Behavior of Solutions of the Cauchy Problem and the Mixed Initial Boundary Value Problem for an Inhomogeneous Hyperbolic Equation with Periodic Coefficients

2020 ◽  
pp. 29-35
Author(s):  
Hovik A. Matevossian ◽  
Giorgio Nordo ◽  
Anatoly V. Vestyak
Keyword(s):  
Cauchy Problem ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Periodic Coefficients ◽  
The Cauchy Problem ◽  
Initial Boundary Value

Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

2020 ◽  
Author(s):  
Georgios Akrivis ◽  
Buyang Li
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inner Product ◽  
Polynomial Degree ◽  
Finite Element Approximations ◽  
The Cauchy Problem ◽  
Initial Boundary Value

Abstract For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order $1\leqslant q\leqslant 5$ and in space by the Galerkin finite element method of polynomial degree $r-1$, with $r\geqslant 2$. We establish an error estimate of $O(\tau ^q\varepsilon ^{-q-\frac 12}+h^{r}\varepsilon ^{-r-\frac 12}+{e}^{-c/\varepsilon })$ with explicit dependence on the small parameter $\varepsilon$ describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal $L^p$-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.

scholarly journals On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

2019 ◽  
Vol 21 (4) ◽  
pp. 413-429
Author(s):  
Pavel N. Burago ◽  
Albert I. Egamov
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Phase Constraint ◽  
Initial Boundary Value

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

scholarly journals INITIAL-BOUNDARY VALUE PROBLEM FOR ONE-DIMENSION HYPERBOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

2017 ◽  
Vol 17 (8) ◽  
pp. 95-101
Author(s):  
M.V. Strigun
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Integral Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Integral Boundary ◽  
Nonlocal Integral Condition ◽  
Initial Boundary Value

In this paper, we study an initial-boundary value problem with nonlocal integral condition for a hyperbolic equation. The existence and uniqueness of a generalized solution of the problem is proved.

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