Explicit Two-Derivative Runge–Kutta Methods Coupled with the Richardson Extrapolation

The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

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.

When estimating the error of numerical solutions of ordinary differential equations, it is proposed to use the regularization of error estimation algorithms for approximate solutions. Regularization is considered as the use of additional a priori information about the numerical solution. A priori information is estimated by the method of the inverse error analysis, which implements the transformation of the statement of the problem of evaluating the accuracy of the numerical solution. In this case, the problem is selected so that the approximate solution is the exact solution of the transformed problem, close to the original problem. It is also possible to add some additional restrictions to the conditions of the problem. This allows us to solve the incorrectly posed problems of estimating errors in numerical solutions, which in most cases are unstable. It is known that earlier in the numerical solution of linear algebra problems the inverse error analysis was used by Wilkinson J. and also Voevodin V.V., application of the inverse error analysis in other areas of numerical analysis can be noted. Using direct and inverse error analysis Voevodin V.V. effectively calculated majorants of rounding errors in the most important methods of linear algebra and significantly developed the method of inverse error analysis.. However, the regularization process associated with the reverse analysis of errors has not been previously studied. The article discusses the characteristics of regularization in the inverse analysis of errors in the numerical solution of ordinary differential equations with initial data. The regularization of error estimation algorithms has advantages in the numerical solution of ordinary differential equations of real models of physics and technology.

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.

A novel and efficient algorithm for the concurrent numerical solution of second, third and fourth orders of ordinary differential equations is studied in this article. Collocation and interpolation technique was employed in the derivation of the method and power series approximate solution was used as the interpolating polynomial. The fourth derivative of the power series was collocated at the entire grid and off-grid points, while the fifth and sixth derivatives of the polynomial were collocated at the endpoint only. Proper investigation of the basic properties of the method was done. The results showed that the new block method applied on a nonlinear second, third and fourth orders of ordinary differential equations were better in terms of accuracy than the existing methods. The proposed method takes away the burden of developing a separate method for the solution of second, third and fourth order initial value problem of ordinary differential equations.

On the numerical solution of second order ordinary differential equations in the high-frequency regime

A novel genetic algorithm has been developed and applied to the solution of ordinary differential equations. The algorithm solves the equations by a process of breeding better candidate solutions from a family of estimates, and learns to retain the best features as it progresses. This self-learning system is intrinsically parallel, and capable of handling linear and nonlinear equations, both stiff and non-stiff. Genetic algorithms are a key element in artificial intelligence and machine learning, playing a significant role in optimization and robotics. In this document, the application of genetic algorithms to the solution of ordinary differential equations is presented as a radically different way of approaching numerical simulations, and is of importance to many disciplines.

L-stable symmetrizers for symmetric Runge–Kutta methods are constructed to preserve the asymptotic error expansion in even powers of the stepsize and to provide necessary damping for the numerical solution of stiff initial value ordinary differential equations. The process is called symmetrization and has the effect of dampening down undesirable oscillations that may arise when the problem is stiff. The symmetry property can be exploited by Richardson extrapolation in increasing the order by two at a time. The effect of the damping is to stabilize the order behaviour of the extrapolation. In this paper, we discuss two types of symmetrizers; the well-known one-step smoothing formula of Gragg, and the new two-step smoothing formula. We construct these symmetrizers for the implicit midpoint and trapezoidal rules and investigate the active mode of application with extrapolation. Finally, we present numerical results in a constant stepsize setting that show two-step symmetrizers having certain advantages over one-step symmetrizers for stiff linear and nonlinear problems.

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

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.

This paper shows how a model of the human cardiovascular system was developed on a digital computer and how this model was used to test some important theories of cardiovascular physiology. An algorithm for the rapid numerical solution of non-linear ordinary differential equations is introduced and applied to the state differential equation of the hydraulic system model. A brief account of the physiological reflexes that control blood pressure, as studied by animal experimentation, is given and shown to suggest the particular hydraulic parameters which are controlled in each refiex, as well as the approximate dynamics of these reflexes. The digital implementation of the reflexes are then outlined and their effect when used to control parameters of the hydraulic system model is studied.Next, it is demonstrated how experimentation in the form of computer simulation trials can be utilized to characterize the above reflexes more accurately so that certain key experimental phenomena observed in the human may be reproduced in the model. These phenomena are indicative of a highly non-linear control system and may be reproduced in the model if “on–off” control of simulated peripheral resistance and venomotor tone is implemented. Thus, it is illustrated how a computer model may be used to assess the nature of a biological control system and even specify important elements in this system.

THIS investigation is primarily concerned with the solution of boundary-value problems by means of automatic computing equipment. Specifically, it is concerned with the solution of ordinary differential equations in which certain conditions (boundary values) are specified at two or more values of the independent variable. Such problems are generally called boundary-value problems; however, it must be emphasized that this report is restricted to the solution of ordinary differential equations, and not to the solution of partial differential equations.

This paper presents solutions of second-order ordinary differential equations on bond problems to estimate crack widths of reinforced concrete (RC) members subjected to repeated cyclic loads. Up to now, bond behaviors under repeated cyclic loads have usually been calculated by a difference scheme. In contrast, this research adopts analytical solutions of the differential equations to calculate bond behaviors in finite element models where a number of concrete cracks occur. The conventional differential equation for the bond problem has been available only under monotonic loading conditions. This paper extends the equation to cases where the reinforcement becomes plastic and the bond is subject to unloading/reloading conditions. The bond stress-slip relationship is modeled by a multilinear model that is composed of an envelope made of five lines, unloading/reloading paths, and a negative friction path. The solution of the differential equation is classified into 25 cases when the bond stress-slip relationship is on the envelope and seven cases when on the unloading/reloading paths or the negative friction path. The solutions are implemented into the smeared crack-based finite-element method (FEM) to evaluate the crack widths of a RC beam subject to repeated cyclic flexural/shear loads. The FEM program is incorporated with the discrete-like crack procedure developed by the authors and calculates stress redistributions due to cracking using the solutions of the differential equations. Typical distributions of reinforcement stress, bond stress, and slip along the reinforcement are presented.

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.