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.
228Numerical solution for high order differential equations using a hybrid neural network—Optimization method
Numerical solution for high order differential equations using a hybrid neural network—Optimization method
9Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem
Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem
862Topological Degree Methods in Nonlinear Boundary Value Problems
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.
Read more5Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations
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.
Read more6All meromorphic solutions of an ordinary differential equation and its applications
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.
Read more17Applications of genetic algorithms to the solution of ordinary differential equations
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.
Read more13Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order
Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order
Read moreSolutions of boundary-value problems on automatic computing equipment
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.
Read moreAnalytical approaches for the approximate solution of a nonlinear fractional ordinary differential equation
The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0⩽t⩽2 and 1<α⩽2.
Read more1TrapezoidalNet: A new network architecture inspired from the numerical solution of ordinary differential equations
TrapezoidalNet: A new network architecture inspired from the numerical solution of ordinary differential equations
87The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations
The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations
1RETRACTED ARTICLE: Data simulation of optimal model for numerical solution of differential equations based on deep learning and genetic algorithm
Calculus equation is an important tool for mathematical research and plays an important role in most natural science research. Since the beginning of the eighteenth century, people have gradually used differential and integral equations to solve physical problems. In general, several different aspects of differential equations in the field of mathematics are concerned and studied by most scholars. However, this paper studies and establishes the optimal model for numerical solution of differential equations through deep learning and genetic algorithm. In this paper, the solution of ordinary differential equations is solved through the use of polynomial function space, while the linear combination of simple function x and its product nx can obtain multinomial function space. The space function form of polynomial is very simple, and the operation ability is very strong. Almost all functions can be approximated, and the function space can be transformed by a simple function. Through data simulation test results, it can be found that the oscillation of neural network output is stronger and stronger with the increase in depth, that is to say, the deeper depth endows the neural network with stronger oscillation properties, so for the oscillation function, the depth neural network fitting effect is better than the shallow neural network. Therefore, by combining deep learning and genetic algorithm, this paper studies and establishes the optimal model for numerical solution of differential equations, and finds that the deep neural network can largely complete data simulation.
Read more7Discrete-Like Crack Simulation of Reinforced Concrete Incorporated with Analytical Solution of Cyclic Bond Model
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.
Read more276Seven common errors in finding exact solutions of nonlinear differential equations
Seven common errors in finding exact solutions of nonlinear differential equations
5Some studies of the numerical solution of ordinary differential equations
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.
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