An approach to deducing approximate solutions to the Cauchy problem for nonlinear differential equations

1992 ◽  
Vol 44 (12) ◽  
pp. 1554-1560 ◽  
Author(s):  
Ya. M. Pelekh
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
The Cauchy Problem

scholarly journals On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

2020 ◽  
Vol 56 (1) ◽  
pp. 51-71
Author(s):  
M. V. Ignatenko ◽  
L. A. Yanovich
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Approximate Solution ◽  
Differential Equations ◽  
Interpolation Problem ◽  
Approximate Solutions ◽  
Hyperbolic Type ◽  
The Cauchy Problem ◽  
Variational Derivatives ◽  
Derivatives Of

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

scholarly journals Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations

2020 ◽  
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Approximate Solutions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Fuzzy Fractional Differential Equations

In this paper, the Cauchy problem of fuzzy fractional differential equationsTγut=Ftut, ut0=u0,

scholarly journals The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures

2018 ◽  
pp. 486-510
Author(s):  
Evgenii B. Kuznetsov ◽  
Sergey S. Leonov ◽  
Ekaterina D. Tsapko
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Computational Time ◽  
Method Of Solution ◽  
The Cauchy Problem ◽  
Contrast Structures ◽  
Interior Layers ◽  
Boundary And Interior Layers ◽  
Best Argument

Introduction. The paper provides an analysis of numerical methods for solving the Cauchy problem for nonlinear ordinary differential equations with contrast structures (interior layers). Similar equations simulate various applied problems of hydro- and aeromechanics, chemical kinetics, the theory of catalytic reactions, etc. An analytical solution to these problems is rarely obtained, and numerical procedure is related with significant difficulties associated with ill-conditionality in the neighborhoods of the boundary and interior layers. The aim of the paper is the scope analysis of traditional numerical methods for solving this class problems and approbation of alternative solution methods. Materials and methods. The traditional explicit Euler and fourth-order Runge-Kutta methods, as well as the implicit Euler method with constant and variable step sizes are used for the numerical solution of the Cauchy problem. The method of solution continuation with respect to the best argument is suggested as an alternative to use. The solution continuation method consists in replacing the original argument of the problem with a new one, measured along the integral curve of the problem. The transformation to the best argument allows obtaining the best conditioned Cauchy problem. Results. The computational difficulties arising when solving the equations with contrast structures by traditional explicit and implicit methods are shown on the example of the test problem solution. These difficulties are expressed in a significant decrease of the step size in the neighborhood of the boundary and interior layers. It leads to the increase of the computational time, as well as to the complication of the solving process for super stiff problems. The authenticity of the obtained results is confirmed by the comparison with the analytical solution and the works of other authors. Conclusions. The results of the computational experiment demonstrate the applicability of the traditional methods for solving the Cauchy problem for equations with contrast structures only at low stiffness. In other cases these methods are ineffective. It is shown that the method of solution continuation with respect to the best argument allows eliminating most of the disadvantages inherent to the original problem. It is reflected in decreasing the computational time and in increasing the solution accuracy. Keywords: contrast structures, method of solution continuation, the best argument, illconditionality, the Cauchy problem, ordinary differential equation For citation: Kuznetsov E. B., Leonov S. S., Tsapko E. D. The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):486–510. DOI: https://doi.org/10.15507/0236-2910.028.201804.486-510 Acknowledgements: This work was supported by the Russian Science Foundation, project no. 18-19-00474.

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