Some applications of Meijer G-functions as solutions of differential equations in physical models

A. Pishkoo, Maslina Darus

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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 Schrodinger 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 Φ, 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 G1 0 , , 0 2, appears in all the mentioned solutions.

Original languageEnglish
Pages (from-to)379-391
Number of pages13
JournalJournal of Mathematical Physics, Analysis, Geometry
Volume9
Issue number3
Publication statusPublished - 2013

Fingerprint

Meijer's G-function
Physical Model
Differential equation
Ordinary differential equation
Symmetric Product
Product Form
Schrodinger Equation
Linear Ordinary Differential Equations
Separation of Variables
G-function
Potential Function
Laplace's equation
Explicit Solution
Diffusion equation
Wave Function
Partial differential equation
Mathematical Model

Keywords

  • Diffusion equation
  • Laplace equation
  • Meijer G-functions
  • Partial differential equations
  • Schrödinger equation
  • Separation of variables

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Mathematical Physics

Cite this

@article{86d63bd95cad4a3d9b6eedbe47063dc0,
title = "Some applications of Meijer G-functions as solutions of differential equations in physical models",
abstract = "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 Schrodinger 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 Φ, 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 G1 0 , , 0 2, appears in all the mentioned solutions.",
keywords = "Diffusion equation, Laplace equation, Meijer G-functions, Partial differential equations, Schr{\"o}dinger equation, Separation of variables",
author = "A. Pishkoo and Maslina Darus",
year = "2013",
language = "English",
volume = "9",
pages = "379--391",
journal = "Journal of Mathematical Physics, Analysis, Geometry",
issn = "1812-9471",
publisher = "ILTPE-B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine",
number = "3",

}

TY - JOUR

T1 - Some applications of Meijer G-functions as solutions of differential equations in physical models

AU - Pishkoo, A.

AU - Darus, Maslina

PY - 2013

Y1 - 2013

N2 - 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 Schrodinger 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 Φ, 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 G1 0 , , 0 2, appears in all the mentioned solutions.

AB - 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 Schrodinger 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 Φ, 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 G1 0 , , 0 2, appears in all the mentioned solutions.

KW - Diffusion equation

KW - Laplace equation

KW - Meijer G-functions

KW - Partial differential equations

KW - Schrödinger equation

KW - Separation of variables

UR - http://www.scopus.com/inward/record.url?scp=84883811930&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883811930&partnerID=8YFLogxK

M3 - Article

VL - 9

SP - 379

EP - 391

JO - Journal of Mathematical Physics, Analysis, Geometry

JF - Journal of Mathematical Physics, Analysis, Geometry

SN - 1812-9471

IS - 3

ER -