Solution of singularly perturbed Cauchy problem for ordinary differential equation of second order with constant coefficients by Fourier method

2017 ◽  
Author(s):  
Amir Sh. Shaldanbayev ◽  
Manat T. Shomanbayeva
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Fourier Method ◽  
Second Order ◽  
Singularly Perturbed ◽  
Constant Coefficients

scholarly journals Scalar Singularly Perturbed Cauchy Problem for a Differential Equation of Fractional Order

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Burkhan Kalimbetov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solution ◽  
Small Parameter ◽  
Fractional Order ◽  
Regularization Method ◽  
Original Problem ◽  
Convergent Series ◽  
Singularly Perturbed

In this paper we consider initial problem for an ordinary differential equation of fractional order with a small parameter for the derivative. S.A. Lomov regularization method is used to construct an asymptotic approximate solution of the problem with accuracy up to any power of a small parameter. Using the computer mathematics system (CMS) Maple, a symbolic solution of the original problem is obtained, and solution schedules are constructed, depending on the initial data and various values of the small parameter. It is shown that the asymptotic solution presented in the form of a specific convergent series and the solution represented by the CMS Maple coincides with the exact solution of the original problem. 

Cauchy problem for second-order ordinary differential equation with continuously distributed differentiation operator

2020 ◽  
Vol 20 (1) ◽  
pp. 15-20
Author(s):  
B.I. Efendiev ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Order Differential Equation ◽  
Differentiation Operator ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation ◽  
The Cauchy Problem

In this paper, we construct the fundamental solution for ordinary second-order differential equation with continuously distributed differentiation operator. With the help of fundamental solution the solution of the Cauchy problem is written out.

scholarly journals ROBUST DIFFERENCE SCHEME FOR THE CAUCHY PROBLEM FOR A SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATION

2018 ◽  
Vol 23 (4) ◽  
pp. 527-537 ◽  
Author(s):  
Lidia Pavlovna Shishkina ◽  
Grigorii Ivanovich Shishkin
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Difference Scheme ◽  
Singularly Perturbed ◽  
Uniform Grid ◽  
Piecewise Uniform Grid ◽  
The Cauchy Problem ◽  
Standard Difference

Grid approximation of the Cauchy problem on the interval D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem under consideration, for small values of the parameter ε, a boundary layer of width O(ε) appears on which the solution varies by a finite value. It is shown that, for such a Cauchy problem, the solution of the standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm; convergence occurs only under the condition h ε, where h = d N −1 , N is the number of grid intervals, h is the grid step-size. Taking into account the behavior of the singular component in the solution, a special piecewise-uniform grid is constructed that condenses in a neighborhood of the boundary layer. It is established that the standard difference scheme on such a special grid converges ε-uniformly in the maximum norm at the rate O(N −1 lnN). Such a scheme is called a robust one. For a model Cauchy problem for a singularly perturbed ordinary differential equation, standard difference schemes on a uniform grid (a classical difference scheme) and on a piecewise-uniform grid (a special difference scheme) are constructed and investigated. The results of numerical experiments are given, which are consistent with theoretical results.

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