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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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1067602016-10-05T03:02:15Z Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models Pishkoo, A. Darus, M. 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. Цель этой статьи - показать, что G-функции Мейера можно использовать для нахождения в явном виде решений уравнений в частных производных, связанных с некоторыми математическими моделями физических явлений, таких как, например, уравнение Лапласа, уравнение диффузии и уравнение Шредингера. Как правило, первым шагом в решении таких уравнений является использование метода разделения переменных для того, чтобы свести их к обыкновенным дифференциальным уравнениям (ОДУ). Очень часто это уравнение оказывается случаем линейного обыкновенного дифференциального уравнения, которое удовлетворяет G-функция и поэтому, правильно выбрав ее порядок m; n; p; q и параметры, мы можем найти решение ОДУ в явном виде. Мы иллюстрируем этот подход, предлагая такие решения, как потенциальная функция Ф, температурная функция T и волновая функция Ψ, все из которых являются видами симметричных произведений G-функций Мейера. Показано, что одна из трех основных однолистных G-функций Мейера, а именно G₀,₂¹’⁰, встречается во всех упомянутых решениях. 2013 Article 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. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106760 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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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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Article |
| author |
Pishkoo, A. Darus, M. |
| spellingShingle |
Pishkoo, A. Darus, M. Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models Журнал математической физики, анализа, геометрии |
| author_facet |
Pishkoo, A. Darus, M. |
| author_sort |
Pishkoo, A. |
| title |
Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models |
| title_short |
Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models |
| title_full |
Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models |
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Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models |
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Some Applications of Meijer G-Functions as Solutions of Differential Equations in Physical Models |
| title_sort |
some applications of meijer g-functions as solutions of differential equations in physical models |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/106760 |
| citation_txt |
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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Журнал математической физики, анализа, геометрии |
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AT pishkooa someapplicationsofmeijergfunctionsassolutionsofdifferentialequationsinphysicalmodels AT darusm someapplicationsofmeijergfunctionsassolutionsofdifferentialequationsinphysicalmodels |
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2025-07-07T18:57:27Z |
| last_indexed |
2025-07-07T18:57:27Z |
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1837015653942296576 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 3, pp. 379–391
Some Applications of Meijer G-Functions as Solutions
of Differential Equations in Physical Models
A. Pishkoo and M. Darus
School of Mathematical Sciences Faculty of Science and Technology
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan, Malaysia
E-mail: apishkoo@yahoo.in
maslina@ukm.my
Received July 1, 2011, revised December 19, 2012
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 func-
tion 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.
Key words: Meijer G-functions; partial differential equations; Laplace
equation; diffusion equation; Schrödinger equation; separation of variables.
Mathematics Subject Classification 2010: 35Q40, 35Q79, 33C60, 30C55.
1. Introduction
In the recent decades, the Meijer G-function has found various applications
in different areas which are close to applied mathematics, such as mathematical
physics (hydrodynamics, theory of elasticity, potential theory, etc.), theoretical
physics, mathematical statistics, queuing theory, optimization theory, sinusoidal
The work here is supported by MOHE:UKM-ST-06-FRGS0244-2010 and
ERGS/1/2013/STG06/UKM/01/2.
c© A. Pishkoo and M. Darus, 2013
A. Pishkoo and M. Darus
signals, generalized birth and death processes and many others. Because of in-
teresting and general properties of G-function, it is possible to represent the
solutions of many problems in these fields in their terms. Stated in this way,
the problems gain a much more general character due to the great freedom of
choice of the orders m;n; p; q and the parameters of G-functions in comparison
to other special functions. Simultaneously, the calculations become simpler and
more unified. Evidence showing the importance of the G-function is given by
the fact that the basic elementary functions and most of the special functions
of mathematical physics, including the generalized hypergeometric functions, fol-
low as its particular cases. Therefore, each result concerning the G-function has
become a key leading to numerous particular results for the Bessel functions,
confluent hypergeometric functions, classical orthogonal polynomials and others
(see [1]).
The Meijer G-function has been useful in mathematical physics because of its
analytical properties, in particular, it can be expressed as a final sum of the gen-
eralized hypergeometric functions with the well-known series expansions. Some
of the differential properties of Meijer G-functions were derived by E.E. Fitchard
and V. Franco (see [2–5], [1, -Appendix]). The G-function is also relatively easy
to compute numerically. Recently, K. Roach has discussed an algorithm for com-
puting the formula representations of instances of the Meijer G-function [6]. The
astrophysical thermonuclear functions I1(z, ν) and I2(z, d, ν) are expressed in
terms of these functions [7]. The Meijer G-function is also used as the weight
function to obtain the Gazeau–Klauder (photon-added) coherent states [8].
In the previous paper we have classified the univalent Meijer G-functions into
three types. Three basic univalent Meijer G-functions are introduced, namely,
G1,0
0,2; G
1,1
1,2;G
1,1
1,1, and by the successive applications of fractional differintegral
transformations, a number of univalent Meijer G-functions could be obtained
and the Erdélyi–Kober operators (m = 1, 2) as the transformations preserving the
univalence of the Meijer G-functions [9]. These classification and transformations
are based on Kiryakova’s studies in representing the generalized hypergeometric
functions as fractional differintegral operators of three basic elementary functions
[10, 11]. It is shown that the function G1,0
0,2, as one of the three basic univalent
Meijer G-functions, enters in all solutions of the three PDEs considered here.
The starting point in this work is the question ”Is it possible to represent
some solutions of physical models explicitly in terms of the Meijer G-functions?”
The question contains two important points that motivate us to answer it:
(i) Finding explicit solutions to mathematical models of various physical,
statistical and even social events through Meijer G-functions was almost unknown
as an idea till the 80s of the last century. But recently there appeared many books,
surveys and papers emphasizing the role of the G-functions not only as kernels
of some integrals.
380 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Some Applications of Meijer G-Functions...
(ii) Studying the ODEs satisfied by the Meijer G-functions hence treats
and solves explicitly each particular ODE arising from a PDE by separation of
variables.
It induces us to find a new method for doing this work. In fact, we will
concentrate on G-function’s ordinary linear differential equation (OLDE), and
by proper selection of its orders m; n; p; q and the parameters, we will equate
G-function’s OLDE and ODEs. By doing this, we can show that G-functions are
the explicit solutions of the PDEs.
Helping idea in obtaining our results is that the Meijer G-functions include
all elementary and special functions, and so the ODE for the G-function can
include many cases of ordinary differential equations whose solutions are exactly
these elementary and special functions. However, we have PDEs but not ODEs
to solve, so the separation of variables is needed.
The laws of physics are almost exclusively written in the form of differential
equations (PDEs). Depending on the geometry of the problem, these differential
equations are separated into ODEs, each involving a single coordinate of a suitable
coordinate system. With the Cartesian, cylindrical and spherical coordinates, the
boundary conditions are important in determining the nature of the solutions of
ODEs obtained from PDEs. These ODEs are usually of the Sturm–Liouville
(S–L) type [12].
The contents of this paper is divided into three sections. In the first section,
we recall the definition of the Meijer G-function and the G-function’s ordinary
linear differential equation. The second section introduces three well-known PDEs
in physics and the method of separation of variables. The third section discusses
the new representation of the solutions in terms of symmetric product forms of
the univalent Meijer G-functions.
Definition 1.1. A definition of the Meijer G-function is given by the following
path integral in the complex plane, called of the Mellin–Barnes type [1, 13–16]:
Gm,n
p,q (a1,...,ap
b1,...,bq
|z) =
1
2πi
∫
L
∏m
j=1 Γ(bj − s)
∏n
j=1 Γ(1− aj + s)∏q
j=m+1 Γ(1− bj + s)
∏p
j=n+1 Γ(aj − s)
zsds. (1.1)
Here, an empty product means unity and the integers m; n; p; q are called the
orders of the G-function, or the components of the order (m; n; p; q), while ap
and bq are called the ”parameters” and, in general, are complex numbers. The
definition holds under the following assumptions: 0 ≤ m ≤ q and 0 ≤ n ≤ p ,
where m,n, p, and q are the integer numbers. aj−bk 6= 1, 2, 3, . . . for k = 1, . . . , n
and j = 1, 2, . . . , m imply that no pole of any Γ(bj − s), j = 1, . . . , m coincides
with a pole of any Γ(1− ak + s), k = 1, . . . , n.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 381
A. Pishkoo and M. Darus
The Meijer G-function y(z) = Gm,n
p,q (z|aj
bk
) satisfies the linear ordinary differ-
ential equation of the generalized hypergeometric type
[(−1)p−m−nz
p∏
j=1
(z
d
dz
− aj + 1)−
q∏
k=1
(z
d
dz
− bk)]y(z) = 0 (1.2)
whose order is equal to max(p, q), (see [3–5], [1, -Appendix]).
2. Three PDEs in Cartesian Coordinates System
A problem most suitable for the Cartesian coordinates has the boundaries
with rectangular symmetry such as boxes or planes. Separation of variables leads
to the ODEs in which certain constants (eigenvalues) appear. Different choices of
signs for these constants can lead to different functional forms of general solution.
The general form of the solution is indeterminate. However, once the boundary
conditions are imposed, the unique solutions will emerge regardless of the initial
functional form of the solution.
In electrostatics, where time-independent scalar fields such as potentials are
studied, the law is described in vacuum by the Laplace equation
∇2Φ = 0.
Writing Φ(x, y, z) as a product of three functions, Φ(x, y, z) = X(x)Y (y)Z(z),
yields three ODEs as follows:
d2X
dx2
+ λX = 0,
d2Y
dy2
+ µY = 0,
d2Z
dz2
+ νZ = 0, (2.1)
where λ + µ + ν = 0.
The Laplace equation describes not only electrostatics, but also heat transfer.
When the transfer (diffusion) of heat takes place with the temperature being
independent of time, the process is known as a steady-state heat transfer. The
diffusion equation ∂T
∂t = a2∇2T becomes the Laplace equation ∇2T = 0, where
T is the temperature and a is a constant characterizing the medium in which heat
is flowing.
The Schrödinger equation, describing non-relativistic quantum phenom-
ena, is as follows:
− ~
2
2m
∇2Ψ + V (r)Ψ = −i~
∂Ψ
∂t
,
where m is the mass of a subatomic particle, ~ is the Plank constant (divided
by 2π), V is the potential energy of the particle, and |Ψ(r, t)|2 is the probability
density of finding the particle at r at time t.
382 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Some Applications of Meijer G-Functions...
In the next section, we obtain the solutions of these three equations in terms
of the Meijer G-functions.
3. Results
The Meijer G-function y(z) = Gm,n
p,q (z|aj
bk
) satisfies the linear ordinary differ-
ential equation of the generalized hypergeometric type
[(−1)p−m−nz
p∏
j=1
(z
d
dz
− aj + 1)−
q∏
k=1
(z
d
dz
− bk)]y(z) = 0 (3.1)
whose order is equal to max(p, q),(see [3–5], [1, Appendix]).
We here consider two cases when (3.1) reduces to first and second order or-
dinary differential equations, respectively:
Case 1. Setting m = 1, n = 0, p = 0, q = 1 in (3.1) yields
[−z − (z
d
dz
− b1)]G
1,0
0,1(z|−b1) = 0.
By changing z to −z and dividing by z, we have
[
d
dz
− 1 +
b1
z
]G1,0
0,1(−z|−b1) = 0.
On the other hand, changing from variable t to z gives
[
d
dz
− 1]T (z) = 0.
Equality condition for these two differential equations leads to b1 = 0, and the
solution in terms of the Meijer G-function is
T (z) = G1,0
0,1(−z|−0 ) = ez.
Case 2. Setting m = 1, n = 0, p = 0, q = 2 in (3.1) yields
[−z − (z
d
dz
− b2)(z
d
dz
− b1)]G
1,0
0,2(z|−b1,b2
) = 0.
Changing variable from z to αz2 gives
[−αz2 − (
z
2
d
dz
− b2)(
z
2
d
dz
− b1)]G
1,0
0,2(αz2|−b1,b2
) = 0,
and dividing by z2 gives
[
d2
dz2
+ (1− 2(b1 + b2))z
d
dz
+ 4α + 4
b1b2
z2
]G1,0
0,2(αz2|−b1,b2
) = 0. (3.2)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 383
A. Pishkoo and M. Darus
On the other hand,
[
d2
dz2
+ 1]Z = 0.
Then an equality condition leads to
1− 2(b1 + b2) = 0, 4α = 1, b1b2 = 0.
If b1 = 1
2 , b2 = 0, α = 1
4 , then the first independent solution in terms of the Meijer
G-function is
Z(z) = sin z = G1,0
0,2(
1
4
z2|−1
2
,0
).
If b1 = 0, b2 = 1
2 , α = 1
4 , then the second independent solution in terms of the
Meijer G-function is
Z(z) = cos z = G1,0
0,2(
1
4
z2|−
0, 1
2
).
Furthermore, an equality condition for (3.2) and the differential equation
[
d2
dz2
− 1]Z = 0
gives 1− 2(b1 + b2) = 0, 4α = −1, b1b2 = 0, that is,
b1 =
1
2
, b2 = 0, α = −1
4
.
If b1 = 1
2 , b2 = 0, α = −1
4 , then the first independent solution in terms of the
Meijer G-function is
Z(z) = sinh z = G1,0
0,2(−
1
4
z2|−1
2
,0
).
If b1 = 0, b2 = 1
2 , α = −1
4 , then the second independent solution in terms of the
Meijer G-function is
Z(z) = cosh z = G1,0
0,2(−
1
4
z2|−
0, 1
2
).
All the obtained solutions are shown in Table 1.
There are three kinds of boundary conditions that can be written in terms of
the Meijer G-functions:
1. Separated boundary conditions (BCs):
α1G
m,n
p,q |a + β1G
′m,n
p,q |a = 0, α2G
m,n
p,q |b + β2G
′m,n
p,q |b = 0. (3.3)
2. Periodic BCs:
Gm,n
p,q |a = Gm,n
p,q |b, G′m,n
p,q |a = G′m,n
p,q |b. (3.4)
384 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Some Applications of Meijer G-Functions...
T a b l e 1. The general solutions in terms of the Meijer G-functions and
elementary functions
Differential equation Elementary functions Meijer G-functions
[ d
dz − 1]T (z) = 0 T (z) = ez T (z) = G1,0
0,1(−z|−0 )
[ d2
dz2 + 1]Z(z) = 0 Z(z) = A sin z + B cos z Z(z) = AG1,0
0,2(
1
4z2|−1
2
,0
)
+BG1,0
0,2(
1
4z2|−
0, 1
2
)
[ d2
dz2 − 1]Z(z) = 0 Z(z) = C sinh z + D cosh z Z(z) = CG1,0
0,2(−1
4z2|−1
2
,0
)
+DG1,0
0,2(−1
4z2|−
0, 1
2
)
3. Generalization of separated and periodic BCs:
α11G
m,n
p,q (a|ap
bq
) + α12G
′m,n
p,q (a|ap
bq
) + α13G
m,n
p,q (b|ap
bq
) + α14G
′m,n
p,q (b|ap
bq
) = 0,
α21G
m,n
p,q (a|ap
bq
) + α22G
′m,n
p,q (a|ap
bq
) + α23G
m,n
p,q (b|ap
bq
) + α24G
′m,n
p,q (b|ap
bq
) = 0. (3.5)
E x a m p l e 3.1. Steady-state heat conducting plate.
Let us consider a rectangular heat conducting plate with sides of length a and b.
Three of the sides are held at T = 0, and the fourth side, at y = b, has a
temperature variation T = f(x). The flat faces are insulated so that they cannot
lose heat to the surroundings. Assuming a steady-state heat transfer, the diffusion
equation ∂T
∂t = a2∇2T becomes the Laplace equation ∇2T = 0. Let us calculate
the variation of T over the plate.
The problem is two-dimensional. The separation of variables T (x, y) =
X(x)Y (y) leads to
d2X
dx2
+ λX = 0,
d2Y
dy2
+ µY = 0, (3.6)
where λ + µ = 0.
The X equation (see Table 1) has a general solution as follows:
X(x) = AG1,0
0,2(
λ
4
x2|−1
2
,0
) + BG1,0
0,2(
λ
4
x2|−
0, 1
2
).
The boundary condition, T (0, y) = 0 for all y, implies that X(0) = 0. Therefore
we get
X(x) = G1,0
0,2(
λ
4
x2|−1
2
,0
) = sin
√
λx.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 385
A. Pishkoo and M. Darus
The X equation and two BCs, T (0, y) = T (a, y) = 0, form an S–L system for
which we obtain the following eigenvalues and eigenfunctions:
• λn = (nπ
a )2 and Xn(x) = G1,0
0,2(
n2π2
4a2 x2|−1
2
,0
) = sin(nπ
a ) for n = 1, 2, . . . .
Therefore, a general solution X(x) can be written as
X(x) =
∞∑
n=1
AnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
).
On the other hand, the Y equation does not form an S−L system. Nevertheless,
we can solve the equation Y ′′ − (nπ
a )2Y = 0 to obtain a general solution
Y (y) = C sinh y + D cosh y = CG1,0
0,2(−
1
4
y2|−1
2
,0
) + DG1,0
0,2(−
1
4
y2|−
0, 1
2
).
The boundary condition T (x, 0) = 0 for all x implies that Y (0) = 0. So,
Y =
∞∑
n′=1
Cn′ sinh
n′πy
a
δnn′ =
∞∑
n′=1
Cn′G
1,0
0,2(−
n′2π2
4a2
y2|−1
2
,0
)δnn′ ,
where sin
√
µy = sin
√−λy (see (3.6)) which means sin
√
µy = sinh
√
λy.
Thus, the most general solution consistent with the three BCs is
T (x, y) = X(x)Y (y) =
∞∑
n=1
EnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(−
n2π2
4a2
y2|−1
2
,0
).
The fourth BC gives
f(x) = T (x, b) =
∞∑
n=1
[EnG1,0
0,2(−
n2π2
4a2
b2|−1
2
,0
)]G1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)
=
∞∑
n=1
FnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)
whose coefficients can be determined from
Fn =
2
a
a∫
0
G1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)f(x)dx.
E x a m p l e 3.2. Rectangular conducting box.
Consider a rectangular conducting box with sides a, b and c. All faces are held at
zero potential except the top face whose potential is given by a function f(x, y)
at z = c. Let us find the potential at all points inside the box.
386 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Some Applications of Meijer G-Functions...
The relevant PDE for this situation is the Laplace equation ∇2Φ = 0. The
problem is three-dimensional. Writing Φ(x, y, z) = X(x)Y (y)Z(z) yields three
ODEs
d2X
dx2
+ λX = 0,
d2Y
dy2
+ µY = 0,
d2Z
dz2
+ νZ = 0, (3.7)
where λ + µ + ν = 0.
The vanishing of Φ at x = 0 and x = a gives
• Φ(0, y, z) = X(0)Y (y)Z(z) = 0 for all y, z , that is, X(0) = 0,
• Φ(a, y, z) = X(a)Y (y)Z(z) = 0 for all y, z , so X(a) = 0.
The X equation and two BCs, X(0) = X(a) = 0, form the S–L system whose
eigenvalues and eigenfunctions are
• λn = (nπ
a )2 and Xn(x) = G1,0
0,2(
n2π2
4a2 x2|−1
2
,0
) for n = 1, 2, ...
Similarly, the second equation in (3.7) means that
• µm = (mπ
b )2 and Ym(y) = G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
) for m = 1, 2, ...
Furthermore, the third equation in (3.7) does not lead to an S–L system,
whose eigenvalues and eigenfunctions consistent with the boundary condition
Z(0) = 0 are
• (γmn)2 = (nπ
a )2 + (mπ
b )2 and Z(z) = AmnG1,0
0,2(−γmn
4 z2|−1
2
,0
).
Consequently, putting everything together, we obtain
Φ(x, y, z) = X(x)Y (y)Z(z)
=
∞∑
n=1
∞∑
m=1
AmnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
)G1,0
0,2(−
γmn
4
z2|−1
2
,0
).
To specify Φ completely, we must determine the arbitrary constants Amn. Im-
posing the last BC, Φ(x, y, c) = f(x, y), yields
f(x, y) =
∞∑
n=1
∞∑
m=1
AmnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
)G1,0
0,2(−
γmn
4
c2|−1
2
,0
)
=
∞∑
n=1
∞∑
m=1
BmnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
),
where Bmn = AmnG1,0
0,2(−γmn
4 c2|−1
2
,0
).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 387
A. Pishkoo and M. Darus
E x a m p l e 3.3. Conducting heat in a rectangular plate.
Consider a rectangular heat-conducting plate with sides of length a and b all held
at T = 0. Assume that at time t = 0 the temperature has a distribution function
f(x, y). Let us find the variation of the temperature for all points (x, y) at all
times t > 0.
The diffusion equation for this problem is
∂T
∂t
= k2∇2T = k2(
∂2T
∂x2
+
∂2T
∂y2
).
A separation of variables T (x, y, t) = X(x)Y (y)g(t) leads to three ODEs:
d2X
dx2
+ λX = 0,
d2Y
dy2
+ µY = 0,
dg
dt
+ k2(λ + µ)g = 0. (3.8)
The BCs, T (0, y, t) = T (a, y, t) = T (x, 0, t) = T (x, b, t) = 0, together with the
three ODEs, give rise to two S − L systems. The solutions to both are easy to
find:
• λn = (nπ
a )2 and Xn(x) = G1,0
0,2(
n2π2
4a2 x2|−1
2
,0
) for n = 1, 2, . . . ,
• µm = (mπ
b )2 and Ym(y) = G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
) for m = 1, 2, . . . .
These give rise to the general solution
X(x) =
∞∑
n=1
AnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
), Y (y) =
∞∑
m=1
BmG1,0
0,2(
m2π2
4b2
y2|−1
2
,0
)
with γmn = k2(λn + µm).
The solution to the g equation can be expressed as
g(t) = Cmne−γmnt = CmnG1,0
0,1(γmnt|−0 ).
Consequently, the most general solution can be expressed as follows:
T (x, y, t) =
∞∑
n=1
∞∑
m=1
AmnG1,0
0,1(γmnt|−0 )G1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
),
where Amn = AnBmCmn is an arbitrary constant.
To determine it, we impose the initial condition T (x, y, 0) = f(x, y). This yields
the following:
f(x, y) =
∞∑
n=1
∞∑
m=1
AmnG1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
),
388 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Some Applications of Meijer G-Functions...
which determines the coefficients Amn
Amn =
4
ab
a∫
0
dx
b∫
0
dyf(x, y)G1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
).
E x a m p l e 3.4. Quantum particle in a box.
The behaviour of an atomic particle of mass µ confined in a rectangular box with
sides a, b and c (an infinite three- dimensional potential well is governed by the
Schrödinger equation for a free particle
i~
∂ψ
∂t
= − ~
2
2m
(
∂2ψ
∂x2
+
∂2ψ
∂y2
+
∂2ψ
∂z2
),
and the BC that ψ(x, y, z) vanishes at all sides of the box for all time.
A separation of variable ψ(x, y, z, t) = X(x)Y (y)Z(z)T (t) yields the ODEs
d2X
dx2
+ λX = 0,
d2Y
dy2
+ σY = 0,
d2Z
dz2
+ νZ = 0, (3.9)
dT
dt
+ iωT = 0, (3.10)
where ω = ~
2µ(λ + σ + ν).
Vanishing of ψ, BCs at x = 0 and x = a, for all y, z; at y = 0 and
y = b for all x, z; at z = 0 and z = c for all x, y, gives
• ψ(0, y, z, t) = ψ(a, y, z, t) = 0, which means X(0) = X(a) = 0,
• ψ(x, 0, z, t) = ψ(x, b, z, t) = 0, which means Y (0) = Y (b) = 0,
• ψ(x, y, 0, t) = ψ(x, y, c, t) = 0, which means Z(0) = Z(C) = 0,
leads to three S–L systems whose solutions (see Table 1) are easily found:
• Xn(x) = G1,0
0,2(
n2π2
4a2 x2|−1
2
,0
), λn = (nπ
a )2 for n = 1, 2, . . . ,
• Ym(y) = G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
), σm = (mπ
b )2 for m = 1, 2, . . . ,
• Zl(z) = G1,0
0,2(
l2π2
4c2
z2|−1
2
,0
), νm = ( lπ
c )2 for l = 1, 2, . . . .
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 389
A. Pishkoo and M. Darus
On the other hand, the time equation has the solution (Table 1) of the following
form:
T (t) = ClmnG1,0
0,1(iωlmnt|−0 ), where ωlmn =
~
2µ
[(
nπ
a
)2 + (
mπ
b
)2 + (
lπ
c
)2].
Therefore, the solution of the Schrödinger equation consistent with the BCs is
ψ(x, y, z, t)
=
∞∑
l,m,n=1
AlmnG1,0
0,1(iωlmnt|−0 )G1,0
0,2(
n2π2
4a2
x2|−1
2
,0
)G1,0
0,2(
m2π2
4b2
y2|−1
2
,0
)G1,0
0,2(
l2π2
4c2
z2|−1
2
,0
).
The constants Al,m,n are determined by the initial shape ψ(x, y, z, 0) of the wave
function.
4. Conclusions
In this article, we illustrated that the Meijer G-functions have many appli-
cations in explicit solutions of the three well-known PDEs, namely the Laplace
equation, the diffusion equation and the Schrödinger equation. These solutions
are very nice because of their symmetric product forms. We believe that the
G-function will be used as a global function in physics and engineering for unifi-
cation.
Acknowledgement. The authors also would like to thank all the referees
for the informative critics to improve the content of the article.
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