The existence of generalized solutions for nonlinear initial-value problems was proved by Ansorge [1] in a general theorem. This paper shows, that a class of finite-difference methods, which approximate the solutions ...
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The existence of generalized solutions for nonlinear initial-value problems was proved by Ansorge [1] in a general theorem. This paper shows, that a class of finite-difference methods, which approximate the solutions of quasilinear equations, fulfills the assumptions of this theorem. The semi-linear theory given by Ansorge-Hass [2] is extended to quasilinear equations.
As well known, Gauss-quadrature formulas are constructed by integration of suitably choosen Hermitian interpolating polynomials. At first, a general linear interpolating operator and its error-term are given. Integrat...
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As well known, Gauss-quadrature formulas are constructed by integration of suitably choosen Hermitian interpolating polynomials. At first, a general linear interpolating operator and its error-term are given. Integrating this operator, one yields quadrature formulas containing some first derivatives of the integrand-function. By the requirement, that the weights of these derivatives should vanish all together, we get general quadrature formulas of Gauss-type. For some special basic functions of the interpolating operator, vanishing of the weights of the derivatives is necessary for minimization of the quadrature error. The general Gauss-quadratures may be determinated algebraically if sufficiently many fix-elements of the interpolating operator are known. As special cases one has, besides the known usual Gauss-quadratures and Wilf's quadrature procedure, a very surprising result.
The solution of the nonlinear boundary value problem$$y''(t) = - f{\text{ }}(t,{\text{ }}y{\text{ }}(t),{\text{ }}y'{\text{ }}(t)),{\text{ }}y{\text{ }}(0) = y{\text{ }}(1) = 0$$ under suitable conditions ...
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The solution of the nonlinear boundary value problem$$y''(t) = - f{\text{ }}(t,{\text{ }}y{\text{ }}(t),{\text{ }}y'{\text{ }}(t)),{\text{ }}y{\text{ }}(0) = y{\text{ }}(1) = 0$$ under suitable conditions onf(t, u, v) is constructed by a variant ofPicards iteration method. The convergence of the iteration sequence follows by considering a second iteration process, of which the convergence is shown by the theory of strictly monotonic operators onHilbert spaces.
New explicit fifth- and seventh-orderRunge-Kutta formulas are derived. They include a stepsize control procedure based on a complete coverage of the leading term of the local truncation error. These formulas require f...
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New explicit fifth- and seventh-orderRunge-Kutta formulas are derived. They include a stepsize control procedure based on a complete coverage of the leading term of the local truncation error. These formulas require fewer evaluations per step than otherRunge-Kutta formulas of corresponding order if the latter ones are also used with stepsize control (richardson's extrapolation to the limit). By a proper choice of some parameters the leading truncation error term of our formulas can be reduced substantially, thereby allowing an increase in the stepsize without loss of accuracy. A numerical example is presented. Our results being of the same accuracy, we save in this example 40% to 60% computer time compared with the knownRunge-Kutta formulas of corresponding order.
Several versions of an algorithm, which is an adaptation of the Todd-Coxeter-algorithm to semigroups, are described. They enumerate a representation by transformations of an abstractly presented semigroups the kernel ...
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Several versions of an algorithm, which is an adaptation of the Todd-Coxeter-algorithm to semigroups, are described. They enumerate a representation by transformations of an abstractly presented semigroups the kernel of this representation being determined by a subsemigroup, which is given by a finite set of generators. The enumeration process stops, if and only if the semigroup faithfully represented in this manner is finite.
A method is described to get the factorisation of big natural numbers, when their representations by certain binary quadratic forms are known. The realisation on a digital computer makes use of a number sieve given by...
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A method is described to get the factorisation of big natural numbers, when their representations by certain binary quadratic forms are known. The realisation on a digital computer makes use of a number sieve given byD. H. Lehmer.
The classical results on state reduction for complete deterministic automata (see [2]) can be proved for incomplete deterministic automata, too. This is especially due to the fact that, in spite of a widespread opinio...
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The classical results on state reduction for complete deterministic automata (see [2]) can be proved for incomplete deterministic automata, too. This is especially due to the fact that, in spite of a widespread opinion (see e. g. [2], p. 45), the notion of equivalence of states of incomplete automata can be defined in a similar way as in the complete case (this has also been remarked bySchmitt [7]), and that the notion of state homomorphism may likewise be generalized. The proof becomes very easy, since, in analogy to the normal one-point compactification of a partial algebra (see [6]), we associate to each incomplete automaton a unique complete automaton such that minimality is preserved.
This paper deals with a characterization of special cases when the distributive law is satisfied in interval arithmetic, a question first formulated byMoore [1]. To find a solution, the basic problem is subdivided int...
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This paper deals with a characterization of special cases when the distributive law is satisfied in interval arithmetic, a question first formulated byMoore [1]. To find a solution, the basic problem is subdivided into eight cases; in three of them distributivity ever holds, while in other three cases the law is failing. The conditions concerning the remaining cases are combined to a general characterization.
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