on the regularity of the initial data, the nature of the control and the parameters in the system. However, when the system is controled via its first component (transversal displacement), we found a new stability condition depending on the parameters in the system. This condition defines a limit between the stability and instability of the system in the sense that, when this condition is staisfied, the system is polynomially stable. Otherwise, when this condition is not satisfied, we prove that the solution does not converge to zero at all. The proofs are based on the energy method and Fourier analysis combined with judicious choices of weight functions.

scholarly journals Problem with data on the characteristics for a loaded system of hyperbolic equations

2021 ◽  
Vol 31 (3) ◽  
pp. 353-364
Author(s):  
A.T. Assanova ◽  
A. Zholamankyzy
Keyword(s):  
Approximate Solution ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Rectangular Domain ◽  
Second Order ◽  
Cauchy Problems ◽  
Equivalent Problem ◽  
New Approach ◽  
Integral Relations

We consider a problem with data on the characteristics for a loaded system of hyperbolic equations of the second order on a rectangular domain. The questions of the existence and uniqueness of the classical solution of the considered problem, as well as the continuity dependence of the solution on the initial data, are investigated. We propose a new approach to solving the problem with data on the characteristics for the loaded system of hyperbolic equations second order based on the introduction new functions. By introducing new unknown functions the problem is reduced to an equivalent family of Cauchy problems for a loaded system of differential with a parameters and integral relations. An algorithm for finding an approximate solution to the equivalent problem is proposed and its convergence is proved. Conditions for the unique solvability of the problem with data on the characteristics for the loaded system of hyperbolic equations of the second order are established in the terms of coefficient's system.

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