On the convergence of multistep methods for nonlinear two-point boundary value problems
Multistep methods for ordinary differential equations with parameters
Convergence of generalized-approximate iterations for an abstract equation
On the convergence of approximate iterations for an abstract equation
In questa Nota si studiano soluzioni approssimanti di una soluzione dell'equazione astratta x = f(x).
Boundary value problems for ODEs
We use the method of quasilinearization to boundary value problems of ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.
One-step methods for two-point boundary value problems in ordinary differential equations with parameters
A general theory of one-step methods for two-point boundary value problems with parameters is developed. On nonuniform nets , one-step schemes are considered. Sufficient conditions for convergence and error estimates are given. Linear or quadratic convergence is obtained by Theorem 1 or 2, respectively.
Boundary value problems for systems of functional differential equations
Algorithms for finding an approximate solution of boundary value problems for systems of functional ordinary differential equations are studied. Sufficient conditions for consistency and convergence of these methods are given. In the last section, a construction of methods of arbitrary order is presented.
Convergence of multistep methods for systems of ordinary differential equations with parameters
The author considers the convergence of quasilinear nonstationary multistep methods for systems of ordinary differential with parameters. Sufficient conditions for their convergence are given. The new numerical method is tested for two examples and it turns out to be a little better than the Hamming method.
Some remarks on numerical solution of initial problems for systems of differential equations
This paper presents a class of numerical methods for approximate solution of systems of ordinary differential equations. It is shown that under certain general conditions these methods are convergent for sufficiently small step size. We give estimations of errors which are better than the known ones.
One-step methods for ordinary differential equations with parameters
In the present paper we are concerned with the problem of numerical solution of ordinary differential equations with parameters. Our method is based on a one-step procedure for IDEs combined with an iterative process. Simple sufficient conditions for the convergence of this method are obtained. Estimations of errors and some numerical examples are given.
On numerical solution of ordinary differential equations with discontinuities
The author defines the numerical solution of a first order ordinary differential equation on a bounded interval in the way covering the general form of the so called one-step methods, proves convergence of the method (without the assumption of continuity of the righthad side) and gives a sufficient condition for the order of convergence to be .
An extension of the method of quasilinearization
The method of quasilinearization is a well–known technique for obtaining approximate solutions of nonlinear differential equations. This method has recently been generalized and extended using less restrictive assumptions so as to apply to a larger class of differential equations. In this paper, we use this technique to nonlinear differential problems.
Multipoint boundary value problems for ODEs. Part II
We apply the method of quasilinearization to multipoint boundary value problems for ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.
On the existence of solutions and one-step method for functional-differential equations with parameters
Functional differential equations
The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. In this paper we apply this technique to functional differential problems. It is shown that linear iterations converge to the unique solution and this convergence is superlinear.
Convergence of numerical methods for systems of neutral functional-differential-algebraic equations
A general class of numerical methods for solving initial value problems for neutral functional-differential-algebraic systems is considered. Necessary and sufficient conditions under which these methods are consistent with the problem are established. The order of consistency is discussed. A convergence theorem for a general class of methods is proved.
On a differential-algebraic problem
The method of quasilinearization is a procedure for obtaining approximate solutions of differential equations. In this paper, this technique is applied to a differential-algebraic problem. Under some natural assumptions, monotone sequences converge quadratically to a unique solution of our problem.
Quadratic convergence of monotone iterations of differential-algebraic equations
On delay differential equations with nonlinear boundary conditions.
Existence of solutions for functional antiperiodic boundary value problems.
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