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Initial value problems with regularinitial functions in quaternionicanalysisNguyen Quoc Hung a & Le Hung Son aa Faculty of Applied Mathematics, Hanoi University of Technology,1 Dai Co Viet, Hanoi, VietnamVersion of record first published: 17 Nov 2009.

To cite this article: Nguyen Quoc Hung & Le Hung Son (2009): Initial value problems with regularinitial functions in quaternionic analysis, Complex Variables and Elliptic Equations: An InternationalJournal, 54:12, 1163-1170

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Complex Variables and Elliptic EquationsVol. 54, No. 12, December 2009, 1163–1170

Initial value problems with regular initial functions in

quaternionic analysis

Nguyen Quoc Hung* and Le Hung Son

Faculty of Applied Mathematics, Hanoi University of Technology,1 Dai Co Viet, Hanoi, Vietnam

Communicated by W. Tutschke

(Received 30 May 2009; final version received 29 June 2009)

In the present article, we consider the initial value problems (IVPs) of type

@w

@t¼ Lw, wð0, :Þ ¼ ’,

where t is the time variable and L is a linear partial differential operator ofthe first order acting with respect to space like variables and taking value inthe quaternion algebra. We formulate conditions on the coefficients of theoperator L under which L is associated with the Cauchy–Riemann operatorin quaternionic analysis. Hence we can construct all linear operators forwhich the IVP with an arbitrary regular function ’ is always solvable.

Keywords: quaternionic analysis; associated operator; Cauchy–Riemannoperator; initial value problem

AMS Subject Classifications: 35F10; 30G35; 35A10

1. Introduction

In this article, we consider the linear partial differential equation of the first order:

@w

@t¼X3j¼0

Að j Þ@w

@xjþX3k¼0

BðkÞwk þ C :¼ Lw, ð1Þ

where w(t, x) is unknown quaternion–valued function, t means the time variable andx runs in a (bounded) domain � of Euclidean space R

4. The coefficients of L aresupposed to depend at least continuously on t, x and they take value in thequaternion algebra. We are interested in finding a solution w(t, x) of (1) whichsatisfies the initial condition

wð0, xÞ ¼ ’ðxÞ: ð2Þ

In [1] Lewy showed that there exists linear first order differential equations withinfinity differentiable coefficients not having any solutions. On the other hand, the

*Corresponding author. Email: hungnq-fami@mail.hut.edu.vn

ISSN 1747–6933 print/ISSN 1747–6941 online

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DOI: 10.1080/17476930903276217

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classical Cauchy–Kovalevskaya Theorem can be proved by considering the

corresponding integro–differential operator in a suitably defined Banach space [2].

By this way, in [3] Son and Tutschke also formulated sufficient and necessary

conditions for associated pairs in complex sense. In this article, the space of

holomorphic functions is replaced by the space of regular functions in the sense of

quaternionic analysis.In order to investigate for which operators the initial value problem (IVP) can be

solved with arbitrary regular initial functions, one has to determine operators (1)

which are associated with the Cauchy–Riemann operator D of quaternionic analysis.

2. Preliminaries and notations

We denote by e0¼ 1, e1¼ i, e2¼ j, e3¼ k where i, j, k are the units of the (real)

quaternion algebra (H).A function f defined in the bounded domain ��R

4 and takes values in H is

a map

f : �! H

and thus f can be represented in a form

f ¼X3j¼0

ej fj ðxÞ

where fj(x) are the real valued functions of x¼ (x0, x1, x2, x3) 2 �. The properties,

such as continuity, differentiability and integrability, which are ascribed to the

function f have to be fulfilled by all components fj.Denoted by CH(�), CK

Hð�Þ, . . . the corresponding space of continuous or k-time

continuously differentiable functions taking values in H.Now let us introduce the Cauchy–Riemann operator as

D ¼X3k¼0

ek@

@xk:

The Cauchy–Riemann operator can act to f (on the left side) as follows:

Df ¼X3k, j¼0

ekej@fj@xk

:

The function f 2C1Hð�Þ is said to be (left) regular if Df¼ 0 in �. We denote by

R(�; H) the set of all regular functions in �.For other definitions concerning with quaternions and regular functions we refer

the reader to [4,5].

Remark 1 (See [4,5]) If u, v2C1Hð�Þ then

DðuvÞ ¼ ðDuÞvþ uðDvÞ þ 2X3k¼1

ekuk@v

@x0�X3j¼1

uj@v

@xj

" #:

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Definition 2.1 (See [6]) Let L be the first order differential operator depending on t,x, w and on the first order derivatives: @w=@xj, while l is a differential operator withrespect to the space variables xj whose coefficients do not depend on time t. Then L issaid to be ‘associated ’ with l if L transforms solutions of lw¼ 0 into solutions of the

same equation for fixedly chosen t, i.e. from lw¼ 0 it follows l[Lw]¼ 0.

3. Sufficient conditions for associated pairs

According to Definition 2.1 one gets that the operator L is associated with theoperator D if Dw¼ 0 implies D(Lw)¼ 0. Sufficient conditions for associated pairscan be obtained in the following way (see [7,8]):

. Note first that D(Lw)¼ 0 contains not only the components wj, j¼ 0, 1, 2, 3,of w, but D(Lw)¼ 0 also contains the first and second order derivatives withrespect to x0, x1, x2, x3. Since w is supposed to be regular, derivatives withrespect to x0 can be replaced by derivatives with respect to the othervariables (using the Cauchy–Riemann equation Dw¼ 0).

. Therefore in the expression for D(Lw) only the first order derivatives remainwith respect to the three variables x1, x2, x3.

. Analogously, in D(Lw)¼ 0 (for each component) only six second orderderivatives remain with respect to x1, x2, x3.

. Equating the coefficients of the obtained expression for D(Lw)¼ 0 (becausew is an arbitrary regular function), one gets sufficient conditions underwhich L is associated with the Cauchy–Riemann operator D.

The described procedure is a possible way for getting the desired sufficientconditions. However, in this article another method will be used which is based onfour lemmas. Using these lemmas, we shall prove the following Theorem 3.1 (the

lemmas will be formulated and proved after the formulation of Theorem 3.1):

THEOREM 3.1 Suppose that Að j Þðt,xÞ 2C2Hð�Þ; B(k)(t, x), Cðt, xÞ 2C1

Hð�Þ ( j, k¼ 0, 1,

2, 3) for each t 2 [0; T ]. The operator L (see (1)) is associated with the operator D if thefollowing conditions are satisfied:

(I) B(0)(t, x), C(t, x) 2 R(�; H) for each t 2 [0; T ].(II)

Bð0Þ0 ¼ B

ð1Þ1 ¼ B

ð2Þ2 ¼ B

ð3Þ3 ,

Bð0Þ1 ¼ �B

ð1Þ0 ¼ B

ð2Þ3 ¼ �B

ð3Þ2 ,

Bð0Þ2 ¼ �B

ð1Þ3 ¼ �B

ð2Þ0 ¼ B

ð3Þ1 ,

Bð0Þ3 ¼ B

ð1Þ2 ¼ �B

ð2Þ1 ¼ �B

ð3Þ0 :

(III)

Að0Þ0 ¼ �A

ð1Þ1 ¼ �A

ð2Þ2 ¼ �A

ð3Þ3 ,

Að0Þ1 ¼ A

ð1Þ0 ¼ A

ð2Þ3 ¼ �A

ð3Þ2 ,

Að0Þ2 ¼ �A

ð1Þ3 ¼ A

ð2Þ0 ¼ A

ð3Þ1 ,

Að0Þ3 ¼ A

ð1Þ2 ¼ �A

ð2Þ1 ¼ A

ð3Þ0 :

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(IV)@Að0Þ0@x0¼@Að0Þ1@x1¼@Að0Þ2@x2¼@Að0Þ3@x3

,

@Að0Þ0@x1¼ �

@Að0Þ1@x0¼@Að0Þ2@x3� B

ð0Þ1 ¼ �

@Að0Þ3@x2� B

ð0Þ1 ,

@Að0Þ0@x2¼ �

@Að0Þ1@x3� B

ð0Þ2 ¼ �

@Að0Þ2@x0¼@Að0Þ3@x1� B

ð0Þ2 ,

@Að0Þ0@x3¼@Að0Þ1@x2� B

ð0Þ3 ¼ �

@Að0Þ2@x1� B

ð0Þ3 ¼

@Að0Þ3@x0

:

One can prove this theorem by using the following four lemmas:

LEMMA 3.2 If the conditions (III) of Theorem 3.1 are satisfied, then one has

(I)

Að0Þe0þ2X3k¼1

ekAð0Þk ¼Að1Þe1�2A

ð1Þ1 e0¼Að2Þe2�2A

ð2Þ2 e0¼Að3Þe3�2A

ð3Þ3 e0:

(II)

Að0Þe1 þ Að1Þe0 þ 2P3m¼1

emAð1Þm � 2A

ð0Þ1 e0

¼ Að0Þe2 þ Að2Þe0 þ 2P3m¼1

emAð2Þm � 2A

ð0Þ2 e0

¼ Að0Þe3 þ Að3Þe0 þ 2P3m¼1

emAð3Þm � 2A

ð0Þ3 e0

¼ Að1Þe2 þ Að2Þe1 � 2Að1Þ2 e0 � 2A

ð2Þ1 e0

¼ Að1Þe3 þ Að3Þe1 � 2Að1Þ3 e0 � 2A

ð3Þ1 e0

¼ Að2Þe3 þ Að3Þe2 � 2Að2Þ3 e0 � 2A

ð3Þ2 e0 ¼ 0:

Proof The proof of Lemma 3.2 is obtained by straight computation. g

LEMMA 3.3 If the conditions (II) of Theorem 3.1 are satisfied, then one has

(I)½DAð0Þe0 þ Bð0Þ�e0 ¼ �½DAð0Þe1 þ Bð1Þ�e1

¼ �½DAð0Þe2 þ Bð2Þ�e2 ¼ �½DAð0Þe3 þ Bð3Þ�e3:

(II)½DAð1Þe0 þ Bð0Þe1 � 2B

ð0Þ1 e0�e0 ¼ �½DAð1Þe1 þ Bð1Þe1 � 2B

ð1Þ1 e0�e1

¼ �½DAð1Þe2 þ Bð2Þe1 � 2Bð2Þ1 e0�e2 ¼ �½DAð1Þe3 þ Bð3Þe1 � 2B

ð3Þ1 e0�e3:

(III)½DAð2Þe0 þ Bð0Þe2 � 2B

ð0Þ2 e0� ¼ �½DAð2Þe1 þ Bð1Þe2 � 2B

ð1Þ2 e0�e1

¼ �½DAð2Þe2 þ Bð2Þe2 � 2Bð2Þ2 e0�e2 ¼ �½DAð2Þe3 þ Bð3Þe2 � 2B

ð3Þ2 e0�e3:

(IV)½DAð3Þe0 þ Bð0Þe3 � 2B

ð0Þ3 e0�e0 ¼ �½DAð3Þe1 þ Bð1Þe3 � 2B

ð1Þ3 e0�e1

¼ �½DAð3Þe2 þ Bð2Þe3 � 2Bð2Þ3 e0�e2 ¼ �½DAð3Þe3 þ Bð3Þe3 � 2B

ð3Þ3 e0�e3:

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Proof (I) We have

½DAð0Þe0 þ Bð0Þ�e0 ¼ DAð0Þe0 þ Bð0Þe0 ð3Þ

�½DAð0Þe1 þ Bð1Þ�e1 ¼ DAð0Þe0 � Bð1Þe1: ð4Þ

From hypotheses of Lemma 3.3, we see B(0)e0¼�B(1)e1.

Then (3) and (4) imply ½DAð0Þe0 þ Bð0Þ�e0 ¼ �½DAð0Þe1 þ Bð1Þ�e1.Similarly, [DA(0)e0þB

(0)]e0¼�[DA(0)e2þB(2)]e2¼�[DA(0)e3þB

(3)]e3.Hence

½DAð0Þe0 þ Bð0Þ�e0 ¼ �½DAð0Þe1 þ Bð1Þ�e1 ¼ �½DAð0Þe2 þ Bð2Þ�e2 ¼ �½DAð0Þe3 þ Bð3Þ�e3:

The proofs of (II), (III) and (IV) are similar. g

LEMMA 3.4 Suppose B(0) is regular, and conditions (II) of Theorem 3.1 are satisfied,

then B(1),B(2),B(3) are also regular functions for each t 2 [0; T ].

Proof Starting from

DBð1Þ ¼ 0,X3i¼0

ei@

@xi

! X3j¼0

ejBð1Þj

!¼X3i¼0

X3j¼0

eiej@Bð1Þj@xi

¼@Bð1Þ0@x0�@Bð1Þ1@x1�@Bð1Þ2@x2�@Bð1Þ3@x3

!e0

þ@Bð1Þ0@x1þ@Bð1Þ1@x0�@Bð1Þ2@x3þ@Bð1Þ3@x2

!e1

þ@Bð1Þ0@x2þ@Bð1Þ1@x3þ@Bð1Þ2@x0�@Bð1Þ3@x1

!e2

þ@Bð1Þ0@x3�@Bð1Þ1@x2þ@Bð1Þ2@x1þ@Bð1Þ3@x0

!e3:

ð5Þ

Substituting conditions (II) of Theorem 3.1 into (5), we have

DBð1Þ ¼ �@Bð0Þ1@x0�@Bð0Þ0@x1�@Bð0Þ3@x2þ@Bð0Þ2@x3

!e0

þ �@Bð0Þ1@x1þ@Bð0Þ0@x0�@Bð0Þ3@x3�@Bð0Þ2@x2

!e1

þ �@Bð0Þ1@x2þ@Bð0Þ0@x3þ@Bð0Þ3@x0þ@Bð0Þ2@x1

!e2

þ �@Bð0Þ1@x3�@Bð0Þ0@x2þ@Bð0Þ3@x1�@Bð0Þ2@x0

!e3:

ð6Þ

On the other hand, B(0) is regular, therefore (6) leads to DB(1)¼ 0.

Analogously, we have DB(2)¼DB(3)

¼ 0 for each t 2 [0; T ]. g

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LEMMA 3.5 Suppose the conditions (III) and (IV) of Theorem 3.1 are satisfied, thenone has then

½DAð0Þe0 þ Bð0Þ�e0 ¼ �½DAð1Þe0 þ Bð0Þe1 � 2Bð0Þ1 e0�e1

¼ �½DAð2Þe0 þ Bð0Þe2 � 2Bð0Þ2 e0�e2

¼ �½DAð3Þe0 þ Bð0Þe3 � 2Bð0Þ3 e0�e3:

Proof We have

�DAð0Þe0 þ Bð0Þ

�e0 ¼

X3j¼0

X3k¼0

ejek@Að0Þk@xjþ B

ð0Þ0 e0 þ B

ð0Þ1 e1 þ B

ð0Þ2 e2 þ B

ð0Þ3 e3 ð7Þ

and

� DAð1Þe0 þ Bð0Þe1 � 2Bð0Þ1 e0

h ie1

¼ �X3j¼0

X3k¼0

ejek@Að1Þk@xj

!e1 þ B

ð0Þ0 e0 þ B

ð0Þ1 e1 � B

ð0Þ2 e2 � B

ð0Þ3 e3: ð8Þ

It is clear that the proof of

DAð0Þe0 þ Bð0Þ� �

e0 ¼ � DAð1Þe0 þ Bð0Þe1 � 2Bð0Þ1 e0

h ie1

leads to the proof of

X3j¼0

X3k¼0

ejek@Að0Þk@xjþ 2B

ð0Þ2 e2 þ 2B

ð0Þ3 e3 ¼ �

X3j¼0

X3k¼0

ejek@Að1Þk@xj

!e1:

We see

X3j¼0

X3k¼0

ejek@Að0Þk@xjþ 2B

ð0Þ2 e2 þ 2B

ð0Þ3 e3

¼@Að0Þ0@x0�@Að0Þ1@x1�@Að0Þ2@x2�@Að0Þ3@x3

!e0 þ

@Að0Þ1@x0þ@Að0Þ0@x1þ@Að0Þ3@x2�@Að0Þ2@x3

!e1

þ@Að0Þ2@x0�@Að0Þ3@x1þ@Að0Þ0@x2þ@Að0Þ1@x3þ 2B

ð0Þ2 e0

!e2

þ@Að0Þ3@x0þ@Að0Þ2@x1�@Að0Þ1@x2þ@Að0Þ0@x3þ 2B

ð0Þ3 e0

!e3, ð9Þ

and

�X3j¼0

X3k¼0

ejek@Að1Þk@xj

!e1 ¼

@Að1Þ1@x0þ@Að1Þ0@x1þ@Að1Þ3@x2�@Að1Þ2@x3

!e0

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�@Að1Þ0@x0�@Að1Þ1@x1�@Að1Þ2@x2�@Að1Þ3@x3

!e1 �

@Að1Þ3@x0þ@Að1Þ2@x1�@Að1Þ1@x2þ@Að1Þ0@x3

!e2

þ@Að1Þ2@x0�@Að1Þ3@x1þ@Að1Þ0@x2þ@Að1Þ1@x3

!e3: ð10Þ

From (III), (IV) of Theorem 3.1, (9) and (10) follows that

X3j¼0

X3k¼0

ejek@Að0Þk@xjþ 2B

ð0Þ2 e2 þ 2B

ð0Þ3 e3 ¼ �

X3j¼0

X3k¼0

ejek@Að1Þk@xj

!e1: ð11Þ

Hence ½DAð0Þe0 þ Bð0Þ�e0 ¼ �½DAð1Þe0 þ Bð0Þe1 � 2Bð0Þ1 e0�e1.

Completely analogous we see

½DAð0Þe0þBð0Þ�e0 ¼�½DAð2Þe0þBð0Þe2� 2Bð0Þ2 e0�e2 ¼�½DAð3Þe0þBð0Þe3� 2B

ð0Þ2 e0�e3:

g

Using four Lemmas, Theorem 3.1 can be proved by straightforward calculation,for details see the dissertation thesis in the Library of Hanoi University ofTechnology (‘Initial value problems in quaternionic analysis’, N.Q. Hung).

Notes: Theorem 3.1 only constructs sufficient conditions, but not necessaryconditions for associated pairs. This can be seen by the following two examples:

Let �(x0) be an arbitrary real-valued function depending on x0.

Example 1 A( j)¼ �(x0)ej, B

( j)� 0( j¼ 0, 1, 2, 3), C� 0.

Example 2 A(0)¼ �(x0)e0, A

(i)¼��(x0) ei, (i¼ 1, 2, 3), B( j)

� 0( j¼ 0, 1, 2, 3), C� 0.

Then L is associated with the operator D. It is clear that the coefficients in Example 2satisfy Theorem 3.1, but the coefficients in Example 1 do not.

4. Solution of the IVPs with regular initial functions

In order to solve the IVPs in the holomorphic case, one can apply the method of

associated space (see [6]). The basic idea of this method is as follows:First, we write the IVP as integro-differential equation

uðt, xÞ ¼ ’ðxÞ þ

Z t

0

Fð�, x, u, @juÞd�

and thus the solutions of the IVP are fixed points of the operators

Uðt, xÞ ¼ ’ðxÞ þ

Z t

0

Fð�, x, u, @juÞd�: ð12Þ

In order to construct fixed points of this operator, we have to estimate the integro-

differential operator on the right-hand side of (12). This is fairly difficult because theintegrand contains derivatives with respect to the space like variables xj. But theestimation is possible in case one uses the method of associated spaces.

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One looks for an associated function space having two properties:

. the operator maps the space into itself;

. for elements of the associated space one has an interior estimate, that is, the

spacelike variables of the associated space can be estimated by the norm of

the elements.

In the case of classical Cauchy–Kovalevskaya theorem the associated space is the

space of holomorphic functions. But, in this article the associated space is the space

of regular functions in the sense of quaternionic analysis. Notice that the

components of regular functions are harmonic. Therefore, the interior estimate

follows from the Poisson formula (see [9,10]). Consequently, the following theorem

has been proved:

THEOREM 4.1 Suppose L is associated with the Cauchy–Riemann operator of the

quanternionic analysis. Then each IVP (1) and (2) is solvable in case ’ is an arbitrary

regular initial function. The solution w(t, x) is regular for each t.

References

[1] H. Lewy, An example of a smooth linear partial differential equation without solution,

Ann. Math. 66 (1957), p. 155–158.[2] W. Walter, An elementary proof of the Cauchy–Kovalevskaya theorem, Amer. Math.

Monthly 92 (1985), p. 115–125.[3] L.H. Son and W. Tutschke, First order differential operators associated to the Cauchy–

Riemann operator in the plane, Complex Variable 48 (2003), p. 797–801.[4] F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitman Advanced Publishing

Program, Boston, 1982.[5] Klaus Gurlebeck and Wolfgang Sprossig, Quaternionic Analysis and Elliptic Boundary

Value Problems, Akademic–Verlag, Berlin, 1989.[6] W. Tutschke, Solution of Initial Value Problems in Classes of Generalized Analytic

Functions, Teubner Leipzig/Springer Verlag, Leipzig, 1989.[7] N. van Thanh, Differential operators transforming regular function of quaternionic analysis

into themselves, in Method of Complex and Clifford Analysis, SAS International

Publications, Delhi, 2006, p. 363–367.

[8] N. van Thanh, First order differential operators associated to the Cauchy–Riemann

operators in a Clifford algebra, Proceedings of the 14th ICFIDCAA, Hue, Vietnam, 2006.

[9] L.H. Son and Q.H. Nguyen, The initial value problems in Clifford and quaternion analysis,

Proceedings of the 15th ICFIDCAA, Vol. 3, Osaka Municipal Universities Press, Osaka,

2008, p. 317–323.[10] N.Q. Hung and N.C. Luong, First order differential operators associated to the Dirac

operator in quaternionic analysis, Proceedings of 2004 International Conference on Applied

Mathematics, SAS International Publications, Delhi, 2006, p. 369–378.

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