scholarly journals Semigroups and evolutionary equations

2021 ◽  
Author(s):  
Sascha Trostorff
Keyword(s):  
Initial Value Problem ◽  
Differential Algebraic Equations ◽  
Delay Equation ◽  
Algebraic Equations ◽  
Infinite Dimensions ◽  
Initial Value ◽  
Evolutionary Equations ◽  
Strongly Continuous Semigroups ◽  
Initial States

AbstractWe show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated within the framework of evolutionary equations, which is done by using the theory of extrapolation spaces. The results are applied to two examples. First, differential-algebraic equations in infinite dimensions are treated and it is shown, how a $$C_{0}$$ C 0 -semigroup can be associated with such problems. In the second example we treat a concrete hyperbolic delay equation.

Multi-Splitting Waveform Relaxation Methods for Solving the Initial Value Problem of Linear Integral-Differential-Algebraic Equations

2011 ◽  
Vol 403-408 ◽  
pp. 1763-1766
Author(s):  
Xiao Lin Lin ◽  
Yuan Sang ◽  
Hong Wei ◽  
Li Ming Liu ◽  
Yu Mei Wang ◽  
...  
Keyword(s):  
Spectral Radius ◽  
Discrete Time ◽  
Initial Value Problem ◽  
Differential Algebraic Equations ◽  
Waveform Relaxation ◽  
Algebraic Equations ◽  
Initial Value ◽  
Relaxation Methods ◽  
Discrete Time Case ◽  
Linear Integral

We present the multi-splitting waveform relaxation (MSWR) methods for solving the initial value problem of linear integral-differential-algebraic equations. Based on the spectral radius of the derived operator by decoupled process, a convergent condition is proposed for the MSWR method. Finally we discussed the convergent condition of discrete-time case of MSWR.

Operator Differential-Algebraic Equations Arising in Fluid Dynamics

2013 ◽  
Vol 13 (4) ◽  
pp. 443-470 ◽  
Author(s):  
Etienne Emmrich ◽  
Volker Mehrmann
Keyword(s):  
Existence And Uniqueness ◽  
Differential Algebraic Equations ◽  
Generalized Solutions ◽  
Operator Matrix ◽  
Algebraic Equations ◽  
Infinite Dimensions ◽  
Initial Value ◽  
Block Operator ◽  
Oseen Problem ◽  
Problem Data

Abstract. Existence and uniqueness of generalized solutions to initial value problems for a class of abstract differential-algebraic equations (DAEs) is shown. The class of equations covers, in particular, the Stokes and Oseen problem describing the motion of an incompressible or nearly incompressible Newtonian fluid but also their spatial semi-discretization. The equations are governed by a block operator matrix with entries that fulfill suitable inf-sup conditions. The problem data are required to satisfy appropriate consistency conditions. The results in infinite dimensions are compared in detail with those known for the DAEs that arise after semi-discretization in space. Explicit solution formulas are derived in both cases.

The initial value problem for Kelvin vortex waves

1997 ◽  
Vol 344 ◽  
pp. 181-212 ◽  
Author(s):  
STEVE ARENDT ◽  
DAVID C. FRITTS ◽  
ØYVIND ANDREASSEN
Keyword(s):  
Initial Value Problem ◽  
Vortex Tube ◽  
Formal Solution ◽  
Initial Perturbation ◽  
Gaussian Function ◽  
Wave Dispersion ◽  
Initial Value ◽  
Physical Mechanisms ◽  
Constant Vorticity ◽  
Initial States

We present a formal solution to the initial value problem for small perturbations of a straight vortex tube with constant vorticity, and show that any initial perturbation to such a tube evolves exclusively as a collection of Kelvin vortex waves. We then study in detail the evolution of the following particular initial states of the vortex tube: (i) an axisymmetric pinch in the radius of the tube, (ii) a deflection in the location of the tube, and (iii) a flattening of the tube's cross-secton. All of these initial states are localized in the direction along the tube by weighting them with a Gaussian function. In each case, the initial perturbation is decomposed into packets of Kelvin vortex waves which then propagate outward along the vortex tube. We discuss the physical mechanisms responsible for the propagation of the wave packets, and study the consequences of wave dispersion for the solution.

Solution Numerical of Stiff ODEs and DAEs in Mechanical and Other Systems

1997 ◽  
Author(s):  
H. Pasic
Keyword(s):  
Formal Solution ◽  
Differential Algebraic Equations ◽  
Single Step ◽  
Algebraic Equations ◽  
Stiff Differential Equations ◽  
Index Zero ◽  
Initial Value ◽  
Polynomial Coefficients ◽  
Multi Stage ◽  
Index 2

Abstract Presented is a formal solution of the initial-value problem of the system of general implicit differential-algebraic equations (DAEs) F(x, y, y’) = 0 of index zero or higher, based on perturbations of the polynomial coefficients of the vector y(x). The equation is linearized with respect to the coefficients and brought into a form suitable for implementation of the weighted residual methods. The solution is advanced by a single-step multi-stage collocation qadrature formula which is stiffly accurate and suitable for solving stiff differential equations and DAEs that arise in many mechanical and other systems. The algorithm is illustrated by two index-2 and index-3 examples — one of which is the well known pendulum problem.

A Finite Element Method for the Solution of Free Boundary Problem

2004 ◽  
Author(s):  
K. N. Rai ◽  
D. C. Rai
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Finite Element ◽  
Thermal Diffusivity ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Initial Value Problem ◽  
Algebraic Equations ◽  
Initial Value

A finite element method is presented for the solution of a free boundary problem which arises during planar melting of a semi-infinite medium initially at a temperature which is slightly below the melting temperature of the solid. The surface temperature is assumed to vary with time. Two different situations are considered (I) when thermal diffusivity is independent of temperature and (II) when thermal diffusivity varies linearly with temperature. The differential equation governing the process is converted to initial value problem of vector matrix form. The time function is approximated by Chebyshev series and the operational matrix of integration is applied, a linear differential equation can be represented by a set of linear algebraic equations and a nonlinear differential equation can be represented by a set of nonlinear algebraic equations. The solution of the problem is then found in terms of Chebyshev polynomial of second kind. The solution of this initial value problem is utilized iteratively in the interface heat flux equation to determine interface location as well as the temperature in two regions. The method appears to be accurate in cases for which closed form solutions are available, it agrees well with them. The effect of several parameters on the melting are analysed and discussed.

scholarly journals Maximal Regularity for Non-autonomous Evolutionary Equations

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Sascha Trostorff ◽  
Marcus Waurick
Keyword(s):  
Differential Equations ◽  
Maximal Regularity ◽  
Normal Operator ◽  
Time Derivative ◽  
Parabolic Type ◽  
Regularity Result ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Evolutionary Equations

AbstractWe discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

scholarly journals THE MATHEMATICAL MODELLING OF THE NONLINEAR HEAT TRANSPORT IN THIN PLATE

1999 ◽  
Vol 4 (1) ◽  
pp. 44-50
Author(s):  
A. Buikis
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Mathematical Modelling ◽  
Heat Transport ◽  
Initial Value Problem ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
First Order ◽  
Nonlinear Algebraic Equations

The approximations of the nonlinear heat transport problem are based on the finite volume (FM) and averaging (AM) methods [1,2]. This procedures allows reduce the nonlinear 2‐D problem for partial differential equation (PDE) to a initial‐value problem for a system of 2 nonlinear ordinary differential equations(ODE) of first order in the time t or to a initial‐value problem for one nonlinear ODE of first order with two nonlinear algebraic equations.

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