On Nordsieck's method for the numerical solution of ordinary differential equations

The connection between the class of methods suggested by Nordsieck and the class of linear multi-step methods is examined. It is shown that the starting procedure suggested by Nordsieck is specially suited to the Adams method.     

A method for solution of nonlinear ordinary differential equations

We present a method for the solution of nonlinear second-order differential equations by using a system of Fredholm equations of the second kind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1254–1260, September, 1995.     

A new PEC algorithm for the numerical solution of ordinary differential equations

Significant time reduction in obtaining numerical solutions of ordinary differential equations for which function evaluations are time consuming can be obtained with PEC methods as compared to PECE methods. In this report we present two PEC methods: a fourth-order algorithm for which stability characteristics and numerical examples are presented, and a second-order algorithm which is just mentioned. It is believed that PEC methods represent a useful addition to the library of solution techniques.     

A multi-step method for the numerical integration of ordinary differential equations

Summary In this paper we develop a multi-step method of order nine for obtaining an approximate solution of the initial value problemy'=f(x,y),y((x₀)=y ₀. The present method makes use of the second derivatives, namely, at the grid points. A sufficient criterion for the convergence of the iteration procedure is established. Analysis of the discretization error is performed. Various numerical examples are presented to demonstrate the practical usefulness of our integration method.

---

Zusammenfassung In dieser Arbeit entwickeln wir eine mehrschrittige Methode der neunten Ordnung, um eine angenäherte Lösung des Anfangswertproblemsy'=f(x, y), y(x ₀)=y ₀. zu erhalten. Diese Methode bedient sich der Ableitungen zweiter Ordnung an den Schnittpunkten, d.h. . Ein hinreichendes Kriterium für die Konvergenz des Iterationsprozesses wird aufgestellt. Eine Analyse des Diskretionsfehlers ist durchgeführt. Verschiedene numerische Beispiele sollen den praktischen Nutzen unserer Integrationsmethode beweisen.

     

The numerical solution of unstable ordinary differential equations

Invariant imbedding, or the Riccati transformation, has been used to solve unstable ordinary differential equations for a few years. This paper compares the above method with parallel or multiple shooting and a method using Chebyshev series. Parallel shooting gives a solution as accurate as that obtained using the Riccati transformation, in a comparable time.     

On the numerical solution of loaded systems of ordinary differential equations with nonseparated multipoint and integral conditions

We propose a numerical method of solving systems of loaded linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. This method is based on the convolution of integral conditions to obtain local conditions. This approach allows one to reduce solving the original problem to solving a Cauchy problem for a system of ordinary differential equations and linear algebraic equations. Numerous computational experiments on several test problems with the formulas and schemes proposed for the numerical solution have been carried out. The results of the experiments show that the approach is reasonably efficient.     

API stepsize control for the numerical solution of ordinary differential equations

A control-theoretic approach is used to design a new automatic stepsize control algorithm for the numerical integration of ODE's. The new control algorithm is more robust at little extra expense. Its improved performance is particularly evident when the stepsize is limited by numerical stability. Comparative numerical tests are presented.     

A method for reduction of differential problems to integral equations

A differential problem in an arbitrary domain is replaced with a problem with the same differential operator posed in a canonical domain for which the solution is either easily written out or is relatively easy to construct in terms of a function which is already known on the boundary of the canonical domain. The function is determined using the boundary conditions of the original problem, which in general leads to an integral equation. Some typical problems of mathematical physics are considered as an example.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 8–14, 1987.     

Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting

Summary The numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.     

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.     

A parametrization method for the numerical solution of singular differential equations

The paper explains the numerical parametrization method (PM), originally created for optimal control problems, for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs) in frame of their regularization. The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution.     

Numerical solution of nonlinear inverse coefficient problems for ordinary differential equations

Parametric identification for a class of nonlinear objects with lumped parameters described by systems of ordinary differential equations is studied. The problem is to recover the coefficients of a dynamical system depending on the phase state. For that purpose, the phase space is subdivided into a finite set of subsets or zones in which the coefficients are assumed to be constant or linear functions of state. Once the coefficients in such a form are obtained, interpolation and approximation can be used to represent the coefficients as functions of the phase variables.     

A block method for the numerical integration of stiff systems of ordinary differential equations

A new approach to the approximate numerical integration of stiff systems of first order ordinary differential equations is developed. In this approach several different formulae are applied in a well defined cyclic order to produce highly accurate integration schemes with infinite regions of absolute stability. The efficiency of these new algorithms, compared with that of certain existing ones, is demonstrated for some particular test problems.     

Modified quasilinearization technique for the solution of boundary-value problems for ordinary differential equations

A new quasilinearization algorithm is presented which essentially eliminates the necessity for computer storage. The representation theorem for the standard quasilinearization procedure is reformulated in terms of the initial value of the solution to a final-value problem, leading to a modification of the successive approximations. Several theorems establishing the convergence properties are proved; as in the original procedure, these convergence properties are both quadratic and monotonic. Finally, the modified approximation scheme is illustrated through several examples.