scholarly journals On Behavior of Solution of Degenerated Hyperbolic Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Tahir Gadjiev ◽  
Rafig Rasulov ◽  
Orkhan Aliev
Keyword(s):  
Hyperbolic Equation ◽  
Nonlinear Equations ◽  
Finite Time ◽  
Hyperbolic Equations ◽  
Initial Conditions ◽  
Local Existence ◽  
Hyperbolic Type ◽  
Nonlinear Hyperbolic Equations

The purpose of this paper is to learn some features of hyperbolic type of nonlinear equations. It is shown that the solution of the equation approaches to the endlessness in the inside of some initial conditions and time of the special marks. The local existence of the equation’s solution has been proved and the problem of unlimited increasing on the solution of nonlinear hyperbolic equations type during the finite time is investigated.

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 On blowing-up solutions for multi-time nonlinear hyperbolic equations and systems

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2599-2609
Author(s):  
B. Ahmad ◽  
A. Alsaedi ◽  
E. Cuesta ◽  
M. Kirane
Keyword(s):  
Nonlinear Equation ◽  
Hyperbolic Equation ◽  
Space Dimension ◽  
Hyperbolic Equations ◽  
Nonlinear Hyperbolic Equation ◽  
Two Dimensional ◽  
Nonlinear Hyperbolic Equations ◽  
Blowing Up ◽  
Time Order ◽  
Blowing Up Solutions

For a two-dimensional time nonlinear hyperbolic equation with a power nonlinearity, a threshold exponent depending on the space dimension is presented. Furthermore, the analysis is extended not only to a system of two equations but also to a two-time fractional nonlinear equation with different time order derivatives.

scholarly journals On Behaviors of the Energy of Solutions for Some Damped Nonlinear Hyperbolic Equations with p-Laplacian

2016 ◽  
Vol 6 ◽  
pp. 13-20
Author(s):  
Soufiane Mokeddem
Keyword(s):  
Decay Rate ◽  
Hyperbolic Equation ◽  
Hyperbolic Equations ◽  
Integral Inequalities ◽  
Nonlinear Hyperbolic Equations ◽  
Rate Estimate

In this paper we are concerned with nonlinear damped hyperbolic equation withp-Laplace of the formutt-Δpu+ σ(t)(ut-Δut)+w|u|m-2u= |u|r-2u.Used the multiplier techniques combined with a nonlinear integral inequalities given by Martinez we established a decay rate estimate for the energy.

Local existence and singularities formation for isothermal relativistic radiation hydrodynamics equations

2016 ◽  
Vol 13 (04) ◽  
pp. 661-683
Author(s):  
Yongcai Geng
Keyword(s):  
Blow Up ◽  
Hyperbolic Equations ◽  
Local Existence ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Smooth Solutions ◽  
Nonlinear Hyperbolic Equations ◽  
The Cauchy Problem ◽  
Finite Propagation Speed ◽  
Hydrodynamics Equations

We study the Cauchy problem for multi-dimensional compressible relativistic hydrodynamics in the presence of a radiation field. First, based on the theory of quasilinear symmetric hyperbolic, we establish the local existence of smooth solutions for both non-vacuum and vacuum cases. Next, in the spirit of Sideris’ work [T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 86 (1984) 369–381; T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985) 475–485], we show that smooth solutions blow-up in finite time if the initial data is compactly supported and large enough. Compared with the previous work, the main difficulties of the first problem lie in two aspects, we must first deal with the source terms relying on radiative quantities, and we also need to solve out the new coefficients matrices under the Lorentz transformation for vacuum case. The second difficulty arises on how to verifying that the smooth solution has finite propagation speed..

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

EXISTENCE OF PERIODIC SOLUTIONS FOR A FREE BOUNDARY PROBLEM OF HYPERBOLIC TYPE

2008 ◽  
Vol 05 (04) ◽  
pp. 785-806
Author(s):  
KAZUAKI NAKANE ◽  
TOMOKO SHINOHARA
Keyword(s):  
Mathematical Model ◽  
Free Boundary ◽  
Hyperbolic Equation ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Physical Phenomenon ◽  
Hyperbolic Type ◽  
Existence Of Periodic Solutions ◽  
A Free Boundary Problem ◽  
Thin Tape

A free boundary problem that arises from the physical phenomenon of "peeling a thin tape from a domain" is treated. In this phenomenon, the movement of the tape is governed by a hyperbolic equation and is affected by the peeling front. We are interested in the behavior of the peeling front, especially, the phenomenon of self-excitation vibration. In the present paper, a mathematical model of this phenomenon is proposed. The cause of this vibration is discussed in terms of adhesion.

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