Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point
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 pro...
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MDPI AG
2020-11-01
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| Online Access: | https://www.mdpi.com/2075-1680/9/4/138 |
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| author | Tatiana Ratnikova |
| author_facet | Tatiana Ratnikova |
| author_sort | Tatiana Ratnikova |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T14:31:17Z |
| format | Article |
| id | doaj.art-9edbf096ca8248f0b214c1d8242c06d4 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T14:31:17Z |
| publishDate | 2020-11-01 |
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| series | Axioms |
| spelling | doaj.art-9edbf096ca8248f0b214c1d8242c06d42023-11-20T22:35:27ZengMDPI AGAxioms2075-16802020-11-019413810.3390/axioms9040138Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning PointTatiana Ratnikova0Moscow Power Engineering Institute, National Research University, 111250 Moscow, RussiaThe 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.https://www.mdpi.com/2075-1680/9/4/138singularly perturbed Cauchy problemparabolic equationasymptotic solutionrational “simple” turning point |
| spellingShingle | Tatiana Ratnikova Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point Axioms singularly perturbed Cauchy problem parabolic equation asymptotic solution rational “simple” turning point |
| title | Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point |
| title_full | Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point |
| title_fullStr | Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point |
| title_full_unstemmed | Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point |
| title_short | Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point |
| title_sort | singularly perturbed cauchy problem for a parabolic equation with a rational simple turning point |
| topic | singularly perturbed Cauchy problem parabolic equation asymptotic solution rational “simple” turning point |
| url | https://www.mdpi.com/2075-1680/9/4/138 |
| work_keys_str_mv | AT tatianaratnikova singularlyperturbedcauchyproblemforaparabolicequationwitharationalsimpleturningpoint |