Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity

2021 ◽  
Vol 19 (1) ◽  
pp. 244-258
Author(s):  
Burkhan T. Kalimbetov ◽  
Olim D. Tuychiev
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
New Approach ◽  
The Cauchy Problem ◽  
Oscillating Coefficients ◽  
Oscillating Coefficient ◽  
The Relationship

Abstract In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

scholarly journals Asymptotic Solutions of Scalar Integro-Differential Equations with Partial Derivatives and with Fast Oscillating Coefficients

2020 ◽  
Vol 13 (2) ◽  
pp. 287-302
Author(s):  
Burkhan Kalimbetov ◽  
Akisher Temirbekov ◽  
Abdimuhan Tolep
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Unique Solvability ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
Integral Term ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
Oscillating Coefficients

In the paper, ideas of the Lomov regularization method are generalized to the Cauchy problem for a singularly perturbed partial integro-differential equation in the case when the integral term contains a rapidly varying kernel. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.

scholarly journals Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 124 ◽  
Author(s):  
Alexander Eliseev ◽  
Tatjana Ratnikova
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Turning Point ◽  
Singularly Perturbed ◽  
Dimensional Case ◽  
Singularly Perturbed Problem ◽  
Two Dimensional ◽  
Limit Operator ◽  
Regularized Solution

By Lomov’s S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a ”simple” turning point was considered, i.e., one eigenvalue vanishes for t = 0 and has the form t m / n a ( t ) (limit operator is discretely irreversible). The regularization method allows us to construct an asymptotic solution that is uniform over the entire segment [ 0 , T ] , and under additional conditions on the parameters of the singularly perturbed problem and its right-hand side, the exact solution.

scholarly journals Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 138
Author(s):  
Tatiana Ratnikova
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Turning Point ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Asymptotic Convergence ◽  
Limit Operator ◽  
Operator Spectrum

The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. The problem with a “simple” turning point is considered in the case, when the eigenvalue vanishes at t=0 and has the form tm/na(t). The asymptotic convergence of the regularized series is proved.

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. 

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