Quasilinear Hyperbolic Equations Research Articles

The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping

The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping

Asymptotic stability of solutions to quasilinear damped wave equations with variable sources

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents: $$ u_{tt}-\operatorname{div}( |\nabla u|^{r(x)-2}\nabla u)+|u_t|^{m(x)-2} u_t-\Delta u_t=|u|^{q(x)-2}u, $$ with homogenous Dirichlet initial boundary value condition. An energy estimate and Komornik's inequality are used to prove uniform estimate of decay rates of the solution. We also show that $u(x, t)=0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.

Asymptotic stability of Solutions to Quasilinear Damped Wave Equations with Variable Sources

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents \[{u_{tt}} - div( {{{| {\nabla u} |}^{r(x) - 2}}\nabla u} ) + {| {{u_t}} |^{m(x) - 2}}{u_t} - \Delta {u_t} = {| u |^{q( x ) - 2}}u\] with homogenous Dirichlet initial boundary value condition. Some energy estimate and Komornik inequality are used to prove some uniform estimate of decay rates of the solution. And then, we prove that $u(x, t) = 0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.

Exterior Stability of Minkowski Space in Generalized Harmonic Gauge

We give a short proof of the existence of a small piece of null infinity for (3+1)-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.

Riemann solutions of two-layered blood flow model in arteries

This study investigates the solutions of the Riemann problem for a two-layered blood flow model which is modeled by a system of quasi-linear hyperbolic partial differential equations (PDEs) obtained by vertically averaging the Euler equations over each layer. We explore the elementary waves, namely shock wave, rarefaction wave and contact discontinuity wave on the basis of method of characteristics. Further, we establish the existence and uniqueness of the corresponding local Riemann solution. Across the contact discontinuity wave, the areas of two nonlinear algebraic equations are determined by using the Newton–Raphson method of two variables in all possible wave combinations. A precise analytical method is used to display a detailed vision of the solution for this model inside a specified space domain and some certain time frame.

High resolution numerical algorithm of order (2, 4, 4) for 2D quasi-linear hyperbolic equations based on half-step discretization on an unequal grid

This study suggests a novel three-level implicit half-step scheme of order two and four in time and space direction respectively, utilizing an unequal grid for the approximation of 2D quasi-linear hyperbolic partial differential equations on an unequal grid. The stability analysis of the model Telegraphic equation for unequal grid has been discussed and it has been demonstrated that the presented method is unconditionally stable for the Telegraphic equation. The alternating direction implicit (ADI) techniques are discussed for linear difference equations on a dissimilar domain. The developed method investigated several well-known physical problems on an unequal mesh to demonstrate its accuracy and efficiency.

On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schrödinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the time-dependent Dirac–Frenkel variational principle. Our main result addresses a general setting with smooth potentials where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations. We also discuss the case of Coulomb potentials, though treated differently (using Strichartz estimates and a classical fixed point argument).

DYNAMIC MODEL OF THERMAL PROCESSES IN A LIQUID FLOWING THROUGH A CHANNEL

The article presents a dynamic thermal process model for an incompressible fluid moving through a channel of arbitrary cross-sectional shape. The study focuses on expressing the non-steady law of conservation of internal energy of the fluid by considering the dependence of parameters on both spatial coordinates and time. The one-dimensional spatial formulation of the problem is simplified by disregarding any changes in flow parameters in the direction perpendicular to the channel axis. The resulting equation represents a non-stationary, non-homogeneous, quasi-linear hyperbolic partial differential equation, which can be solved with the method of characteristics in the case of constant coefficients. The solution consists of two distinct solutions that are glued along the line x/t=v, with isolines shown in the figure provided. The research is relevant for the design of heat exchangers, rocket and aircraft engines, and other heat engineering devices.

High-precision numerical method for 1D quasilinear hyperbolic equations on a time-graded mesh: application to Telegraph model equation

High-precision numerical method for 1D quasilinear hyperbolic equations on a time-graded mesh: application to Telegraph model equation

On the Existence, Decay and Blowup of Solutions for a Quasilinear Hyperbolic Equations Involving the Weighted $$p-$$Laplacian with Source Terms

On the Existence, Decay and Blowup of Solutions for a Quasilinear Hyperbolic Equations Involving the Weighted $$p-$$Laplacian with Source Terms

Начально-краевая задача о течении в кровеносном сосуде

The solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed. It is assumed that the side surface of the domain is a free (soft, compliant) boundary on which the kinematic condition is set. The dynamic condition at the boundary is modeled by some constitutive relation – the dependence of the pressure P on the cross-sectional area S of the cylindrical domain. To study the flow an asymptotic model based on the shallow water theory is used. When constructing the solution of the initial-boundary (as well as initial and boundary) problem for a system of two quasilinear partial differential equations of the first order, the hodograph method based on the conservation law is used. A variant of the polynomial constitutive relation is chosen for the study: P~S2β (β>0). In the case of the initial data problem which specified at the initial time the Riemann-Green function, allowing to construct an implicit analytical solution, and an algorithm (numerical) for constructing an explicit solution of the original problem are given. The main attention is focused on the special case P~S2, for which all the formulas necessary to construct a solution are written in explicit form. For P~S2, several variants of conditions (initial and boundary value) are considered for the initial boundary value problem, which allow us to trace the evolution of the solution in detail. Numerical calculations demonstrating the motion of shock waves and wave fronts are given. The results obtained, in practice, can be used to describe flows in blood vessels, as well as reliable tests for testing compu tational algorithms intended to solve such problems, in particular, systems of quasi-linear hyperbolic equations.

Similarity solution for isothermal flow behind the magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas with the effect of the gravitational field

In this paper, we investigate a system of quasilinear hyperbolic partial differential equations, which describes the propagation of cylindrical shock waves in a rotating non-ideal gas with the effects of the gravitational field and the axial magnetic field. It is assumed that the flow is isothermal. The Lie group of transformations is used to generate the self-similar solutions of the considered problem in the medium of uniform density. The axial and azimuthal components of fluid velocity and magnetic field are supposed to be varying. We find the generators of the Lie group of transformations by employing the invariant surface criteria. We discovered four alternative solutions by selecting the arbitrary constants indicated in the generators' phrase. Only in three out of these four cases, the self-similar solutions exist. Two types of shock paths appear while solving the above cases. The power-law shock path appears in the first and third cases, while the exponential-law shock path appears in the second case. To find self-similar solutions, these cases have been solved numerically. We graphically show the distributions of flow variables behind the shock wave so that we can observe the effect on flow variables of the various values of the non-ideal parameter, Alfvén Mach number, adiabatic exponent, gravitational parameter, and ambient azimuthal velocity exponent. For the computational task, we used “MATLAB” software.

Goursat’s problem on the plane for a quasilinear hyperbolic equation

A classical solution of the problem for a quasilinear hyperbolic equation in the case of two independent variables with given conditions for the desired function on the characteristic lines is obtained. The problem is reduced to a system of equations with a completely continuous operator. We constructed the unique solution by the method of successive approximations and showed the necessary and sufficient smoothness and matching conditions on given functions.

High precision implicit method for 3D quasilinear hyperbolic equations on a dissimilar domain: Application to 3D telegraphic equation

In this paper, we recommend a novel high accuracy compact three-level implicit numerical method of order two in time and four in space using unequal mesh for the solution of 3D quasi-linear hyperbolic equations on an irrational domain. The stability analysis of the model Telegraphic equation for unequal mesh has been discussed and it has been shown that the proposed method for Telegraphic equation is unconditionally stable. Operator splitting technique is used to solve 3D linear model equations. In this technique, we use very well-known tri-diagonal solver technique to solve a set of tri-diagonal matrices on an irrational domain. The projected scheme is examined on numerous physical problems on irrational domain to demonstrate the accurateness and efficiency of the suggested scheme.

A parallel type decomposition scheme for quasi-linear abstract hyperbolic equation

Abstract The Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is the sum of operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . Each summand is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval the classical difference problems are solved in parallel (independently from each other) with the operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . The weighted average of the obtained solutions is announced as an approximate solution at the right end of the local interval. The convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.

Global Smooth Sonic-Supersonic Flows in a Class of Critical Nozzles

This paper concerns the global existence of smooth sonic-supersonic potential flows in a two-dimensional expanding nozzle with the critical geometry at the inlet. The flow is sonic and its velocity is along the normal direction at the inlet, which is a segment vertical to the walls of the nozzle, and is supersonic in the nozzle. Such a sonic-supersonic model is governed by a quasilinear nonstrictly hyperbolic equation with degeneracy at the inlet, and the degeneracy is strong in the sense that all characteristics from the inlet coincide with the inlet and never approach the domain. Furthermore, the asymptotic behaviors of the average speed on the cross section and the normal acceleration on the walls are of the same order with respect to the distance to the inlet, and of different order with respect to the height of the inlet. We seek the flows whose asymptotic behavior near the inlet is the same as the average speed on the cross section. An interesting phenomenon is that the existence of such sonic-supersonic flows depends on the height of the inlet. More precisely, it is shown by means of a method of characteristics with many elaborate estimates that there exists uniquely such a flow if it is suitably small, while it does not if it is suitably large. The results in the paper, together with other works, describe completely the geometry of the de Laval nozzles where there are smooth transonic flows of Meyer type whose sonic points are all exceptional.

Numerical Method for 2D Quasi-linear Hyperbolic Equation on an Irrational Domain: Application to Telegraphic Equation

We develop a novel three-level compact method (implicit) of second order in time and space directions using unequal grid for the numerical solution of 2D quasi-linear hyperbolic partial differential equations on an irrational domain. The stability analysis of the model problem for unequal mesh is discussed and it is revealed that the developed scheme is unconditionally stable for the Telegraphic equation. For linear difference equations on an irrational domain, the alternating direction implicit method is discussed. The projected technique is scrutinized on several physical problems on an irrational domain to exhibitthe accuracy and effectiveness of the suggested method. Dhaka Univ. J. Sci. 69(2): 116-123, 2021 (July)

High resolution operator compact implicit half-step approximation for 3D quasi-linear hyperbolic equations and ADI method for 3D telegraphic equation on an irrational domain

We propose a novel three-level implicit half-step method of order two in time and four in space using unequal mesh for the solution of three-space dimensional quasi-linear hyperbolic equations on an irrational domain. The proposed method is directly applicable to 3D hyperbolic equation with singular coefficients, which is the key charisma of our scheme and we do not need any fictitious points for computation. The stability analysis of the model problem for unequal mesh has been discussed and it has been shown that the proposed method for Telegraphic equation is unconditionally stable. Alternating direction implicit (ADI) method by the help of a tri-diagonal solver is used to solve 3D linear schemes in three sweeps. The proposed method is scrutinized on several physical problems on dissimilar domain to exhibit the accuracy and effectiveness of the proposed method.

Exact solution for 1D deep bed filtration with particle capture by advection and dispersion

We consider a one-dimensional (1D) non-linear 2x2 hyperbolic problem of suspension-colloidal-nano transport in porous media, so-called deep bed filtration. The particle capture rate is proportional to the overall particle flux, which includes dispersion. The captured-particle concentration dependency on the proportionality coefficient, which is typical for large captured concentrations, causes the non-linearity of the problem. However, the 1D initial–boundary value problem allows for an exact solution. The introduction of the Riemann invariant converts the 2x2 system into a scalar quasi-linear hyperbolic equation, which is implicitly solved using the method of characteristics. The non-linear equation for particle capture rate yields a limited travelling wave that approximates the exact solution for the initial–boundary problem.

Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

We consider compact difference schemes of approximation order 4+2 on a three-point spatial stencil for theKlein–Gordon equations with constant and variable coefficients. New compact schemes areproposed for one type of second-order quasilinear hyperbolic equations. In the case of constantcoefficients, we prove the strong stability of the difference solution under small perturbations ofthe initial conditions, the right-hand side, and the coefficients of the equation. A priori estimatesare obtained for the stability and convergence of the difference solution in strong mesh norms.