The Cauchy problem (constant coefficients)

1969 ◽  
pp. 114-155
Author(s):  
Lars Hörmander
Keyword(s):  
Cauchy Problem ◽  
The Cauchy Problem ◽  
Constant Coefficients

Approximation of the solution of the Cauchy problem with constant coefficients

1983 ◽  
Vol 34 (4) ◽  
pp. 337-340
Author(s):  
V. P. Burlachenko ◽  
Yu. I. Romanenko
Keyword(s):  
Cauchy Problem ◽  
The Cauchy Problem ◽  
Constant Coefficients

Exact Difference Schemes for Hyperbolic Equations

2007 ◽  
Vol 7 (4) ◽  
pp. 341-364 ◽  
Author(s):  
P. Matus ◽  
A. Kołodyńska
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Difference Scheme ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Cauchy Problem ◽  
Exact Difference ◽  
Constant Coefficients ◽  
Initial Boundary Value

AbstractIn the present paper, an exact difference scheme for the initial boundary- value problem of the third kind for an inhomogeneous hyperbolic equation of the second order with constant coefficients has been constructed on ordinary rectangular grids with constant space and time steps, where the Courant number γ=1. Later we proved a priori estimates of the stability in energy norm. For a quasi-linear wave equation on the moving characteristic grid a difference scheme has been constructed, which has the second order of approximation for the initial boundary-value problem and is exact for the Cauchy problem. The computational results for smooth functions and for a weak solution confirm the high accuracy of the introduced algorithm. We have also constructed exact difference schemes for the Cauchy problem for a system of two hyperbolic equations of the first order with constant coefficients on grids with constant space and time steps. Stability in energy norm for one of the constructed schemes has been proved. Using a method analogous to that used for the nonlinear wave equation a difference scheme for a nonlinear gas dynamic system has been constructed.

On the solution of Kac-type partial differential equations

1994 ◽  
Vol 31 (A) ◽  
pp. 311-324
Author(s):  
Mátyás Arató
Keyword(s):  
Stochastic Process ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Explicit Form ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
The Cauchy Problem ◽  
Constant Coefficients

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

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