scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

scholarly journals Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

2020 ◽  
Vol 40 (6) ◽  
pp. 725-736
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Time Dependent ◽  
Energy Solution ◽  
Noncylindrical Domain ◽  
Finite Energy Solution ◽  
Initial Boundary Value

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

scholarly journals Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
I. Amirali ◽  
G. M. Amiraliyev ◽  
M. Cakir ◽  
E. Cimen
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Energy Estimates ◽  
Fully Discrete ◽  
Discrete Norm ◽  
Explicit Finite Difference ◽  
Initial Boundary Value

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

2013 ◽  
Vol 5 (04) ◽  
pp. 548-568 ◽  
Author(s):  
Tao Lin ◽  
Yanping Lin ◽  
Xu Zhang
Keyword(s):  
Finite Element ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Interface Problem ◽  
Cartesian Mesh ◽  
Moving Interface ◽  
Immersed Finite Elements ◽  
Initial Boundary Value ◽  
Moving Interface Problems

AbstractThis article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

scholarly journals A high order splitting method for time-dependent domains

2008 ◽  
Vol 197 (51-52) ◽  
pp. 4763-4773 ◽  
Author(s):  
Tormod Bjøntegaard ◽  
Einar M. Rønquist
Keyword(s):  
Splitting Method ◽  
High Order ◽  
Time Dependent

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

1992 ◽  
Vol 121 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
Dao-Qing Dai ◽  
Wei Lin
Keyword(s):  
Lipschitz Constant ◽  
Complex Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Lipschitz Continuous ◽  
Two Space Dimensions ◽  
Image Position ◽  
Initial Boundary Value ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

SynopsisAn initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variablesThe complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

On an initial boundary value problem involving Beltrami-Moses fields in electromagnetic theory

1993 ◽  
Vol 344 (1671) ◽  
pp. 235-248 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Electromagnetic Theory ◽  
Time Dependent ◽  
Covariant Formulation ◽  
Beltrami Fields ◽  
Initial Boundary Value ◽  
Infinite Media

The so-called Harmuth ansatz consists of including autonomous magnetic sources in the time-dependent Maxwell postulates. The Beltrami fields are eigenfunctions of the curl operator, and have been used by Moses for propagation in infinite media. These developments are of relatively recent provenances in electromagnetic theory. We discuss an initial-boundary value problem (IBVP) within the framework of a manifestly covariant electromagnetic formalism by using the Harmuth ansatz. We also show how a covariant formulation of the Beltrami-Moses fields may be used for solving electromagnetic IBVPS.

scholarly journals NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD

2013 ◽  
Vol 18 (1) ◽  
pp. 80-96
Author(s):  
Andrejs Cebers ◽  
Harijs Kalis
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Rotating Magnetic Field ◽  
Droplet Dynamics ◽  
Ill Posed ◽  
Initial Boundary Value

Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).

Aspects of Osillations of Beams and Plates

2005 ◽  
Author(s):  
Mariya A. Zarubinska ◽  
W. T. van Horssen
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boundary Values ◽  
Internal Resonances ◽  
Specific Parameter ◽  
Plate Equations ◽  
Initial Boundary Value ◽  
Parameter Values

In this paper some initial boundary value problems for beam and plate equations will be studied. These initial boundary values problems can be regarded as simple models describing free oscillations of plates on elastic foundations or describing coupled torsional and vertical oscillations of a beam. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For the plate on an elastic foundation it turns out that complicated internal resonances can occur for specific parameter values.

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