Fast algorithm for finding approximate solutions to the Cauchy problem for ODE

2018 ◽  
pp. 41-49
Author(s):  
Gasan Akniyev ◽  
Ramis Gadzhimirzaev
Keyword(s):  
Cauchy Problem ◽  
Fast Algorithm ◽  
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,

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