A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation
The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mro...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/7/1094 |
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| author | Zharasbek Baishemirov Abdumauvlen Berdyshev Ainur Ryskan |
| author_facet | Zharasbek Baishemirov Abdumauvlen Berdyshev Ainur Ryskan |
| author_sort | Zharasbek Baishemirov |
| collection | DOAJ |
| description | The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mfenced><mi>u</mi></mfenced><mo>≡</mo><msup><mi>y</mi><mi>m</mi></msup><msup><mi>z</mi><mi>k</mi></msup><msup><mi>t</mi><mi>l</mi></msup><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>z</mi><mi>k</mi></msup><msup><mi>t</mi><mi>l</mi></msup><msub><mi>u</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mi>m</mi></msup><msup><mi>t</mi><mi>l</mi></msup><msub><mi>u</mi><mrow><mi>z</mi><mi>z</mi></mrow></msub><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mi>m</mi></msup><msup><mi>z</mi><mi>k</mi></msup><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>≡</mo><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> are studied in the finite domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>R</mi><mn>4</mn><mo>+</mo></msubsup></mrow></semantics></math></inline-formula>, where the values of normal derivatives are set on the piecewise smooth part of the boundary and the values of the desired function are set on the remaining part of the boundary. The main results of the work are the proof of the uniqueness of the considered problem solution by using an energy integral’s method and the construction of the solution of counterpart Holmgren’s boundary value problem in explicit form by means of Green’s function method, containing the hypergeometric Lauricella’s function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>F</mi><mi>A</mi><mrow><mfenced><mn>4</mn></mfenced></mrow></msubsup></mrow></semantics></math></inline-formula>. Using the corresponding fundamental solution for the considered generalized Gellerstedt equation of elliptic type, we construct Green’s function. In addition, formulas of differentiation, some adjacent relations, decomposition formulas, and various properties of Lauricella’s hypergeometric functions were used to establish the main results for the aforementioned problem. |
| first_indexed | 2024-03-09T11:37:55Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:37:55Z |
| publishDate | 2022-03-01 |
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| series | Mathematics |
| spelling | doaj.art-34c973be25f942e7a8b6562a7ebfc7672023-11-30T23:37:05ZengMDPI AGMathematics2227-73902022-03-01107109410.3390/math10071094A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic EquationZharasbek Baishemirov0Abdumauvlen Berdyshev1Ainur Ryskan2Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Tole bi Str., 86, Almaty 050012, KazakhstanInstitute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Tole bi Str., 86, Almaty 050012, KazakhstanInstitute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Tole bi Str., 86, Almaty 050012, KazakhstanThe solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mfenced><mi>u</mi></mfenced><mo>≡</mo><msup><mi>y</mi><mi>m</mi></msup><msup><mi>z</mi><mi>k</mi></msup><msup><mi>t</mi><mi>l</mi></msup><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>z</mi><mi>k</mi></msup><msup><mi>t</mi><mi>l</mi></msup><msub><mi>u</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mi>m</mi></msup><msup><mi>t</mi><mi>l</mi></msup><msub><mi>u</mi><mrow><mi>z</mi><mi>z</mi></mrow></msub><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mi>y</mi><mi>m</mi></msup><msup><mi>z</mi><mi>k</mi></msup><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>≡</mo><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> are studied in the finite domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>R</mi><mn>4</mn><mo>+</mo></msubsup></mrow></semantics></math></inline-formula>, where the values of normal derivatives are set on the piecewise smooth part of the boundary and the values of the desired function are set on the remaining part of the boundary. The main results of the work are the proof of the uniqueness of the considered problem solution by using an energy integral’s method and the construction of the solution of counterpart Holmgren’s boundary value problem in explicit form by means of Green’s function method, containing the hypergeometric Lauricella’s function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>F</mi><mi>A</mi><mrow><mfenced><mn>4</mn></mfenced></mrow></msubsup></mrow></semantics></math></inline-formula>. Using the corresponding fundamental solution for the considered generalized Gellerstedt equation of elliptic type, we construct Green’s function. In addition, formulas of differentiation, some adjacent relations, decomposition formulas, and various properties of Lauricella’s hypergeometric functions were used to establish the main results for the aforementioned problem.https://www.mdpi.com/2227-7390/10/7/1094degenerate partial differential equationcounterpart Holmgren’s boundary value problemfundamental solutionLauricella’s hypergeometric functionGreen’s function |
| spellingShingle | Zharasbek Baishemirov Abdumauvlen Berdyshev Ainur Ryskan A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation Mathematics degenerate partial differential equation counterpart Holmgren’s boundary value problem fundamental solution Lauricella’s hypergeometric function Green’s function |
| title | A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation |
| title_full | A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation |
| title_fullStr | A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation |
| title_full_unstemmed | A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation |
| title_short | A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation |
| title_sort | solution of a boundary value problem with mixed conditions for a four dimensional degenerate elliptic equation |
| topic | degenerate partial differential equation counterpart Holmgren’s boundary value problem fundamental solution Lauricella’s hypergeometric function Green’s function |
| url | https://www.mdpi.com/2227-7390/10/7/1094 |
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