Error Estimates for General Linear Methods for Ordinary Differential Equations

Many classes of methods for numerical problems can be characterized by parameters. For example, Adams methods for integrating ordinary differential equations can be characterized by the step size and order. In many cases it is known that a method is stable for fixed values of those parameters, whereas many programs vary those parameters in order to reduce computation time. This paper examines the effects on stability of changes to two kinds of parameters—“discrete” and “continuous” — corresponding to the parameters of order and step size in the numerical solution of ordinary differential equations. Conditions are given that are sufficient to ensure that these changes do not affect stability. These results are applied to two classes of automatic methods for ordinary differential equations—one based on unequal interval quadrature formulas, the other based on interpolation and equal interval quadrature formulas.

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge---Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge---Kutta stability is also briefly discussed

This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are local error estimates in the sense of the local error in the numerical solution of ordinary differential equations. They represent a different approach to the general problem of error estimation, an approach which involves only a single discretized equation instead of a converging family of such equations.

This paper describes some methods for the solution of ordinary differential equations based on the Lanczos "selected points" or collocation principle. Some properties of the methods are examined theoretically, and some examples of their use given.

Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem

Introduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.

Dedicated to Professor Yuzan He on the Occasion of his 80th BirthdayIn this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.

With the numerical solution of ordinary differential equations(ODE), scientists engaged in the Middle Ages, beginning with the work of Clairaut. The domain of the numerical methods involved in many famous mathematicians - Euler, Runge, Kutta, Adams, Laplace, and others. They have constructed methods with different properties. In this paper we consider the construction of numerical methods with high accuracy and to this end is proposed to use multi-step multi-derivative and hybrid methods. As well as specific methods are constructed with a certain accuracy.

In this paper, we derive new method for solving particular solution of linear second order ordinary differential equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct second solution of the associated homogeneous differential equation from this new method. Moreover, we have general solution method of linear second order ordinary differential equations without applying the two famous methods, undetermined coefficients method and variation of parameters method, for solving their particular solution

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

We consider adaptive time discretization methods for ordinary differential equations where one aims to control the error in a quantity of interest of the form , with j : ℝd → ℝ. In this setting we propose a new timestep controller based on local error estimates of the quantity of interest. The new method converges when the tolerance goes to zero.We experimentally compare the new scheme with the classic norm‐based time‐adaptivity based on local error estimates as well as the dual‐weighted residual (DWR) method. The results show significantly lower efficiency for the DWR method. The local error based schemes are similarly efficient, with the new scheme showing significant improvement in some cases. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical solution for high order differential equations using a hybrid neural network—Optimization method

We prove new optimal a priori error estimates for a class of implicit one-step methods for stiff ordinary differential equations obtained by using the discontinuous Galerkin method with piecewise polynomials of degree zero and one. Starting from these estimates we propose a new algorithm for automatic time-step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations.

In this article, we have presented the details of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solutions of ordinary differential equations (ODEs). In these hybrid BDF, one additional stage point (or off-step point) has been used in the first derivative of the solution to improve the absolute stability regions. Stability domains of our presented methods have been obtained showing that all these new methods, we say HBDF, of order p, p = 2,4,..., 12, are A(α)-stable whereas they have wide stability regions comparing with those of some known methods like BDF, extended BDF (EBDF), modified EBDF (MEBDF), adaptive EBDF (A-EBDF), and second derivtive Enright methods. Numerical results are also given for five test problems.