the fundamental solution for shallow circular cylindrical ...dma/pdf/fundamental solution 2000...

The fundamental solution for shallow circular cylindrical shellsPart I: derivations

Goong Chen a,*,1, Matthew P. Coleman b, Daowei Ma c, Philip J. Morris d,Puhong You a

a Department of Mathematics, Texas A&M University, College Station, TX 77843, USAb Department of Mathematics and Computer Science, Fair®eld University, Fair®eld, CT 06430, USA

c Department of Mathematical Science, Wichita State University, Wichita KS 67226, USAd Department of Aerospace Engineering, Pennsylvania State University, University Park, PA 16802, USA

Received 2 December 1996; accepted 7 July 1999

(Communicated by I. STAKGOLD)

Abstract

The equations which model the elastostatic shallow circular cylindrical shell (see, e.g., [6,15,23,29])constitute an important elliptic partial di�erential equation (PDE) system in the study of shell structures.When the system is subjected to a concentrated point load, the response is described by a fundamentalsolution of the PDE system. We have found some mathematical inconsistencies in the existing literature.Therefore, in this paper, we discuss these errors, then we use partial fractions and Fourier transformtechniques to determine the fundamental solution. Explicit expressions in terms of special functions andconvolution integrals are derived and simpli®ed so that the formulas are suitable for algorithmic evaluationand for application elsewhere. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Shallow circular cylindrical shell; Fundamental solution; Fourier transforms

1. Introduction

Shells are important engineering structures. Their mathematical models are commonly de-scribed by systems of coupled partial di�erential equations (PDEs) of higher order. The funda-mental solution is the response of the PDE system when there is a concentrated point load or

International Journal of Engineering Science 38 (2000) 1235±1257www.elsevier.com/locate/ijengsci

*Corresponding author. Tel.: +1-409-862-4190; fax: +1-409-845-7336.

E-mail addresses: [email protected] (G. Chen), [email protected]®eld.edu (M.P. Coleman), dma@cs.

twsu.edu (D. Ma), [email protected] (P.J. Morris), [email protected] (P. You).1 Supported in part by NSF grants DMS 94-04380 and 96-10076 and NATO Grant CRG 940369.

0020-7225/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0020-7225(99)00078-6

excitation to the system. The fundamental solution is known to be extremely useful in boundaryelement computations of structures, as far as the authors' interest is concerned (see, for e.g., [3, ch.8]; [12,19,30]).

Many models of both shallow and nonshallow thin shells of conical, cylindrical and sphericalshape can be found in the literature. The obvious advantage in using a shallow shell model, whenapplicable, is that a shallow model is invariably simpler mathematically than the correspondingnonshallow model ± in fact, in many shallow shell models, the coe�cients are constant. Further, ashallow shell model seems advantageous in the determination of fundamental solutions forproblems involving concentrated loads since, in these problems, the dominant part of the stressdistribution in the neighborhood of the singular point is the same as for the correspondingproblem for the plate [4,23].

Therefore, the problem of ®nding singular solutions to shallow shell equations is signi®cant tovarious areas of engineering, as is witnessed by the many papers of the 1960s and early 1970swhich treated the topic (see, e.g., [5,7±10,14,16,17,23±27]), along with some more recent ones, suchas [18,32].

Among the various types of shells, the cylindrical shell seems to be the most intensively studiedshell structure, due to its technological signi®cance and mathematical elegance. It is the limitedobjective of this paper to ®nd explicit representations of fundamental solutions for shallow cir-cular cylindrical shells. We must mention that, although two papers claim to have solved thisproblem [23,24], we have found an inconsistency in the model which leads to what we believe is anincorrect expression for the fundamental solution. A critical examination will be given below.

In this paper, we will use partial fraction expansions and Fourier transforms to derive ourfundamental solution. In the sequel, Part II, we delineate algorithms which realize the funda-mental solution in numerics. The ultimate goal we hope to reach is to be able to actually use thefundamental solution in the boundary element computation of shell structures.

The organization of this paper is as follows: in Section 2, we examine the work in [23,24]. InSection 3 we describe the model we have chosen for a shallow circular cylindrical shell, andcompare it to other models. In Section 4, we use the Fourier transform and partial fraction ex-pansions to ®nd the fundamental solution.

We have tried to simplify the fundamental solution, as much as possible, into a form suitablefor computer implementation. It will serve as the basis of our boundary element method for-mulations and computations for shallow circular cylindrical shells in forthcoming papers. We alsohope that the explicit construction here will be useful to other researchers in structural mechanics.We have veri®ed that our fundamental solution has all the requisite properties. The calculationsare too lengthy to be included in this paper, but are available from the authors.

2. A critique of the complex model

Finding the fundamental solution for various shell models has been of great interest to re-searchers in structural mechanics, as is evidenced by the work published in [5,8±10,14,16±18,22,27,31], to mention just a few. Among these, the papers by Sanders [23] and Sanders andSimmonds [24] merit special attention as they seem to be of the most relevance to our work here.However, we have found that the shallow circular cylindrical shell in complex form subject to

1236 G. Chen et al. / International Journal of Engineering Science 38 (2000) 1235±1257

concentrated forces, as adopted in [23,24], is mathematically inconsistent. We shall thereforerigorously derive the fundamental solution in the consistent model we have chosen.

Our critique of [23,24] is articulated below, based on our careful reading of their papers.

2.1. Inconsistency of the complex model in [22±24]

Let us clarify what we believe are the inconsistencies in [23,24]. First, let us list the equations ofthe (real) model for the shallow circular cylindrical shell given in [23, (1)±(9), p. 362]. Using thesame notation therein, we consider the shallow shell with middle surface z � ÿ1

2x2

2 in Cartesianx1 ÿ x2 ÿ z space. The fundamental equations of shallow shell theory then become:

Equilibrium equations:

oN11

ox1

� oN12

ox2

� ÿp1; �2:1�oN12

ox1

� oN22

ox2

� ÿp2; �2:2�

lN22 ÿ o2M11

ox21

�� 2

o2M12

ox1ox2

� o2M22

ox22

�� p: �2:3�

Strain displacement relations:

E11 � ouox1

; E22 � ovox2

� lw; E12 � 1

2

ouox2

�� ov

ox1

�; �2:4�

K11 � ÿ o2wox2

1

; K22 � ÿ o2wox2

2

; K12 � ÿ o2wox1ox2

: �2:5�

Constitutive relations:

E11 � N11 ÿ mN22; E22 � N22 ÿ mN11; E12 � �1� m�N12; �2:6�M11 � K11 � mK22; M22 � K22 � mK11; M12 � �1ÿ m�K12: �2:7�

Compatibility conditions:

oK22

ox1

ÿ oK12

ox2

� 0; �2:8�oK12

ox1

ÿ oK11

ox2

� 0; �2:9�

lK11 � o2E11

ox21

ÿ 2o2E12

ox1ox2

� o2E22

ox22

� 0: �2:10�

In the above, Nij;Mij;Eij and Kij represent the forces, moments, strains and curvatures, respec-tively; m is the Poisson ratio; l � �L2

��12�1ÿ m2�p �=�ah�, where a is the radius of the cylindrical

G. Chen et al. / International Journal of Engineering Science 38 (2000) 1235±1257 1237

shell; L is a ``reference length'', h the shell's constant thickness; u; v and w are, respectively, the x1-,x2- and z-direction displacements of the shell; and p1; p2 and p are, respectively, the forces in the x1-,x2- and z-directions.

In [23, (10)±(12), p. 362], Sanders gave, in addition, stress-stress-function relations, and statedthat his Eqs. (10) and (11) are not to be used in loaded regions. Since these relations are notactually needed by us here, we leave them out.

We ®rst remark that the above model is consistent. In fact, we will see below that it is equivalentto the model which yields Eqs. (3.1)±(3.10) and, further, that it is a limiting case of the (real) modeldeveloped by Novozhilov [21]. We also note that, if the loads are su�ciently smooth, then so arethe other expressions which appear in (2.1)±(2.3) ± speci®cally, the Nij will also be su�cientlysmooth.

Next, Sanders [23] converts the above into a complex model, by allowing all of the quantities tobe complex-valued, and by adjoining to Eqs. (2.1)±(2.7) the conditions

N11 � iK22; N22 � iK11; N12 � iK12: �2:11�

These, in conjunction with (2.5), require that we have

N11 � ÿio2wox2

2

; N22 � ÿio2wox2

1

; N12 � io2w

ox1ox2

: �2:12�

Now, inserting the expressions (2.12) into (2.1) and (2.2) yield, respectively,

ÿiwx2x2x1� iwx1x2x2

� ÿp1; �2:13�

and

iwx1x2x1ÿ iwx1x1x2

� ÿp2: �2:14�

Now, the smoothness of the Nij implies the smoothness of w which, in turn, means that the orderof di�erentiation on the left sides of (2.13) and (2.14) does not matter. Therefore, (2.13) and(2.14) yield, respectively,

0 � ÿp1; 0 � ÿp2; �2:15�

each of which is a contradiction, of course, unless the corresponding force is zero. If the forces p1

and p2 in (2.13) and (2.14) are not pointwise de®ned smooth functions (in the classical sense), say,concentrated forces such as delta functions and their derivatives, then (2.13) and (2.14) must beinterpreted in the sense of distributions. Since delta functions and/or their derivatives can be``approximated'' by sequences of smooth functions (such as a ``d-sequence''), we again conclude(2.15). This also follows simply from applying the theory of generalized functions [11,13].

The conclusion is that Eqs. (2.1) and (2.5), along with the additional requirements (2.11), forman inconsistent system (and, similarly, for (2.2), (2.5) and (2.11)). Further, since there obviously isno inconsistency in the system (2.1) and (2.5) (similarly, in (2.2) and (2.5)), it is the introduction ofthe additional set of requirements (2.11) which leads to the inconsistency in Sanders' complex model.


(We mention here that the smoothness of w causes similar problems in Sanders' complex com-patibility conditions [23, Eq. (8)0].) We also note here (and elaborate below) that Novozhilov [21]does not use (2.11) in the derivation of his complex model, but, instead, takes

N11 � ÿi~cK22 � N �11; N22 � ÿi~cK11 � N �22; N12 � i~cK12 � N �12 �~c � constant�; �2:16�

where the extra degrees of freedom N �11;N�22 and N �12 are necessary to circumvent the di�culties in

Sanders' model. See more discussion in Section 3, particularly (3.17), later.Finally, we mention that Sanders' equations (2.1), (2.2) and (2.5), along with the relation (2.11)

remain the same regardless of the geometry of the shell under consideration. Therefore, the abovearguments show that Sanders' complex model is inconsistent not only in the circular cylindricalcase, but in every case to which his model applies.

2.2. Inconsistency of the shallow cylindrical shell equations in complex form

Let us now address expressly the shallow circular cylindrical PDE system. The PDEs given bySanders ([23, (42) and (43), p. 365]) from his complex model are

D2wÿ ilwx1x1� p � im�p1;x1

� p2;x2�; �2:17�

ux1� i�1� m�wx1x1

ÿ iDw; �2:18�vx2� i�1� m�wx2x2

ÿ iDwÿ lw; �2:19�ux2� vx1

� 2i�1� m�wx1x2: �2:20�

Now, if we take o2=ox22 of (2.18) and o2=ox2

1 of (2.19), add these results, and compare witho2=ox1ox2 of (2.20), we see that w must satisfy

D2wÿ ilwx1x1� 0 �D2 � the biharmonic operator�; �2:21�

which contradicts (2.17) except when

p � im�p1;x1� p2;x2

� � 0: �2:22�

This contradiction, in fact, is a direct consequence of the inconsistency of Sanders' complex modelwhich was exhibited in Section 2.1. In the case where p1; p2 and p involve delta functions, (2.22)holds pointwise except at the singular points, but it does not hold in the sense of distributions;hence it is still inconsistent. In determining the Fourier transform of their PDEs in [23,24], theauthors have ignored Eq. (2.20), which might explain why the inconsistency issue was not dis-covered earlier. The model we shall use is consistent and the Fourier transforms can be performedrigorously to derive the fundamental solution.

3. Choice of model

In this paper we treat the model which appears in Kraus [15, (6.8)±(6.10), pp. 200±201]. Thismodel is equivalent to Sanders' equations (2.1)±(2.3) and it is easy to show that it also satis®es


conditions (2.8)±(2.10). Moreover, this model is also equivalent to that given by Timoshenko andWoinowsky-Krieger [29, (300), (302) and (304), pp. 512±514], after including the simpli®cationsdue to Vlasov (see [29, the paragraph following (303), p. 513]). Finally, these models are alsoequivalent to the model of a ``shallow cylindrical shell with small de¯ections'' given by Donnell [6,ch. 6]. Although the compatibility conditions are not expressed explicitly in any of these threeworks, it is easy to show that all these models satisfy Eqs. (2.8)±(2.10).

The (real) model for a circular cylindrical shell treated by Novozhilov [21], in notation con-sistent with Section 2.1, consists of:


oN11

ox1

� oN12

ox2

� ÿp1; �3:1�oN12

ox1

� oN22

ox2

� 1

aoM22

ox2

�� 2

oM12

ox1

�� ÿp2; �3:2�

1

aN22 ÿ o2M11

ox21

�� 2

o2M12

ox1ox2

� o2M22

ox22

�� p: �3:3�

Strain±displacement relations:

E11 � ouox1

; E22 � ovox2

� 1

aw; E12 � 1

2

ouox2

�� ov

ox1

�; �3:4�

K11 � ÿ o2wox2

1

; K22 � ÿ o2wox2

2

� 1

aovox2


� 1

aovox1

: �3:5�

Constitutive relations:

E11 � 1

Eh�N11 ÿ mN22�; E22 � 1

Eh�N22 ÿ mN11�; E12 � 2�1� m�

EhN12; �3:6�

M11 � Eh3

12�1ÿ m2� �K11 � mK22�; M22 � Eh3

12�1ÿ m2� �K22 � mK11�; M12 � Eh3

12�1� m�K12: �3:7�

Compatibility conditions:

oK22

ox1

ÿ oK12

ox2

� 0; �3:8�oK11

ox2

ÿ oK12

ox1

� 1

aoE12

ox1

�ÿ oE11

ox2

�� 0; �3:9�

1

aK11 � o2E11

ox21

ÿ 2o2E12

ox1ox2

� o2E22

ox22

� 0; �3:10�

where, a is the shell's radius, m the Poisson ratio, h the constant shell thickness and E the modulusof elasticity.


On the other hand, the model given in [15], after the simpli®cations included therein, is:


oN11

ox1

� oN12

ox2

� ÿp1; �3:11�oN12

ox1

� oN22

ox2

� ÿp2; �3:12�1

aN22 ÿ o2M11

ox21

�� 2

o2M12

ox1ox2

� o2M22

ox22

�� p: �3:13�

Strain±displacement relations:

E11 � ouox1

; E22 � ovox2

� 1

aw; E12 � 1

2

ouox2

�� ov

ox1

�; �3:14�

K11 � ÿ o2wox2

1

; K22 � ÿ o2wox2

2


: �3:15�

Constitutive relation:

Same as �3:6� and �3:7�: �3:16�

(Kraus' model also satis®es the compatibility conditions (3.8)±(3.10), even though he does not listthem.) We see that Kraus' model is a ``special case'' or ``simpli®ed version'' of Novozhilov's, inthat Kraus has neglected certain terms which appear in (3.2) and (3.5). Further, if we make thevariables dimensionless in Kraus' model (as does [23, p. 362] and [24, p. 367]), and choose the``reference length'' and ``reference radius of curvature'' appropriately, we arrive at Sanders' (real)system of equations. Therefore, the systems of Kraus and of Sanders are equivalent.

Finally, it is easy to see that Novozhilov's model is a special case of the more general Morley±Koiter equations ([20, ch. 11, Section 2]).

We conclude with a discussion of Novozhilov's complex model [21]. Novozhilov proceeds byde®ning the complex expressions eNij and eKij as follows:

eN11 � N11 ÿ icEhK22; eK11 � K11 � i

cEh�N22 ÿ N �22�;

eN22 � N22 ÿ icEhK11; eK22 � K22 � i

cEh�N11 ÿ N �11�; �3:17�

eN12 � N12 � icEhK12; eK12 � K12 ÿ i

cEh�N12 ÿ N �12�;

where c � h=��12�1ÿ m2�p

, and where the quantities N �ij are introduced in order to ensure that themodel is consistent. Novozhilov shows that these quantities must be solutions of the corre-sponding equations of the membrane theory.


Note that the equations in (3.17) imply Eq. (2.16), where ~c � cEh. Finally, Eqs. (3.1)±(3.17),after some simpli®cation, lead to the complex PDEs

oeN11

ox1

� oeN12

ox2

� ÿp1;

oeN12

ox1

� oeN22

ox2

� ih

a��12�1ÿ m2�p o

ox2

�eN11 � eN22� � ÿp2; �3:18�

1

aeN22 ÿ i

h��12�1ÿ m2�p D�eN11 � eN22� � p � i

�1� m�h��12�1ÿ m2�p op1

ox1

�� op2

ox2

�;

where D � o2=ox21 � o2=ox2

2 is the Laplacian operator. We may then use (3.17), along with (3.4)±(3.7), to derive from (3.18) three PDEs in the six unknowns ~u; ~v; ~w, N �11;N

�22 and N �12, where the

complex displacements ~u; ~v and ~w are de®ned via (3.4)±(3.7) and (3.17).While this approach is consistent and, in fact, quite elegant, we believe that the introduction of

the three new unknowns N �ij minimizes the advantages of using the complex model to ®nd thefundamental solution.

4. Fundamental solution of the shallow circular cylindrical shell model and its Fourier transform

Using (3.14)±(3.16), we convert the equilibrium equations (3.11)±(3.13) into a system of PDEsin terms of the displacement variables u; v and w. Consistent with Kraus' notation, we rewrite xand s for x1 and x2, respectively, and obtain the resulting PDE system

uxx � 1� m2

vxs � 1ÿ m2

uss � ma

wx � ÿ 1ÿ m2

Ehp1;

1� m2

uxs � 1ÿ m2

vxx � vss � 1

aws � ÿ 1ÿ m2

Ehp2; �4:1�

h2

12D2w� 1

a1

aw

�� vs � mux

�� 1ÿ m2

Ehp:

Although the highest order of partial derivative appearing in (4.1) has only order four, byDonnell's procedures [2,10,15], we can decouple w from u and v and obtain

D4w� 12�1ÿ m2�a2h2

wxxxx � terms involving only p1; p2; p and their partial derivatives; �4:2�

which is actually a single PDE of order eight. Note that it is the lack of symmetry of the x and svariables on the left-hand side of (4.2) that has caused the lengthy derivations required in thispaper. (We note here that our system (4.1) is equivalent to the system of PDEs given in Tim-oshenko and Woinowski±Krieger [29, (304), p. 514].)

Our objective now is to determine the fundamental solution of the elastostatic shallow circularcylindrical shell. We let �U�x; s�; V �x; s�;W �x; s�� denote the fundamental solution, which satis®es


Uxx � 1� m2

Vxs � 1ÿ m2

Uss � ma

Wx � ÿk1d�x�d�s�;1� m

2Uxs � 1ÿ m

2Vxx � Vss � 1

aWs � ÿk2d�x�d�s�; �4:3�

h2

12D2W � 1

a1

aW

�� Vs � mUx

�� ÿk3d�x�d�s�;

namely �U ; V ;W � satis®es (4.1) with a concentrated point loading at �x; s� � �0; 0�, where in (4.3)d denotes the Dirac delta function in R, and k1; k2 and k3 are arbitrary nonnegative constants. TheFourier transform of a tempered distribution always exists [3,11,13] and can be written symbol-ically as

f̂ �a; b� �F�f ��a; b� �Z

R2

eÿi�ax�bs�f �x; s�dxds; �4:4�

where a and b are the (Fourier transformed) dual variables of x and s, respectively. Then theinversion formula gives

f �x; s� �Fÿ1�f̂ ��x; s� � 1

4p2

ZR2

ei�ax�bs�f̂ �a; b�dadb; �x; s� 2 R2: �4:5�

A useful list of Fourier transform and inversion formulas is compiled in Appendix A for theconvenience of the reader. We now perform the Fourier transform on (4.3). By using (A.3) and(A.6), after a slight simpli®cation we obtain

a2

�� 1ÿ m

2b2

�bU � 1� m2

� �abbV ÿ m

a�ia� bW � k1;

1� m2

abbU � 1ÿ m2

a2

�� b2

�bV ÿ 1

a�ib� bW � k2; �4:6�

ma�ia�bU � 1

a�ib�bV � h2

12�a2

�� b2�2 � 12

a2h2

� bW � ÿk3:

We use Cramer's rule to solve the linear algebraic system (4.6) and obtain the Fourier transforms

bU�a; b� � 12i

h2ama3 ÿ ab2

�a2 � b2�21

�a2 � b2�4 � l4a4�a1 � a2 � a3�

� 1

�a2 � b2�2 k1a2

�� 2k1

1ÿ mb2 ÿ 1� m

1ÿ mk2ab

�; �4:7�

bV �a; b� � 12i

h2a�2� m�a2b� b3

�a2 � b2�21

�a2 � b2�4 � l4a4�a1 � a2 � a3�

� 1� m1ÿ m

1

�a2 � b2�22

1� mk2a

2

�� 1ÿ m

1� mk2b

2 ÿ k1ab

�; �4:8�


bW �a; b� � 12

h2

1

�a2 � b2�4 � l4a4�a1 � a2 � a3�; �4:9�

where

l4 � 12�1ÿ m2�h2a2

;

a1 � ÿ k3�a2 � b2�2;a2 � ÿ i

ak1�ma3 ÿ ab2�; �4:10�

a3 � ÿ i

ak2��2� m�a2b� b3�:

There are no readily available Fourier inversion formulas for most of the terms on the right sideof (4.7)±(4.9). Inverse Fourier transforms can be performed on (4.7)±(4.9) only after their ex-pansion. We now proceed to do this. First, note that

1

�a2 � b2�4 � l4a4� ÿ 1

4l3

1

a3

X� xj

a2 � b2 ÿ lxja; �4:11�

where

x � e�p=4�i;X��

Xj�1;3;5;7

: �4:12�

From (4.9)±(4.11), we rewrite bW as

bW �a; b� � ÿ 3

h2l3�

(ÿ k3�

X� 1

a3

xj�a2 � b2�2a2 � b2 ÿ lxja

ÿ i

ak1

X� 1

a3

xj�ma3 ÿ ab2�a2 � b2 ÿ lxja

ÿ i

ak2

X� 1

a3

xj��2� m�a2b� b3�a2 � b2 ÿ lxja

)

� ÿ 3

h2l3

�ÿ k3 A1 ÿ i

ak1 A2 ÿ i

ak2 A3

�: �4:13�

Next, note that

�a2 � b2�2 � �a2 � b2�2 ÿ �lxja�2 � l2x2ja2

� �a2 � b2 ÿ lxja��a2 � b2 � lxja� � l2x2ja2

� D�a; b; j�D�a;b; 4� j� � l2x2ja2; �4:14�

where

D�a;b; j� � a2 � b2 ÿ lxja: �4:15�


Utilizing (4.14) in A1 as de®ned through (4.13), we obtain

A1 �X� xj

a3�a2 � b2 � lxja� �

X�l2x3j 1

aD�a; b; j��X�

l2x3j 1

aD�a;b; j� ; �4:16�

where we have used the ®rst two of the following equalities inX�xj � 0;

X�x2j � 0;

X�x3j � 0: �4:17�

We will be using (4.17) still a few more times in the subsequent calculations.Similarly, for A2 and A3, from (4.13) we have

A2 �X� xj

a3

�ÿa��a2 � b2 ÿ lxja� � �1� m�a3 ÿ lxja2

D�a; b; j�� 1� m�

X�xj 1

D�a;b; j� ÿX�

lx2j � 1

aD�a;b; j� ; �4:18�

where the ®rst summation in (4.18) vanishes because of (4.17), as well as

A3 �X� xj

a3

�1� m�a2b� b�a2 � b2 ÿ lxja� � lxjabD�a; b; j�

�X��1� m�xj b

aD�a; b; j� �X� b

a2

lx2j

D�a; b; j� : �4:19�

We next expand the following into partial fractions:

1

aD�a; b; j� �1

b2

1

a

�� ÿa� � lxj

D�a; b; j��; �4:20�

1

a2 D�a; b; j� � �lxj� 1a

1

b4� 1

a2

1

b2� �ÿlxj� 1=b4

ÿ �a� l2x2j 1=b4

ÿ �ÿ 1=b2� �

D�a; b; j� : �4:21�

Using (4.20) and (4.21) in (4.16), (4.18) and (4.19), we obtain

A1 �X� a

b2

�ÿl2x3j�D�a;b; j� �

X� 1

b2

�ÿ1�jl3

D�a; b; j� ; �4:22�

A2 � �1� m�X� xj

D�a; b; j� ÿX� a

b2

�ÿlx2j�D�a; b; j� ÿ

X� 1

b2

l2x3j

D�a; b; j� ; �4:23�

A3 � �1� m�X� a

b�ÿxj�

D�a; b; j� � �1� m�X� 1

blx2j

D�a; b; j��X� a

b3

�ÿl2x3j�D�a;b; j� �

X� 1

b3

�ÿl3�D�a;b; j� ÿ

X� 1

blx2j

D�a; b; j� : �4:24�


Remark 4.1. Note that in (4.22),(4.23),(4.24), the partial fraction expansions for A1;A2 and A3, wehave allowed bk D�a;b; j�, k � 1; 2; 3, in the denominators on the right side, but not ak D�a; b; j�,j � 1; 2; 3. The reason is that Eqs. (A.21), (A.22), and (A.24),(A.25),(A.26),(A.27) yield conver-gent integrals on their right side. In contrast, (A.28) and (A.29) yield divergent integrals (for j � 5and 7) on their right side (see Appendix A). This observation is critical in deciding the proper formof partial fraction expansions involving two variables.

The partial fraction expansions for A1;A2 and A3 above have prepared bW �a;b� in (4.13) to be inthe right form for Fourier inversion. Using (A.17)±(A.27) in Appendix A (and by noting that F0j

and F1j are de®ned in (A.19) and (A.23)), and the abbreviated notation

C�j �lxj

2F0j�x; r� � F1j�x; r�

both here and subsequently, we obtain

Fÿ1�A1� �X� 1

2l2x3j

ZR

jsÿ rjCÿj dr�X� 1

2l3

ZR

jsÿ rj F0j�x;r�dr

�X� �

ÿ 1

2l2x3j

�ZR

jsÿ rjC�j dr �* x4j � ÿ1 for j � 1; 3; 5; 7� �4:25�

Fÿ1�A2� � �1� m�X�

xjF0j�x; s� ÿX� 1

2lx2j

� �ZR

jsÿ rjCÿj dr

�X� 1

2l2x3j

ZR

jsÿ rjF0j�x; r�dr

� �1� m�X�

xjF0j�x;r� �X� 1

2lx2j

� �ZR

jsÿ rjC�j dr; �4:26�

Fÿ1�A3�

� �1� m�X� �

ÿ i

2xj

�ZR

sgn�sÿ r�Cÿj dr� mX� i

2lx2j

� �ZR

sgn�sÿr�F0j�x;r�dr

�X� i

4l2x3j

� �ZR

�sÿ r�2 sgn�sÿr�Cÿj dr�X� i

4l3

� �ZR

�sÿr�2 sgn�sÿr�F0j�x;r�dr

� �1� m�X� i

2xj

� �ZR

sgn�sÿ r�C�j dr�X� �

ÿ i

2lx2j

�ZR

sgn �sÿr�F0j�x;r�dr

�X� �

ÿ i

4l2x3j

�ZR

�sÿ r�2 sgn �sÿr�C�j dr: �4:27�


Substituting (4.25)±(4.27) into (4.13), we obtain

W �x; s� �Fÿ1�W � � 3

h2l3k3F

ÿ1�A1��

� i

ak1F

ÿ1�A2� � i

ak2F

ÿ1�A3��

) �simplification� )

� 3

h2l3

�1� m�k1

aiX�

xjF0j�x; s�8<: � k2l

2a

X�x2j

ZR

sgn �sÿ r�F0j�x; r�dr

ÿ k2�1� m�2a

X�xj

ZR

sgn �sÿ r�C�j dr

�X� 1

2

�ÿ k3l

2x3j � ik1lx2j

a

�ZR

jsÿ rjC�j dr

�X� k2l2x3j

4a

ZR

�sÿ r�2 sgn �sÿ r�C�j dr

9=;: �4:28�

Therefore, the component W �x; s� of the fundamental solution has been determined. We must stilldetermine U�x; s� and V �x; s�, which require a little extra work because from (4.7)±(4.9), we seethat bU and bV are more complicated than bW . But the procedures are quite similar. We leave out thedetails and directly state the outcomes below. (The authors will be happy to provide technicaldetails to any interested reader upon request.) They are

U�x; s� � 3i

h2al3�1�� m�

X��k3x

j ÿ 2iaÿ1k1lÿ1�F0j�x; s�

�� i

ak1

� �X��1� m�2lÿ2xÿjCÿj

� i

ak2

� �X��1� m�2lÿ2xÿj

�ÿ sjlxj

8

s��x2 � s2p e�ilxj=2�xH �1�1

sjlxj

2

��x2 � s2

p� ��cf : �A:20� and �A:23� for sj and H �1�1 �

� i

ak2

� �X� i

2xj

ZR

sgn �sÿ r�F0j�x; r�dr

� lX� �

ÿ k3lx3j � i

ak1x

2j

��ÿ 1

2

�ZR

jsÿ rjF0j�x;r�dr

�X�

k3lx2j

�ÿ i

ak1x

j

��ÿ 1

2

�ZR

jsÿ rjCÿj dr

� i

ak2

� �X��ÿl2x3j�

�ÿ i

4

�ZR

�sÿ r�2 sgn �sÿ r�F0j�x;r�dr

� i

ak2

� �X��lx2j�

�ÿ i

4

�ZR

�sÿ r�2 sgn �sÿ r�Cÿj dr; �4:29�


V �x; s� � 3i

h2al3

i

ak2�m2

�ÿ 1�lÿ1

X�F0j�x; s�

�ÿ i

ak2�1� m�2

X�lÿ2xÿjCÿj

� i

ak1�1� m�2

X�lÿ2xÿj

�ÿ sjlxj

8

s��x2 � s2p e�ilxj=2�xH �1�1

sjlxj

2

��x2 � s2

p� ��X�

k3mlx2j

�� i

ak1x

j

�i

2

ZR

sgn �sÿ r�F0j�x;r�dr

�X��ÿk3�1� m�xj� i

2

� �ZR

sgn �sÿ r�Cÿj dr

�X� 2i

ak2mlx2j

�ÿ 1

2

�ZR

jsÿ rjF0j�x; r�dr�X� i

2ak2�1� 2m�xj

ZR

jsÿ rjCÿj dr

�X� �

ÿ k3l3 ÿ i

ak1l

2x3j

��ÿ i

4

�ZR

�sÿ r�2 sgn �sÿ r�F0j�x; r�dr

�X� �

ÿ k3l2x3j � i

ak1lx2j

��ÿ i

4

�ZR

�sÿ r�2 sgn �sÿ r�Cÿj dr

�X� �

ÿ i

ak2l

3

�1

12

ZR

jsÿ rj3F0j�x;r�dr

�X� �

ÿ ia

k2l2x3j

�� 1

12

ZR

jsÿ rj3Cÿj dr: �4:30�

Now, we have obtained the fundamental solution �U ; V ;W � as given in (4.29), (4.30) and (4.28),each of which involves lengthy expressions. Let us count the number of terms involved. Note thateach summation

P�contains four terms. We see that U ; V and W contain, respectively, 32, 44

and 20 terms. Each improper integral involving Fij can be further simpli®ed; see Appendix B.

Acknowledgements

The authors wish to thank the anonymous referee and the handling editor, Prof. I. Stakgold,for many helpful comments.

Appendix A. Fourier transform and inversion formulas

Elementary material on the Fourier transform of Schwartz and tempered distributions may befound in [3, ch. 3]; [13], for example. The N-dimensional Fourier transform of a distribution T canbe written symbolically as

bT �n1; . . . ; nN� �F�T ��n1; . . . ; nN�

�Z

RNexp

ÿ iXN

j�1

xjnj

!T �x1; . . . ; xN �dx1 . . . dxN ; �n1; . . . ; nN � 2 RN : �A:1�


Then the inversion formula

T �x1; . . . ; xN� �Fÿ1�bT ��x1; . . . ; xN�

� 1

�2p�NZ

RNexp i

XN

j�1

xjnj

!bT �n1; . . . ; nN�dn1 . . . dnN �A:2�

holds. The Fourier transform simpli®es (distributional) derivatives:

Fo

ox1

� �j1

� � � ooxN

� �jN

T

!�n1; . . . ; nN� � �in1�j1 � � � �inN�jNF�T ��n1; . . . ; nN�; �A:3�

oon1

� �j1

� � � oonN

� �jN

�F�T ��n1; . . . ; nN�� ÿi�j1�j2��jNF�xj1

1 . . . xjNN T � �n1; . . . ; nN� �A:4�

for any nonnegative integers j1; . . . ; jN . The following convolution formula is useful: for twodistributions T1 and T2,

F��T1 � T2��n1; . . . ; nN� � bT1�n1; . . . ; nN�bT2�n1; . . . ; nN�; �A:5�

where

�T1 � T2��x1; . . . ; xN� �Z

RNT1�x1 ÿ y1; . . . ; xN ÿ yN �T2�y1; . . . ; yN�dy1 . . . ; dyN

is the convolution product.Back to the notation (4.4) and (4.5). In R2, the Dirac distribution d�x; s� � d�x�d�s� satis®es

d̂�a;b� � 1: �A:6�

Let H�x� be the Heaviside function in R and let sgn x be the sign function of the scalar variablex:

H�x� � 0 if x6 01 if x > 0;

�sgn x � ÿ1 if x < 0

1 if x > 0:

��A:7�

Then [3, ch. (3.4)]

d

dxH�x� � d�x�; sgn x � H�x� ÿ H�ÿx�; �A:8�

where d=dx is the distributional derivative, and equality holds in the sense of distributions. From(A.2)±(A.4) and (A.6)±(A.8), we have (the single variable Fourier transform and inversion)

�xn sgn x�^�a� �Z

R

eÿiax�xn sgn x�dx � �ÿ2i��ÿi�n�n!�aÿ�n�1�; n � 0; 1; 2; . . . ; �A:9�

1

2p

ZR

eiax 1

an�1da � in�1

2�n!� xn sgn x; n � 0; 1; 2; . . . �A:10�


It is also known [13, ch. 6] that

Fÿ1 1

a2 � b2

� �� ÿ 1

2pln

��x2 � s2

p; �A:11�

Fÿ1 1

�a2 � b2�2 !

� 1

8p�x2 � s2� ln

��x2 � s2

p: �A:12�

Let k2 2 C. The fundamental solution E�x; s� of the Helmholtz equation in R2

o2

ox2

�� o2

os2� k2

�E�x; s� � ÿd�x�d�s� �A:13�

is known [3, ch. 7]; [28] to be

E�x; s� � i

4H �1�0 �k

��x2 � s2

p�; �A:14�

where k is a square root of k2 such that Im k P 0 and H �1�0 �� the Hankel function of order 0 ofthe ®rst kind [1]. On the other hand, the Fourier transform of (A.13) satis®es

bE�a;b� � 1

a2 � b2 ÿ k2: �A:15�

The inverse transform of bE should give back the right side of (A.14). Therefore

1

4p2

ZR2

ei�ax�bs� 1

a2 � b2 ÿ k2dadb � i

4H �i�0 �k

��x2 � s2

p�; �A:16�

From (A.16), we can derive the following important inversion formula. Let g 2 C; we have

1

4p2

ZR2

ei�ax�bs� 1

a2 � b2 ÿ gadadb � 1

4p2

ZR2

ei�ax�bs� 1

aÿ �g=2�� 2 � b2 ÿ g=2� �2 dadb

) change of variable a�

ÿ g27!a

�)

� 1

4p2e�ig=2�x

ZR2

ei�ax�bs� 1

a2 � b2 ÿ g=2� �2 dadb � e�ig=2�x i

4H �1�0

g�

2

��x2 � s2

p� �; �A:17�

where g� is a square root of g2 satisfying Im g�P 0.Let x be the eighth root of unity as given in (4.12). By (A.17), for any constant l > 0, we have

(see (4.15))


Fÿ1 1

D�a;b; j��

�x; s�

�Fÿ1 1

a2 � b2 ÿ xjla

� ��x; s� � e�ix

jl=2�x i

4H �1�0

sjxjl2

��x2 � s2

p� ��A:18�

� F0j�x; s�; j � 1; 3; 5; 7; �A:19�

where we let

sj �1 j � 1; 3;

ifÿ1 j � 5; 7;

8<: �A:20�

so that

Im sjxj > 0; j � 1; 3; 5; 7

are satis®ed.From (A.3)±(A.5) and (A.9), we get

Fÿ1 1

b1

D�a;b; j��

�x; s� � i

2

ZR

sgn �sÿ r� � F0j�x;r�dr; �A:21�

Fÿ1 ab

1

D�a;b; j��

�x; s� � 1

i

oox

� �i

2

ZR

sgn �s24 ÿ r�F0j�x; r�dr

35� 1

2

ZR

sgn �sÿ r� oox

F0j�x;r�dr

� i

2

ZR

sgn �sÿ r�Cÿj dr; �A:22�

where we have de®ned F1j�x;r� through the following relations:

oox

F0j�x;r� � ixjl2

F0j�x; r� ÿ e�ixjl=2�x i

4

sjxjl2

x��x2 � s2p H �1�1

sjxjl2

��x2 � s2

p� �� iCÿj ; �A:23�

with

F1j�x;r� � sjxjl8

x��x2 � r2p e�ix

jl=2�xH �1�1

sjxjl2

��x2 � r2

p� �;

and

H �1�1 �� the Hankel function of order 1 of the first kind �1�:


Similarly, we obtain

Fÿ1 1

b2

1

D�a;b; j��

�x; s� � ÿ 1

2

ZR

jsÿ rjF0j�x; r�dr; �A:24�

Fÿ1 a

b2

1

D�a;b; j��

�x; s� � i

2

ZR

jsÿ rj oox

F0j�x;r�dr

� ÿ 1

2

ZR

�sÿ r� sgn �sÿ r�Cÿj dr; �A:25�

Fÿ1 1

b3

1

D�a;b; j��

�x; s� � ÿ i

4

ZR

�sÿ r�2 sgn �sÿ r�F0j�x; r�dr; �A:26�

Fÿ1 a

b3

1

D�a;b; j��

�x; s� � ÿ 1

4

ZR

�sÿ r�2 sgn �sÿ r� oox

F0j�x;r�dr

� ÿ i

4

ZR

�sÿ r�2 sgn �sÿ r�Cÿj dr: �A:27�

It is not di�cult to show that all of the convolution integrals in (A.21), (A.22), and (A.24)±(A.27) are convergent integrals for any given �x; s� 2 R2. On the other hand, if one were to performthe Fourier inversion of �1=a��1=D�a;b; j��, then by (A.3), (A.5) and (A.9), one would get

Fÿ1 1

a1

D�a; b; j��

�x; s� � i

2

ZR

sgn �xÿ n�F0j�n; s�dn: �A:28�

This last integral diverges for j � 5; 7, because the decay of the Hankel function does not com-pensate for the growth of the exponential term and is therefore unsuitable for computationalpurposes. The same argument applies to the inversion formula

Fÿ1 bk

a`1

D�a; b; j��

�x; s� � 1

i

oos

� �ki`�1

2�`!�Z

R

�x24 ÿ n�` sgn �xÿ n�F0j�n; s�dn

35: �A:29�

Appendix B. Simpli®cations of improper integrals

The expressions for U ; V and W as derived in (4.29), (4.30) and (4.28), respectively, involvemany improper integrals. There are a total of eight types for j � 1; 3; 5; 7,


�I1�Z

R

sgn �sÿ f�F0j�x; f�df;

�I2�Z

R

sgn �sÿ f�F1j�x; f�df;

�I3�Z

R

jsÿ fjF0j�x; f�; df;

�I4�Z

R

jsÿ fjF1j�x; f�df;

�I5�Z

R

�sÿ f�2 sgn �sÿ f�F0j�x; f�df;

�I6�Z

R

�sÿ f�2 sgn �sÿ f�F1j�x; f�df;

�I7�Z

R

jsÿ fj3F0j�x; f�df;

�I8�Z

R

jsÿ fj3F1j�x; f�df:

We reduce them to integrals on bounded intervals suitable for numerical quadrature. Using theevenness in s of F0j and F1j, as well as basic properties of the Hankel functions, we obtain

�I1�Z

R

sgn �sÿ f�F0j�x; f�df � 2

Z s

0

F0j�x; f�df; �B:1�

�I2�Z

R

sgn �sÿ f�F1j�x; f�df � 2

Z s

0

F1j�x; f�df; �B:2�

�I3�Z

R

jsÿ fjF0j�x; f�df

� 2sZ s

0

F0j�x; f�dfÿ i

sjlxje�ilxj=2�x

��x2 � s2

pH �1�1

sjlxj

2

��x2 � s2

p� �; �B:3�

�I4�Z

R

jsÿ fjF1j�x; f�df � 2sZ s

0

F1j�x; f�df� x2

e�ilxj=2�xH �1�0

sjlxj

2

��x2 � s2

p� �; �B:4�


�I5�Z

R

�sÿ f�2 sgn �sÿ f�F0j�x; f�df

� 2s2

Z s

0

F0j�x; f�df� 2

Z s

0

f2F0j�x; f�df

ÿ 2i

sjlxjse�ilxj=2�x

��xs � s2

pH �1�1

sjlxj

2

��x2 � s2

p� �: �B:5�

�I6�Z

R

�sÿ f�2 sgn �sÿ f�F1j�x; f�df

� 2s2

Z s

0

F1j�x; f�dfÿ 2ixZ s

0


2xse�ilxj=2�xH �1�0

sjlxj

2

��x2 � s2

p� �; �B:6�

�I7�Z

R

jsÿ fj3F0j�x; f�df � 2s3

Z s

0

F0j�x; f�df� 6sZ s

0

f2F0j�x; f�df

ÿ 4i

sjlxje�ilxj=2�xs2

��x2 � s2

pH �1�1

sjlxj

2

��x2 � s2

p� �

ÿ 4i

�sjlxj�2 e�ilxj=2�x�x2 � s2�H �1�0

sjlxj

2

��x2 � s2

p� �

� 16i

�sjlxj�3 e�ilxj=2�x��x2 � s2

pH �1�1

sjlxj

2

��x2 � s2

p� �; �B:7�

�I8�Z

R

jsÿ fj3F1j�x; f�df

� 2s2

Z s

0


2xs2e�ilxj=2�xH �1�0

sjlxj

2

��x2 � s2

p� �

ÿ 6ixsZ s

0

F0j�x;r�drÿ 2

sjlxjxe�ilxj=2�x

��x2 � s2

pH �1�1

sjlxj

2

��x2 � s2

p� �: �B:8�

We illustrate the calculations for (I3):ZR

jsÿ fjF0j�x; f�df � sZ

R

sgn �sÿ f�F0j�x; f�dfÿZ

R

f sgn �sÿ f�F0j�x; f�df

� 2sZ s

0

F0j�x; f�dfÿZ

R

f sgn �sÿ f�F0j�x; f�df:


But ZR

f sgn �sÿ f�F0j�x; f�df

�Z s

ÿ1fF0j�x; f�dfÿ

Z 1

sfF0j�x; f�df

�Z 1

ÿs�ÿf�F0j�x;ÿf�dfÿ

Z 1

sfF0j�x; f�df

� ÿZ s

ÿsfF0j�x; f�dfÿ 2

Z 1

sfF0j�x; f�df

� ÿ2

Z 1

sfF0j�x; f�df:

Further,Z 1

sfF0j�x; f�df � i

4e�ilxj=2�x

Z 1

sH �1�0

sjlxj

2

��x2 � f2

q� �fdf

�change of variables t2 � x2 � f2 )�

� i

4e�ilxj=2�x

Z 1��x2�s2p H �1�0

sjlxj

2t

� �t dt;

and since H �1�0 �kr� satis®es the Bessel di�erential equation

�rF 0�0 � k2rF � 0; F �r� � H �1�0 �kr�;

from (B.5), with k � sjlxj=2, we getZ 1��x2�s2p tH �1�0 �kt�dt � ÿ kÿ2

Z 1��x2�s2p d

dtt

d

dtH �1�0 �kt�

� �dt

� ÿ kÿ2td

dtH �1�0 �kt� 1 ��

x2�s2p��

� kÿ2��x2 � s2

pkhÿ H �1�1 k

��x2 � s2

p� �i� ÿ kÿ1

��x2 � s2

pH �1�1 k

��x2 � s2

p� �� ÿ 2

sjlxj

��x2 � s2

pH �1�1 k

��x2 � s2

p� �:


Combining the above we obtainZR

�sÿ f� sgn �sÿ f�F0j�x; f�df

� 2sZ s

0

F0j�x; f�dfÿ i

sjlxje�ilxj=2�x

��x2 � s2

pH �1�1

sjlxj

2

��x2 � s2

p� �:

Remark B.1. In conclusion, all integrals (I1)±(I8) are representable as simple functions, plus threebasic integralsZ s

0

F0j�x; f�df;Z s

0

F1j�x; f�df and

Z s

0

f2F0j�x; f�df: �B:9�

Some of the actual evaluation of (B.9) will be studied in the sequel, Part II.

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