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

• Main Author: Tatiana Ratnikova
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Axioms
• Subjects: singularly perturbed Cauchy problem; parabolic equation; asymptotic solution; rational “simple” turning point
• Online Access: https://www.mdpi.com/2075-1680/9/4/138
• ISSN: 2075-1680

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 <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and has the form <inline-formula><math display="inline"><semantics><mrow><msup><mi>t</mi><mrow><mi>m</mi><mo>/</mo><mi>n</mi></mrow></msup><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The asymptotic convergence of the regularized series is proved.