scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

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.

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