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On asymptotic expansions of global errors

100%
Pfeifer E.
Mathematica Applicanda
|
1990
|
tom 18
|
nr 32
PL
.
EN
We consider a linear partial differential equation with constant coefficients and the corresponding difference equation. Conditions are given under which the error possesses an asymptotic expansion with respect to h. These results are applied to obtain asymptotic expansions of some quadratures and of the approximate solution of a boundary value problem for a second order differential equation.
2
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The optimal explicit unconditionally stable box scheme

63%
Moszyński K.
Mathematica Applicanda
|
1997
|
tom 26
|
nr 40
PL
.
EN
The finite difference “box” scheme, (see also [1],[2]), is considered on the simplest possible model of single first order linear hyperbolic equation: ut+mux=0 with constant, coefficient m, and one space variable. The optimal version of the scheme, which is nonoscilating and unconditionally stable with respect to the initial and boundary conditions, is derived in the class of box schemes of the order at, least one. If apropriately iterated, this ścinane may be applied to general systems of quasilinear first order hyperbolic equations in one space variable, as an explicit, unconditionally stable solver. For more than one space variable this solver is applicable via splitting (see [3]).
3
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Difference method for a nonlinear parabolic equation of second order in two space variables

63%
Kawecki R.
Mathematica Applicanda
|
1990
|
tom 18
|
nr 32
PL
.
EN
A.C. Reynolds in his paper (1972) proposed a difference parametric method for solving the Fourier problem for a nonlinear parabolic equation of second order in one space variable. The paper presents a generalization of Reynolds’ method for the problem in two space variables with mixed derivatives. In this paper, Fourier problems for a general class of nonlinear parabolic equation, in QT = Q x [0, T], are studied. To solve this problem we construct a finite difference scheme with a real parameter. We prove that the solutions of certain associated finite difference equations are unique and converge to the solution of the initial-boundary value problem with 0(h^2) rate of convergence.
4
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Numerical methods for the multiplicative partial differential equations

51%
Yazıcı M., Selvitopi H.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1344-1350
EN
We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.
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