Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models
In this paper, we aim to show that the Meijer G-functions can serve to find explicit solutions of partial differential equations (PDEs) related to some mathematical models of physical phenomena, as for example, the Laplace equation, the diffusion equation and the Schrödinger equation. Usually, the f...
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| Date: | 2013 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
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| Series: | Журнал математической физики, анализа, геометрии |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/106760 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models / A. Pishkoo, M. Darus // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 379-391. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | In this paper, we aim to show that the Meijer G-functions can serve to find explicit solutions of partial differential equations (PDEs) related to some mathematical models of physical phenomena, as for example, the Laplace equation, the diffusion equation and the Schrödinger equation. Usually, the first step in solving such equations is to use the separation of variables method to reduce them to ordinary differential equations (ODEs). Very often this equation happens to be a case of the linear ordinary differential equation satisfied by the G-function, and so, by proper selection of its orders m; n; p; q and the parameters, we can find the solution of the ODE explicitly. We illustrate this approach by proposing solutions as: the potential function Ф, the temperature function T and the wave function Ψ, all of which are symmetric product forms of the Meijer G-functions. We show that one of the three basic univalent Meijer G-functions, namely G₀,₂¹’⁰, appears in all the mentioned solutions. |
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