Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras
Some systems of differential equations with partial derivatives are studied by using the properties of Gateaux differentiable functions on commutative algebras. The connection between solutions of systems of partial differential equations and components of monogenic functions on the corresponding co...
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irk-123456789-1693982020-06-13T01:27:05Z Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras Kolomiiets, T. Pogorui, A. Rodriguez-Dagnino, R.M. Some systems of differential equations with partial derivatives are studied by using the properties of Gateaux differentiable functions on commutative algebras. The connection between solutions of systems of partial differential equations and components of monogenic functions on the corresponding commutative algebras is shown.We also give some examples of systems of partial differential equations and find their solutions. 2018 Article Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras / T. Kolomiiets, A. Pogorui, R.M. Rodriguez-Dagnino // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 210-219. — Бібліогр.: 8 назв. — англ. 1810-3200 2000 MSC. Primary 35C99; Secondary 32W50. http://dspace.nbuv.gov.ua/handle/123456789/169398 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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Some systems of differential equations with partial derivatives are studied by using the properties of Gateaux differentiable functions on commutative algebras. The connection between solutions of systems of partial differential equations and components of monogenic functions on the corresponding commutative algebras is shown.We also give some examples of systems of partial differential equations and find their solutions. |
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Kolomiiets, T. Pogorui, A. Rodriguez-Dagnino, R.M. |
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Kolomiiets, T. Pogorui, A. Rodriguez-Dagnino, R.M. Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras Український математичний вісник |
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Kolomiiets, T. Pogorui, A. Rodriguez-Dagnino, R.M. |
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Kolomiiets, T. |
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Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras |
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Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras |
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Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras |
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Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras |
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Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras |
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solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras |
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Інститут прикладної математики і механіки НАН України |
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2018 |
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Solution of systems of partial differential equations by using properties of monogenic functions on commutative algebras / T. Kolomiiets, A. Pogorui, R.M. Rodriguez-Dagnino // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 210-219. — Бібліогр.: 8 назв. — англ. |
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Український математичний вісник |
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AT kolomiietst solutionofsystemsofpartialdifferentialequationsbyusingpropertiesofmonogenicfunctionsoncommutativealgebras AT pogoruia solutionofsystemsofpartialdifferentialequationsbyusingpropertiesofmonogenicfunctionsoncommutativealgebras AT rodriguezdagninorm solutionofsystemsofpartialdifferentialequationsbyusingpropertiesofmonogenicfunctionsoncommutativealgebras |
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2025-07-15T04:08:25Z |
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Український математичний вiсник
Том 15 (2018), № 2, 210 – 219
Solution of systems of partial differential
equations by using properties of monogenic
functions on commutative algebras
Tamila Kolomiiets, Anatoliy Pogorui,
Ramón M. Rodrı́guez-Dagnino
(Presented by V. Ya. Gutlyanskii)
Abstract. In this paper some systems of differential equations with
partial derivatives are studied by using the properties of Gâteaux dif-
ferentiable functions on commutative algebras. The connection between
solutions of systems of differential equations in partial derivatives and
components of monogenic functions on corresponding commutative al-
gebras is shown. We also give some examples of systems of differential
equations with partial derivatives and find their solutions.
2000 MSC. Primary 35C99; Secondary 32W50.
Key words and phrases. PDE systems; Monogenic functions; Gâ-
teaux derivative; commutative algebra.
1. Introduction
The study of partial differential equations (PDE) by using the prop-
erties of monogenic (Gâteaux differentiable) functions on commutative
algebras sometimes makes it possible to effectively find solutions of these
equations and investigate their characteristics. The best-known example
of such an approach is the fact that the real and imaginary parts of a
complex analytic function are harmonic functions, that is, solutions of
the two-dimensional differential Laplace equation.
In [1–4] solutions of the multidimensional Laplace equation were stud-
ied. A generalization of this method to a wide class of partial differen-
tial equations with constant coefficients was implemented in [5]. The
method consists in finding the commutative algebra associated with the
differential equation and constructing monogenic functions on the cor-
responding subspace of this algebra. It is proved that in the case of a
finite-dimensional algebra all the components of such a monogenic func-
tion are solutions of the corresponding equation.
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
T. Kolomiiets, A. Pogorui, R. M. Rodŕıguez-Dagnino 211
In [6] and [7] this method is applied to the study of one-dimensional
distribution of the particles moving in Erlang-2 and Erlang-3 semi-Mar-
kov media.
In [8] the method is generalized to a PDE with linearly dependent
variable coefficients.
In this paper the method of finding solutions by using monogenic
functions is used to study the solutions of systems of partial differential
equations with constant coefficients.
2. Monogenic functions on commutative algebra
Let A be a n-dimensional commutative algebra over a field K of
characteristic 0. Denote by e⃗1, . . . , e⃗n a basis of A. Consider an m-
dimensional subspace B of the algebra A, where m ≤ n, m,n ∈ N with
the basis e⃗1, e⃗2, . . . , e⃗m. Suppose we have a function f⃗ : B → A of the
following form
f⃗ (x⃗) =
n∑
k=1
uk(x⃗)e⃗k,
where uk(x⃗) = uk(x1, x2, . . . , xm) are K-valued functions of m variables
xi ∈ K.
Thus,
B ∋ x⃗ =
m∑
i=1
xie⃗i
f⃗−→
n∑
k=1
uk(x1, x2, . . . , xm)e⃗k,
such that uk : Km → K, k = 1, . . . , n.
Definition 1. A function f⃗ is called differentiable at a point x⃗0 ∈ B if
there exists a unique element f ′ (x⃗0) ∈ A such that for any h⃗ ∈ B
f ′(x⃗0)⃗h = lim
K∋ε→0
f(x⃗0 + εh⃗)− f(x⃗0)
ε
,
where f ′(x⃗0)⃗h is the product of two elements f ′(x⃗0) and h⃗ of algebra A.
Definition 2. A function f⃗ : B → A is said to be monogenic if it is
differentiable at every point x⃗ ∈ B.
Remark 1. It is easily seen that if A = B = C then a monogenic function
is differentiable in the complex sense.
In papers [5, 8] the following two theorems were proved.
212 Solution of systems of partial differential equations...
Theorem 1. A function f⃗(x⃗) =
∑n
k=1 e⃗kuk(x⃗) is differentiable at point
x⃗0 if and only if there exist partial derivatives ∂f⃗(x⃗0)
∂xi
, i = 1, . . . ,m, which
satisfy the following conditions
uk(x1 + εh1, x2 + εh2, . . . , xm + εhm)− uk(x1, x2, . . . , xm)
= ε
m∑
i=1
∂uk
∂xi
hi + o(ε), K ∋ ε→ 0, ∀(h1, h2, . . . , hm) ∈ Km,
k = 1, 2, . . . , n,
and
e⃗i
∂f⃗ (x⃗0)
∂xj
= e⃗j
∂f⃗ (x⃗0)
∂xi
, i, j = 1, . . . ,m.
Remark 2. The conditions e⃗i
∂f⃗(x⃗0)
∂xj
= e⃗j
∂f⃗(x⃗0)
∂xi
, i, j = 1, . . . ,m can be
written in the following form
n∑
k=1
∂uk
∂xj
e⃗ie⃗k =
n∑
k=1
∂uk
∂xi
e⃗j e⃗k. (2.1)
Conditions (2.1) are called the generalized Cauchy–Riemann equa-
tions since when we have the special case A = B = C they are the
Cauchy–Riemann equations.
For positive integers r, m we introduce the following polynomial
P (ξ1, ξ2, . . . , ξm)=
∑
i1+i2+···+im=r
Ci1,i2,...,im (x1, x2, . . . , xm) ξ
i1
1 ξ
i2
2 . . . ξ
im
m ,
(2.2)
where Ci1,...,im (x1, . . . , xm) are K-valued continuous functions of the m
variables x1, . . . , xm ∈ K.
Now, let us consider the following differential equations
P (∂1, ∂2, . . . , ∂m) [u (x1, x2, . . . , xm)] = 0, (2.3)
where ∂k = ∂
∂xk
.
Theorem 2. Let P be a polynomial as in Eq. (2.2), and let a function
f⃗ : B → A be monogenic. In addition,
f⃗(x⃗) =
n∑
k=1
e⃗kuk(x⃗),
where e⃗1, e⃗2, . . . , e⃗m is a basis of the subspace B of the algebra A such
that
P (e⃗1, e⃗2, . . . , e⃗m) = 0,
then the functions uk(x⃗), k = 1, . . . , n are solutions of Eq. (2.3).
T. Kolomiiets, A. Pogorui, R. M. Rodŕıguez-Dagnino 213
3. Differential functions proving solution to systems
of PDEs.
Consider the following system of PDEs
D11u1 (x⃗) +D12u2 (x⃗) + · · ·+D1NuN (x⃗) = 0,
D21u1 (x⃗) +D22u2 (x⃗) + · · ·+D2NuN (x⃗) = 0,
...
...
...
Dn1u1 (x⃗) +Dn2u2 (x⃗) + · · ·+DNNuN (x⃗) = 0,
(3.4)
where Dij are differential operators which satisfy commutative conditions
DijDkl = DklDij , i, j, k, l = 1, 2, . . . , N .
Let us introduce the matrix of differential operators as follows
D =
D11 D12 · · · D1N
D21 D22 · · · D2N
...
...
. . .
...
DN1 DN2 · · · DNN
. (3.5)
Denote by det (D) the determinant of D and by Aij the algebraic
complements of the matrix of elements of D, that is det (D) = DN1AN1+
DN2AN2 + · · ·+DNNANN .
Theorem 3. Suppose ui(x⃗), i = 1, . . . , N is a solution of system (3.4),
then for all λ1, λ2, . . . , λN ∈ K, we have
det (D) (λ1u1(x⃗) + λ2u2(x⃗) + · · ·+ λNuN (x⃗)) = 0. (3.6)
Proof. The system of PDEs (3.4) can be represented in the vector-matrix
form:
DU⃗ (x⃗) = 0⃗, (3.7)
where D is the matrix (3.5), and U⃗ (x⃗) =
u1(x⃗)
u2(x⃗)
...
uN (x⃗)
, 0⃗ =
0
0
...
0
.
Denote by D̃ = (Aij)
T the transpose matrix composed of the com-
plements Aij .
Multiplying Eq. (3.7) by the matrix D̃, i.e.,
(
D̃D
)
U⃗ = D̃
(
DU⃗
)
=
D̃0⃗ = 0⃗, we obtain
214 Solution of systems of partial differential equations...
det (D) 0 · · · 0
0 det (D) · · · 0
...
...
. . .
...
0 0 · · · det (D)
u1(x⃗)
u2(x⃗)
...
uN (x⃗)
=
0
0
...
0
.
Hence,
det (D)u1(x⃗) = 0,
det (D)u2(x⃗) = 0,
...
det (D)uN (x⃗) = 0,
or
det (D)
(
N∑
i=1
λiui(x⃗)
)
= 0 (3.8)
for all λ1, λ2, . . . , λN ∈ K.
Remark 3. It is easily seen that by solving Eq. (3.8) we obtain a lin-
ear combination of the system (3.4) with solutions ui(x⃗), i = 1, . . . , N .
Hence, if we have m linear independent solutions vi(x⃗), i = 1, . . . ,m of
det (D) v(x⃗) = 0 then we can look for a solution of the system (3.4) in
the form of a linear combination as follows
ui(x⃗) = ci1v1(x⃗) + ci2v2(x⃗) + · · ·+ cimvm(x⃗), i = 1, 2, . . . ,m.
4. Examples
4.1. Example 1.
Consider the following system of PDEs
∂2
∂x2
u1(x, y) +
∂3
∂y3
u2(x, y) = 0,
∂
∂y
u1(x, y) +
∂2
∂x2
u2(x, y) = 0,
(4.9)
where x, y ∈ R and uk, k = 1, 2 are real functions.
Let us write Eqs. (4.9) in the matrix form
DU⃗ (x, y) = 0⃗,
T. Kolomiiets, A. Pogorui, R. M. Rodŕıguez-Dagnino 215
where
D =
∂2
∂x2
∂3
∂y3
∂
∂y
∂2
∂x2
, U⃗ =
(
u1(x, y)
u2(x, y)
)
, 0⃗ =
(
0
0
)
.
In this case we have the following equation
det (D) v(x, y) =
(
∂4
∂x4
− ∂4
∂y4
)
v(x, y) = 0. (4.10)
Here we have the following corresponding polynomial P (ξ1, ξ2) =
=
(
ξ21
)2 − (ξ22)2. Then, we should find an algebra with basis e⃗1, e⃗2 satis-
fying
P (e⃗1, e⃗2) =
(
e⃗ 21
)2 − (e⃗ 22 )2 = 0. (4.11)
It is easily seen that the basis {1, i} of complex numbers satisfies Eq.
(4.11), hence, we consider the case where A = B = C.
Let us consider function f (z) = ez = ex (cos y + i sin y). It is easily
verified that u1(x, y) = Cex cos y − Cex sin y, u2(x, y) = Cex cos y +
Cex sin y, where C ∈ R, is a solution of the system (4.9). It means that
c11 = −c12 = c21 = c22 = C.
We should notice that for the system
∂
∂x u1(x, y) +
∂2
∂y2
u2(x, y) = 0,
∂2
∂y2
u1(x, y) +
∂3
∂x3
u2(x, y) = 0.
(4.12)
In this case we also have det (D) =
(
∂4
∂x4
− ∂4
∂y4
)
. Hence, we can
consider the same function f (z) = ex (cos y + i sin y) to obtain a solution
for (4.12).
It can be easily verified that u1(x, y) = Cex cos y+Cex sin y, u2(x, y)=
u1(x, y), is a solution of the system (4.12). This is also equivalent to say
that c11 = c22 = c12 = c21 = C.
4.2. Example 2.
Let us solve the following system
∂
∂z
u1(z, w) +
∂2
∂w2
u2(z, w) = 0,
∂2
∂w2
u1(z, w) +
∂
∂z
u2(z, w) = 0,
(4.13)
where z, w ∈ C and uk, k = 1, 2, are complex functions.
216 Solution of systems of partial differential equations...
Consider the matrix form of system (4.13)
DU⃗ = 0⃗,
where
D =
(
∂
∂z
∂2
∂w2
∂2
∂w2
∂
∂z
)
, U⃗ =
(
u1 (z, w)
u2 (z, w)
)
, 0⃗ =
(
0
0
)
.
For this case we have
det (D) v(z, w) =
(
∂2
∂z2
− ∂4
∂w4
)
v(z, w) = 0. (4.14)
In order to apply Theorem 2, instead of Equation (4.14), we solve the
following equation (
∂4
∂ω2∂z2
− ∂4
∂w4
)
V (ω, z, w) = 0. (4.15)
Let us find a solution of Eq. (4.15) in the form V (ω, z, w) = eωv (z, w).
It is easily seen that a function v0 (z, w) is a solution of Eq. (4.14) if and
only if the function V (ω, z, w) = eωv0 (z, w) is a solution of Eq. (4.15).
The polynomial P for Eq. (4.15) is as follows P (ξ1, ξ2, ξ3) = ξ21ξ
2
2−ξ43 .
Hence, we should use a commutative algebra whose basis contains vectors
e⃗1, e⃗2, e⃗3 such that
P (e⃗1, e⃗2, e⃗3) = e⃗ 21 e⃗
2
2 − e⃗ 43 = 0. (4.16)
For this case we can use bicomplex Segre numbers over a complex
field as follows A = {a0 + a1j + a2k + a3f | a0, a1, a2, a3 ∈ C}, j2 = k2 =
−1, f2 = 1, jk = kj, jf = fj, kf = fk, where j, k, f commute with
i ∈ C.
Let us denote ij = ji = p, ik = ki = q, if = fi = r. Then, A can be
represented in the following form
A = {a0 + a1i+ a2j + a3k + a4f + a5p+ a6q + a7r| ai ∈ R}
with the following Cayley table
1 i j k f p q r
1 1 i j k f p q r
i i −1 p q r −j −k −f
j j p −1 f −k −i r −q
k k q f −1 −j r −i −p
f f r −k −j 1 −q −p i
p p −j −i r −q 1 −f k
q q −k r −i −p −f 1 j
r r −f −q −p i k j −1
T. Kolomiiets, A. Pogorui, R. M. Rodŕıguez-Dagnino 217
It is easily seen that A is associative and commutative. As the sub-
space B of the algebra A we consider
B = {a0 + a1f + a2j| ai ∈ C} .
The basis {1, f, j} of B satisfies Eq. (4.16).
Consider f⃗ : B → A of the following form
f⃗(ω, z, w) = eω+fz+jw
= V1(ω, z, w) + V2(ω, z, w)j + V3(ω, z, w)k + V4(ω, z, w)f.
According to Theorem 1 the function f⃗ is monogenic since
f
∂
∂ω
f⃗(ω, z, w) =
∂
∂z
f⃗(ω, z, w) = feω+fz+jw,
j
∂
∂ω
f⃗(ω, z, w) =
∂
∂w
f⃗(ω, z, w) = jeω+fz+jw,
j
∂
∂z
f⃗(ω, z, w) = f
∂
∂w
f⃗(ω, z, w) = jfeω+fz+jw.
It follows from Theorem 2 that the components V 0
l : C3 → C,
l = 1, 2, 3, 4, are solutions of Eq. (4.15). Hence, the components of
the function f⃗(z, w) = efz+jw are solutions of Eq. (4.14). Let us write
these components in more details
f⃗(z, w) = ejw+fz = ejw · efz = (cos(w) + j sin(w)) (cosh(z) + f sinh(z))
= cos(w) cosh(z) + j sin(w) cosh(z)− k sin(w) sinh(z) + f cos(w) sinh(z).
Hence, we have the following solutions of Eq. (4.14){
v1(z, w) = cos(w) cosh(z), v2(z, w) = sin(w) cosh(z),
v3(z, w) = − sin(w) sinh(z), v4(z, w) = cos(w) sinh(z).
(4.17)
Let us find a solution of system (4.13) in the following form
u1(z, w) = c11 cos(w) cosh(z) + c12 sin(w) cosh(z)
−c13 sin(w) sinh(z) + c14 cos(w) sinh(z),
u2(z, w) = c21 cos(w) cosh(z) + c22 sin(w) cosh(z)
−c23 sin(w) sinh(z) + c24 cos(w) sinh(z).
(4.18)
where cij ∈ C, i, j = 1, 2, 3, 4.
218 Solution of systems of partial differential equations...
Substituting Eqs. (4.18) into Eq. (4.13), and simplifying these equa-
tions we obtain:
(c11 − c24) cos(w) sinh(z) + (c12 + c23) sin(w) sinh(z)
− (c13 + c22) sin(w) cosh(z) + (c14 − c21) cos(w) cosh(z) = 0;
(c11 − c24) cos(w) cosh(z) + (c12 + c23) sin(w) cosh(z)
− (c13 + c22) sin(w) sinh(z) + (c14 − c21) cos(w) sinh(z) = 0.
(4.19)
It follows from (4.19) that
c11 = c24, c12 = −c23, c13 = −c22, c14 = c21.
Acknowledgments
We thank the anonymous reviewer for his/her thorough review and
highly appreciate the comments and suggestions.
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T. Kolomiiets, A. Pogorui, R. M. Rodŕıguez-Dagnino 219
Contact information
Tamila Kolomiiets Department of Mathematical Analysis,
Zhytomyr State University,
Zhytomyr, Ukraine
E-Mail: tamila.kolomiiets@gmail.com
Anatoliy Pogorui Department of Mathematical Analysis,
Zhytomyr State University,
Zhytomyr, Ukraine
E-Mail: pogor@zu.edu.ua
Ramón M.
Rodŕıguez-Dagnino
School of Engineering and Sciences,
Tecnológico de Monterrey,
Monterrey, México
E-Mail: rmrodrig@itesm.mx
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