The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient
In this paper, we prove the Hyers–Ulam stability and generalized Hyers–Ulam stability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub>...
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MDPI AG
2022-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/22/4355 |
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| author | Fang Wang Ying Gao |
| author_facet | Fang Wang Ying Gao |
| author_sort | Fang Wang |
| collection | DOAJ |
| description | In this paper, we prove the Hyers–Ulam stability and generalized Hyers–Ulam stability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Δ</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with an initial condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mi>T</mi></mrow></semantics></math></inline-formula>; the corresponding conclusions of the standard heat equation can be also derived as corollaries. All of the above results are proved by using the properties of the fundamental solution of the equation. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:10:40Z |
| publishDate | 2022-11-01 |
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| series | Mathematics |
| spelling | doaj.art-39c109b6cccf49e39397b0829df16d082023-11-24T09:10:14ZengMDPI AGMathematics2227-73902022-11-011022435510.3390/math10224355The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent CoefficientFang Wang0Ying Gao1School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, ChinaSchool of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, ChinaIn this paper, we prove the Hyers–Ulam stability and generalized Hyers–Ulam stability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Δ</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with an initial condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mi>T</mi></mrow></semantics></math></inline-formula>; the corresponding conclusions of the standard heat equation can be also derived as corollaries. All of the above results are proved by using the properties of the fundamental solution of the equation.https://www.mdpi.com/2227-7390/10/22/4355generalized Hyers–Ulam stabilityHyers–Ulam stabilityheat equations with time-dependent coefficientfundamental solutions |
| spellingShingle | Fang Wang Ying Gao The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient Mathematics generalized Hyers–Ulam stability Hyers–Ulam stability heat equations with time-dependent coefficient fundamental solutions |
| title | The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient |
| title_full | The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient |
| title_fullStr | The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient |
| title_full_unstemmed | The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient |
| title_short | The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient |
| title_sort | analysis of hyers ulam stability for heat equations with time dependent coefficient |
| topic | generalized Hyers–Ulam stability Hyers–Ulam stability heat equations with time-dependent coefficient fundamental solutions |
| url | https://www.mdpi.com/2227-7390/10/22/4355 |
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