On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets
The regular, real-valued solutions of the second-order elliptic partial differential equation\beq \frac{\prt^2F}{\prt x^2} + \frac{\prt^2F}{\prt y^2} + \frac{2\alpha+1}{x} \frac{\prt F}{\prt y} + \frac{2\beta+1}{y} \frac{\prt F}{\prt x} =0, \alpha,\beta>\frac{-1}{2}\eeq are known as generaliz...
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2018-12-01
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| Series: | Communications in Advanced Mathematical Sciences |
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| Online Access: | https://dergipark.org.tr/tr/download/article-file/603960 |
| _version_ | 1797294194306318336 |
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| author | Devendra Kumar |
| author_facet | Devendra Kumar |
| author_sort | Devendra Kumar |
| collection | DOAJ |
| description | The regular, real-valued solutions of the second-order elliptic partial differential equation\beq \frac{\prt^2F}{\prt x^2} + \frac{\prt^2F}{\prt y^2} + \frac{2\alpha+1}{x} \frac{\prt F}{\prt y} + \frac{2\beta+1}{y} \frac{\prt F}{\prt x} =0, \alpha,\beta>\frac{-1}{2}\eeq are known as generalized bi-axially symmetric potentials (GBSP's). McCoy \cite{17} has showed that the rate at which approximation error $E^{\frac{p}{2n}}_{2n}(F;D),(p\ge 2,D$ is parabolic-convex set) tends to zero depends on the order of $GBSP$ F and obtained a formula for finite order. If $GBSP$ F is an entire function of infinite order then above formula fails to give satisfactory information about the rate of decrease of $E^{\frac{p}{2n}}_{2n}(F;D)$. The purpose of the present work is to refine above result by using the concept of index-q. Also, the formula corresponding to $q$-order does not always hold for lower $q$-order. Therefore we have proved a result for lower $q$-order also, which have not been studied so far. |
| first_indexed | 2024-03-07T21:26:44Z |
| format | Article |
| id | doaj.art-43c505a3dfac46c7a52e8c2b867d5f39 |
| institution | Directory Open Access Journal |
| issn | 2651-4001 |
| language | English |
| last_indexed | 2024-03-07T21:26:44Z |
| publishDate | 2018-12-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Communications in Advanced Mathematical Sciences |
| spelling | doaj.art-43c505a3dfac46c7a52e8c2b867d5f392024-02-27T04:36:36ZengEmrah Evren KARACommunications in Advanced Mathematical Sciences2651-40012018-12-011215616210.33434/cams.4399771225On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex SetsDevendra KumarThe regular, real-valued solutions of the second-order elliptic partial differential equation\beq \frac{\prt^2F}{\prt x^2} + \frac{\prt^2F}{\prt y^2} + \frac{2\alpha+1}{x} \frac{\prt F}{\prt y} + \frac{2\beta+1}{y} \frac{\prt F}{\prt x} =0, \alpha,\beta>\frac{-1}{2}\eeq are known as generalized bi-axially symmetric potentials (GBSP's). McCoy \cite{17} has showed that the rate at which approximation error $E^{\frac{p}{2n}}_{2n}(F;D),(p\ge 2,D$ is parabolic-convex set) tends to zero depends on the order of $GBSP$ F and obtained a formula for finite order. If $GBSP$ F is an entire function of infinite order then above formula fails to give satisfactory information about the rate of decrease of $E^{\frac{p}{2n}}_{2n}(F;D)$. The purpose of the present work is to refine above result by using the concept of index-q. Also, the formula corresponding to $q$-order does not always hold for lower $q$-order. Therefore we have proved a result for lower $q$-order also, which have not been studied so far.https://dergipark.org.tr/tr/download/article-file/603960parabolic-convex setindex-qq-orderlower q-ordergeneralized bi-axially symmetric potentials and elliptic partial differential equation |
| spellingShingle | Devendra Kumar On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets Communications in Advanced Mathematical Sciences parabolic-convex set index-q q-order lower q-order generalized bi-axially symmetric potentials and elliptic partial differential equation |
| title | On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets |
| title_full | On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets |
| title_fullStr | On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets |
| title_full_unstemmed | On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets |
| title_short | On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets |
| title_sort | on growth and approximation of generalized biaxially symmetric potentials on parabolic convex sets |
| topic | parabolic-convex set index-q q-order lower q-order generalized bi-axially symmetric potentials and elliptic partial differential equation |
| url | https://dergipark.org.tr/tr/download/article-file/603960 |
| work_keys_str_mv | AT devendrakumar ongrowthandapproximationofgeneralizedbiaxiallysymmetricpotentialsonparabolicconvexsets |