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Final Report Of
Minor ResearchProject Entitled
“STUDY OF SOME HANKEL TYPE
TRANSFORMATIONS AND THEIR
APPLICATIONS”
[DISTRIBUTIONAL GENERALIZED MODIFIED
STRUVE TRANSFORM AND THEIR APPLICATIONS]
Submitted to
Western Regional Office
University Grants Commission
Ganesh-Khind, PUNE-7
BY
Dr. GAJANAN SUKHADEO GAIKAWAD
KarmaveerBhauraoPatilMahavidyalaya,
Pandharpur – 413304.
[2017]
Acknowledgement
My sincere thanks to UGC (WRO) Ganesh Khind-7 for financial assistant to me in the
form of Minor Research Project and giving me the opportunity for better research.
I am also very much thankful to Principal, K. B. P. Mahavidyalaya, Pandharpur (Rayat
Shikshan Sanstha, Satara) for giving me the facilities in the college for the completion of this
project.
My thanks to my Research Guide to give me encouragement during the research
project and also my departmental colleagues, my teacher friends and non-teaching staff for
giving the help during research project.
(Dr. G. S. Gaikawad)
Head, Department of Mathematics,
K.B.P. Mahavidyalaya, Pandharpur.
WORK DONE IN MINOR RESEARCH PROJECT :
Introduction:
Generalized integral transformations have been of ever increasing interesting to mathematicians
working in several different branches of mathematics. The theory of integral transformation is a classical
subject in mathematics whose literature can be traced back through at least 160 years. The theory of
generalized functions in the present form, was used explicitly by S.L. Sobolev [1936] in his study of the
uniqueness of solutions of the Cauchy problem for linear hyperbolic equations. In 1950-1951 appeared
Laurent Schwartz’s monograph “Theorie des Distributions” in which he systematized the theory of
generalized functions, basing it on the theory of linear topological spaces. He unified all the earlier
approaches; his results were deep and of far reaching significance.
Due to wide-spread applicability as a powerful tool in solving certain ordinary and partial differential
equations involving distributional boundary conditions, the extensions to generalized functions of
Fourier transformation including several other integral transformations have become an active area of
research in the last thirty years. In the middle of 1960’s Zemanian *1968+ extended the Laplace, Mellin,
Hankel, Meijer, Weierstrass and convolution transformations to certain classes of generalized functions.
The works of Koh and Zemanian [1968], Koh [1970], Lee [1975], Pandey [1996], Erdelyi [1953] ,Pathak
[1975], Zayed [2000] and Malgonde [Feb.2000] on some transformations of generalized functions are
worth mentioning.
The repeated extension of the same integral transformation to generalized functions should create
no confusion. In fact, in each case, either the approach of extension in one is different from the other or
the latter extension is presented for a different generalized function space, possibly larger than the
previous one. Mostly, in all the works in this area, the inversion theorems corresponding to the
generalized integral transformations have been of paramount importance.
It is a fact that in the classical theory of integral transformation every time one states a set of
conditions under which a transformation can be applied to a function, indeed one specifies a space of
functions in the domain of the transformation. However there is no need to denote these function
spaces by symbols as compared to the generalized theory.
The extension of different integral transformations to generalized functions requires different
testing function spaces, which is tailored to suit the kernel function of the transformation. However
there is one unifying concept for all the cases, considered in the present Paper. Let I be the open interval
of integration for the conventional integral transformation under consideration, and let D(I) be the
space of smooth functions with compact support on I. The topology of D(I) is that which makes its dual
)(' ID , the space of Schwartz’s distributions *1957 and 1959,Vol.I, p.65+.
In last twenty four years since the publication of Zemanian’s work, interest has continued for
generalized integral transformations. It appears that so far very few workers have extended the
conventional generalizations of integral transformations to generalized functions. This very situation
motivated us to unify and study some of the scattered results in the theory of integral transformations.
To achieve this we have studied in present Paper generalized Hankel type transformation, generalized
cut Hankel type transformation, generalized Hardy transformations and generalized modified Struve
transformation within a frame work of generalized functions.
Throughout this Paper, we follow the terminology and notations of Zemanian [1968]. We freely use
various properties of special functions, which appear in such standard reference works as Erdelyi [1953]
and Watson [1958]. We also use without proving number of classical results from the conventional
theory of integral transformation.
Objectives of the Project:
To develop the theory of Distributional generalized modified Struve transform by Kernal Method with its application which is different from Adjoint method developed by Dr. Pathak R. S.[1975].
Work done:
The investigations in the present paper are spread over seven articles. The article 1 is introductory
which surveys the historical background and incorporates a few relevant basic concepts from the
standard texts. Article 2 and 3 differential operator deal with the comprehensive study of the
Distributional generalized modified Struve transformation respectively in distributional sense testing
function spaces with their dual spaces. Article 4 and 5 covers the properties of testing function spaces
and distributional generalized modified Struve transform with study of analyticity theorerm. Aticle 6 is
devoted to prove the inversion and uniqueness theorem for distributional generalized modified Struve
transformation and lastly article 7 deals with Application of Distributional generalized modified Struve
transform to solve certain class of differential equation viz. certain boundary value problems
respectively.
Conclusion:
The theory thus developed for the distributional generalized modified Struve transform is shown to be useful to solve certain class of differential equations viz. certain boundary value problems.
Principal Investigator
(Dr. Gaikawad G. S.) Department of Mathematics, K.B.P. Mahavidyalaya, Pandharpur.
DISTRIBUTIONAL GENERALIZED MODIFIED STRUVE TRANSFORM
Abstract.
In this Paper a transform involving generalized modified Struve function in the kernel, is extended
to a class of (generalized functions) distributions and the properties of a testing function space and its
dual are studied. Transformable generalized functions are defined and an analyticity theorem,
inversion theorem and uniqueness theorems are also proved for the generalized modified Struve
transform of generalized functions. It is shown that several classes of differential equations can be
solved with the help of this generalized modified Struve transform of generalized functions.
1. Introduction.
The generalized modified Struve transform is studied by Pathak R.S.[1975]
0
,, )();,()()}()( dttfstcaestfFsF qst
caq
(1.1)
under two cases 11 and 1 qq which are based on the following two integrals Babister
[1967,p.120]
0
11 );;,()();,( kpcabBpbdzkzcaze bbpz ,
(1.2)
p.167].[1967,Babister function richypegeomet
eousnonhomogen theis )zc; b;B(a, and ,ReRe,0Re,0Re kpkppbwhere
.
0
1 ;;,))(();,(pk
kcacbBkpbdzkzcaze bbpz ,
(1.3)
),ReRe,0Re,0(Re kkpkppb , where zca ;, is the generalized modified Struve
function explored by Babister [1967, p.96] and is defined as below:
)1()()1()2();,( 1 cacaeizca ai
(1.4)
Where
)1(
0
112/22/)( )1()1()1[()/2( duuueeeei acazuiazcaic
)1(
0
112/)(2 ])1()1(}1{ duuuee acazuaci.
It satisfies a nonhomogeneous differential equation and, for c=2a, we have
),2/()4/)(2/1();2,( 2/1
2/2/1 zLezazaa a
za
(1.5)
where )(zL is modified Struve function Bromwich [1965,p.38].
By specializing parameters various other particular cases of zca ;, have been obtained by Babister
[1967, pp.104-107 and 113-114]. An inversion formula in the classical sense is given by Pathak R.S.
[1975] as given by the following theorem:
Theorem 1.1 Inversion Theorem.
Let dxxfpxKpfF caq )()()()(0
,,
(1.6)
where );,()exp()( pxcaqpxpxK then
dsssMxi
xfxfiT
iT
s
T )()()2(
1)()(
2
1 lim
,
(1.7)
where, 11)1( ;;,1)].1(/[)( qcasBsqsM s , 1q
11)1( )1(;;,1)].1(/)1([ qcacsBsq s , 11 q
(1.8)
and
0
)()( dppps s
(1.9)
provided that
i) the integrals
0
1 )( dttft and
0)( dttFt s
are absolutely convergent
,( iTs )0 T ,
ii) f(t) is of bounded variation in the neighbourhood of point t = x.,
iii) f(t)= )1(O , for small t, ,0)1Re( s 1q or 11 q ,
iv) )()( 0 tftpK is bounded for 0t and 0Re 0 pp .
The aim of the chapter is to extend (1.1) to a certain class of (generalized functions) distributions and
discuss the properties of a testing function space and its dual with analyticity theorem, the inversion and
uniqueness theorems. For this we first we define a differential operator and examine the behavior of
the result of its nth operation on the kernel of our transform, as this will be needed in our study.
2. Differential Operator.
Let );,();,( )1( xcaeexcaeu xxqqx where zca ;, is the generalized modified Struve
and is defined by (Babister [1967], p.96)
)1()()1()2();,( 1 cacaeizca ai
and
)1(
0
112/22/)( )1()1()1[()/2( duuueeeei acazuiazcaic
)1(
0
112/)(2 ])1()1(}1{ duuuee acazuaci.
Then );,()1( xcaeue xxq
);,()1( xcaedx
due
dx
d xxq );1,(.)(
)()( xcae
c
ac x
);,(.)(
)()1();,(
)(
)(zmcae
c
aczcae
dz
d z
m
mmz
m
m
);,(.)()(
)()()1( zmcae
mcac
cmac zm
for m=1,2,3,… (see Babister *1967+, p.103).
);2,(
)(
)(
)2(
)2()1( xcaec
acue
dx
d
dx
d xxq
);2,(.
)(
)(
)2(
)2()1()1( xcaec
acue
dx
d
dx
de qxxqxq
Let us define an operator xqA , by
)]}([{)( )1()1(
, xeDDexA xq
xx
xq
xq ])])1[(])1[(2[ 22212 xxqDxxqD xx
])1[(])1[(2 22222 xqxDxqDxx xx
(2.1)
where xD =D= d/dx . Therefore, we have
);2,..
)(
)()];,([ 2
2
)2(
, stcaesc
acstcaeA qstqst
tq
);4,(
)(
)()];,([ 4
)4(
)4(2
, stcaesc
acstcaeA qstqst
tq
…….,,. ……….
);2,(
)(
)()];,([ 2
)2(
)2(
, stncaesc
acstcaeA qstn
n
nqstn
tq , for n = 0,1,2,3,…
(2.2)
For large t,
qstn
n
nqstn
tq
ncastqn
n
nqstn
tq
esc
acstcaeA
tsmallfor
stesc
ac
a
cstcaeA
.)(
)( )];,([
, Also
)(.)(
)(
)(
)()];,([
)2(
)2(
,
2)1(2
)2(
)2(
,
since for large z, )](1[)()(
)(),,(
1
zOze
a
czca caz (see Erdelyi [1953]),
and for small z , )1();,( Ozca (see Babister [1967], p.108, 4.105).
)];,([, stcaeA qstn
tq when, Re s > 0, q >1, and Re c >Re a > 0.
3. Testing Function Space and its Dual.
Let us define functionals q
n,, ; n = 0,1,2,3… on certain smooth functions )0(),( tt by
.)()( ,
2sup
0,, tAte n
tq
nt
t
q
n
Let )(, IH be the space of all those complex-valued smooth functions )(t defined on ),0( I
for which )(,, q
n is finite for all n = 0,1,2,3… where , are suitably fixed real numbers and 1q
.11 qand For any complex number , we have
)(.)( ,,,, q
n
q
n ; and )()()( ,,,,,, q
n
q
n
q
n , ).(, , IH
q
n,, is a semi norm on )(, IH .
Again )(0)(0,, tq is zero element in )(, IH . Hence q
0,, is a norm. So the collection
0,, n
q
n is a countable multinorm on )(, IH equipped with topology generated by
0,, n
q
n , )(, IH is a countably multinormed space.
Lemma 3.1 For every fixed s such thatq
s
1
Re0
, 11or 1 qq and 0 ,
)()( , IHstK where ).;,()( stcaestK qst
Proof. We have from (2.2)
);2,(
)(
)()1();,(
)2(
)2(2
, stncaesc
acstcaeA qstn
n
nnqstn
tq , for n = 0,1,2,3…
.)()]([ ,
2sup
0,, stKAtestK n
tq
nt
t
q
n
.);2,(..)(
)(
)2(
)2(2sup
0 stncaesc
acte qstn
n
nnt
t
Now, for large t and fixed s we have st and
);2,(..)(
)(
)2(
)2(2 stncaesc
acte qstn
n
nnt
ncastqstn
n
nnt steesc
acte 2
)2(
)2(2 )( ...)(
)(
can
n
ntsq stsc
acte
).(.
)(
)(.
)2(
)2(])1([
which tends to zero as t since s is fixed with q
s
1
Re0
, 11or 1 qq and Re c >Re
a > 0.
For small t,
);2,(..)(
)(
)2(
)2(2 stncaesc
acte qstn
n
nnt
nn
n
ntqy stsc
acte 2
)2(
)2(][ ).(.)(
)(.
= a finite number, as 0t , since 0 , ,1q
)]([,, stKq
n , for n = 0,1,2,3….This shows that )()( , IHstK .
Theorem 3.1 )(, IH is a complete countably multinormed space i.e. Frechet space.
Proof. Let 1 be a Cauchy sequence in )(, IH and be any arbitrary compact subset of
),0( I .
Let us define an operator D-1 by t
dxD
1 where is the fixed point in I.
Thus for any smooth function )(t on ],0[
).()()(1 ttDD
By definition of tqA , we have, )]([..)( )1()1(
, teDDetA tq
tt
tq
tq .
In view of seminorm q
n,, we see that )(, tA xq converges uniformly on as .
Moreover, we have
)]([..)( )1()1()1(1
,
)1(1 teDDeeDtAeD tq
tt
tqtq
tq
tq
)]([. )1(1 teDDD tq
tt
)()]([ )1()1(
qtq
t eDteD
(3.1)
)]([ ,
)1(11)1( tAeDDe tq
tqtq
)]()()()([ )1()1()1()1(
qqtqtq eDtetee
)()()()( )1())(1(
qtq eDtet
)()()()( )1())(1(
qtq eDtGet
(3.2)
Since multiplication by a power of t or multiplication by 1D preserves the property of convergence of a
converging function, the left hand side of (3.1) and (3.2) also converges uniformly on as .
Thus we see that the left hand side and first two terms of right hand side in (3.2) converges uniformly on
. Hence as 0)( tG , )]([ )1(
qeD must converges as . This with (3.1) implies that
)]([ )1( teD tq
t also converges uniformly on every which, in turn, implies that )(tDt does the
same.
Next by virtue of definition of tqA , it follows that )(2 tDt also converges uniformly on every compact
subset of I.
We repeat this argument with replaced by n
tqA , and tqA , by 1
,
n
tqA .
This shows that for every non-negative integer n, )(xDn
x converges uniformly on every .
Consequently there exist a smooth function )(t on I such that for each n and t, )()( tDtD nt
nt as
. It follows easily that
0)(,, q
n as , n = 0,1,2,3…
(3.3)
Finally, there exist a constant Cn not depending on such that n
q
n C)(,,
(since )(, IH ). Therefore from (3.3),
n
q
n
q
n
q
n C )()()( ,,,,,, .
This implies that )(, IH and is the limit in )(, IH of 1 . Thus )(, IH is a sequentially
complete countably multinormed space or a Frechet space. Thus,
(I) Members of )(, IH are complex-valued smooth functions defined on I.
(II) )(, IH is a complete countably multinormed space or a Frechet space.
(III) If 1 converges in )(, IH to zero, then for every non-negative integer m
1mD
converges to zero function uniformly on every compact subset of I, and so )(, IH is a testing function
space satisfying all necessary
conditions for it to be such a space. The collection of all continuous linear functional on )(, IH is
called of )(, IH and is denoted by )(, IH . Members of )(, IH are (distributions) generalized
functions. Since )(, IH is complete, )(', IH is also complete.
4. Properties of )(, IH .
As in (Zemanian [1968], pp. 32-36) D(I) is a space which contains those complex-valued smooth
functions )(t defined on t0 which have compact supports and E(I) is the space of all complex-
valued smooth functions on I. We now compare )(', IH and )(, IH with D(I) and E(I) and their
duals and list some properties:
Property4.1 From the definition of spaces D(I) and E(I) we see that )()()( , IEIHID . Since D(I)
is dense in E(I), it follows that )(, IH is dense in E(I).
Proof. Let 1 converges to in D(I). Let the supports of and be contained in the closed
interval [a,b],
0 < a < b < , we have
)()( ,
2sup
0,,
n
tq
nt
t
q
n Ate )(),(2
0
sup
rmn
r
r
r
t
bta DttqBte
)(),( 22
0
sup
rnr
r
tn
r
bta DttqBte
)(),( 22
0
sup
rn
r
rtn
r
bta DtqBte
Where ,])1[(),( 2 r
r
n
r tqCtqB and so on.
(4.1)
If we take ),(max tqBteC r
rt
btar
, we see that
)(.),,()( 2sup
0
2
0
,,
rn
t
n
r
r
q
n DtqC ,
for N , where N is a large positive integer. This is true from the property of convergence of 1
in D( I ). We see that the convergence in D( I ) implies convergence in )(, IH . Consequently, the
restriction of )('
, IHf to D(I) is in D’(I).
Property 4.2 If 0< 1 < 2 then )()(,21 , IHIH and the topology of )(,1
IH is stronger than
the topology induced on it by the topology of )(,2IH . Hence the restriction of any )('
,2IHf to
)(,1IH is in )(,
'
1IH . Also convergence in )('
,2IH implies convergence in )(,
'
1IH .
Property 4.3 )(, IH is a dense subspace of E (I), whatever be the choice of and . Indeed
)()()( , IEIHID and since D(I) is dense in E(I) so in )(, IH .
Moreover, in the proof of Theorem 3.2, we have seen that convergence of any sequence in )(, IH
implies its convergence in E(I). Consequently, by cor. 1.8-(2a) (Zemanian [1967], p.21) )(IE is a
subspace of )(,' IH for any permissible values of and .
Property 4.4 The differential operator r
tq
r At , (r = 1,2,3…) are continuous linear mapping of )(, IH
into itself.
Proof. Since ][ ,,
2 r
tq
rn
tq
nt AtAte = ][),( ,
22
0
2 r
tq
rrnrn
r
r
nt AtDttqBte
r
tq
rn
n
n
n
rnr
r
nnnt AttBDtBDtBDtBDBte ,
2
2
12
12
2121
1
2
0
2 ]}......[{
r
tq
r
nn
rr
rn
nnnnnnt AtBtDBDtBDtBDtBtte ,2122
1212
1
22
0
22 ]}......[{.
rr
rn
nnnnt DtBDtBDtBte
2
1212
1
22
0 ...[{
nr
tq
rntr
tq
r
nn AteAtBtDB
,
2
,212 ]}...
nr
tq
rntr
tq
r
rnnnn
t AteAtrBrrBrBBte
,
2
,222122 ]!....)1(..[
Hence for any )(, IH , we have
)]([ ,,
2sup
0 r
tq
rn
tq
nt
t AtAte
)()(),( ,
2sup
0,
sup
0 rn
tq
rnt
t
r
tqr
rt
tAteAtqBte
where ),( tqBr is as defined in (4.1) for all n = 0,1,2,3… and r = 0,1,2,3,…,2n is a constant with
respective to q and for t0 .
The adjoint operator r
tqB , of r
tq
r At , is a generalized differential operator on )(,' IH into
)(,' IH and is defined by .)(,, ,, r
tq
rr
tq AtffB
5. The Distributional Generalized Modified Struve Transform.
We shall call f a distributional generalized modified Struve transformable generalized function if f is a
member of )(,' IH for some suitably fixed real number and for some positive real number .
According to §4, Property
4.2, f is then a member of )(,' IH if for every 0 . This implies that there exists a positive
real number f (possibly f ) such that )(,' IHf every f
q
10 and
)(,' IHf for every
qf
10
.
Definition 5.1 Let )(,' IHf for some fixed real numbers and with Re s > 0
and 0Re . The distributional generalized modified Struve transformation of
generalized function f denoted by )()}()( '
,,
' stfFsF caq , is defined by
)(),()()}()( '
,,
' stKtfstfFsF caq where );,()( stcaestK qst and fs where
sss ff arg,Re0: .
Lemma 5.1 Let and be real numbers with , then for q>1, zzz arg,0,Re
and t0 , we have
)1.();2,()( Re2 zCtzncaezte qtznt
where C is constant with respective to t and z and ca .
Proof. Since z 0, and zarg , from the series expansion and asymptotic properties of
generalized modified Struve function, we see that for 1z , there exist a constant caM , independent
of z such that ca
qzn Mzncaez ,
2 );2,(
and another constant caN , independent of z such that for 1z
zq
ca
zqca
ca
qzn ezNezNzncaez )1Re(Re
,
)1Re()Re(
,
2 ....);2,(
where ca . Consequently, for zRe and t0 , there exist constant caB ,
independent of z and x such that
zq
ca
qtznt etzBtzncaezte )1Re(ReRe
,
2 ).1)(1.();2,()(
Also for q>1, zRe ; zqet )1Re(Re ).1( is uniformly bounded on t0 by another
constant caC , and so
)1.();2,()( Re
,
2 zCtzncaezte ca
qtznt . This completes the proof of lemma.
Theorem 5.2 Analyticity Theorem.
Let )(),()()}()( '
,, stKtfstfFsF caq for fy . Then )(,, sF caq is analytic function on f
and
)(),()(,, stK
stfsF
scaq .
(5.1)
Proof. Let s be an arbitrary but fixed point in f . Let us choose real numbers ,
and positive r and 1r such that .ReReRe 1 srsrs Let C be a circle with center at s
and radius equal to 1r . We restrict 1r and hence and r , in such a way that C lies entirely within f .
Let y be a nonzero complex increment in s such that rs and let us consider the expression
)(),()(),(
)()( ,,,,ttfstK
stf
s
sFssFs
caqcaq
(5.2)
where )()(),(
stKss
stKsstKs
The differentiation formula (Slater [1960], p. 25), the series expansion and asymptotic behavior of
);,( zca shows that and hence equations (5.1) and (5.2) are meaningful.
To prove the theorem we will have to show that in )(, IH as 0s . Using Cauchy
integral formula we can write
dzszsszs
tzncaezc
ac
itA
C
qtzn
n
n
s
n
tq
2
)2(
)2(
,)(
111.);2,(
)(
)(
2
1)(
C
qtzn
n
ndz
sszsz
tzncaez
c
ac
i
s
)()(
);2,(
)(
)(
2 2
)2(
)2(
C
ntqtzn
n
n
s
n
tq
nt dztesszsz
ztncaez
c
ac
i
stAte 2
2
)2(
)2(
,
2 .)()(
);2,(
)(
)(
2)(
C
qtzntn
n
ndz
sszsz
ztncaeztez
c
acs
)()(
);2,()(
)(
)(
2 2
2
)2(
)2(
Let caQ , be the constant bound on [ );2,()( 2 ztncaezte qtznt ] for t0 and Cz
(Lemma 5.1). Then we may write
)(,
2 tAte s
n
tq
nt
C
n
n
n
ca dzsszsz
z
c
acQ
s
)()()(
)(
2 2
)2(
)2(
,
n
n
n
Cz
caz
c
acr
rrr
Qs
.
)(
)(.2
)(2 )2(
)2(sup
1
1
2
1
,
)()( , IHstKs
0s
which tends to zero as 0s . This proves the theorem.
6. Inversion theorem for the distributional generalized modified Struve transform.
In this section we establish an inversion formula for the distributional generalized modified Struve
transformation, which determines the restriction to D(I) of any Struve-transformable generalized
function from its generalized modified Struve transform. From this we will obtain a uniqueness theorem,
which states that two generalized Struve-transformable generalized functions having the same
transformation must have the same restriction to D(I). First we shall prove some lemmas and theorems
which will be used for proving the main inversion theorem.
Lemma 6.1 The function 1su as a function of u is a member of )(, IH if 1Re s and 0 .
Proof. It is clear that 1su is differentiable function of s. Consider
1
,
2sup
0
sn
xq
nu
u uAue 122
0
22sup
0 ])1[(
srnrn
r
r
r
nnu
u uDxuuqCue
rnsrn
r
r
r
nnu
u usrn
suuqCue
21
2
0
22sup
0)!12(
)!1(])1[(
)!12(
)!1(])1[(
2
0
21sup
0srn
suqCue
n
r
r
r
nsu
u
)( 1
,,
sq
n u , under the conditions stated in Lemma for n = 0,1,2,3…
Lemma 6.2 Let be suitably fixed real number and f )(, IH , then
dxxuKxufdxxuKufx ss )(),()(),(00
.
(6.1)
Proof. By using the technique of Riemann sums we can easily prove that
dxxuKxufdxxuKufx
Rs
Rs )(),()(),(
00
(6.2)
There is nothing to prove if 0sx . So assume that 0sx . We shall first show that
Rs dxxuKxuI
0)()(
(6.3)
is a member of )(, IH , this will insure that the right hand side of (6.2) has a sense.
By the smoothness of the integrand of (6.3) we may carry the operator n
xqA , for each
n = 0,1,2,3… under the integral sign in .6.3) to write
dxxuKxAxeuIAxe sn
xq
nxR
n
xq
nx )()( ,
2
0,
2
)(.. ,
sup
0
2sup
00
xuKxAxedxx sn
xqRx
nx
u
Rs
Since the last expression is finite because of
);2,()(
)()(
)2(
)2(2lim
,
2lim xuncaeuc
acxexuKAxe qxun
n
nnx
u
n
xq
nx
u
xuncaqxun
n
nnx
u exueuc
acxe 2
)2(
)2(2lim )()(
)(
0.)(
)(.
)2(
)2(])1([lim
nca
n
ncaxuq
u uc
acxe
(6.4)
provided that 0,1,0ReRe acqac uniformly for Rx 0 and
Rs dxx
0 is finite for
10 s , by Bromwich J. T. Ia [1965]. This proves that )()( , IHuI .
Next we consider the following Riemann sum for the integral (6.3) as
m s
um
RK
m
R
m
RmuJ
1
,),(
Upon applying f(u) to this term, we get another Riemann sum which converges to the left hand side of
(6.2) as m by virtue of the continuity of the integrand on Rx 0 .
Since )('
, IHf , our Lemma 6.2 will therefore be proven when we show that J(u,m) converges
in )(,
, IH ca
to (6.3) as m .
Set )],()([),( ,
2 muJuIAuemuB n
xq
nu
(6.5)
We have to show that 0),( muB as m uniformly on u0 .
In view of (6.4), given 0 , there exist a T such that, for u>T and Rx 0 ,
1
0,
2
3)(
dxxxuKAue
Rsr
xq
nu . Consequently, 3
),(,
2sup
muIAue r
xq
nu
Tu .
Also, for all m,
m
r
sR
sr
xq
nu
Tum
R
m
RdxxmuJAue
1
1
0,
2sup
3),(
(6.6)
Thus, there exist 0m such that for all 0mm the right hand side of (6.6) is bounded by 3
2. We have
thus shown that, for 0mm and Tu , ),( muB .
Next, set nu
Tu ueK 2sup
. Then, for Tu 0
R mr
xq
r
xq
s um
RKA
m
R
m
RdxxuKAxKmuB
01
,, ,)(),(
(6.7)
Since )(, xuKAx r
xq
s is uniformly continuous for all (u,x) such that Tu 0 and Rx 0 , there
exist an 1m such that for all 1mm , the right hand side of (6.7) is bounded by on Tu 0 . Thus,
when ),max( 10 mmm , then ),( muB on u0 . This completes the proof of Lemma 6.2.
Again we shall prove that
R
s dxxuKx 0)( in )(, IH as R .
Consider,
R
sq
n dxxuKx )(,, =
R
sr
xq
nu
u dxxuKxAue )(,
2sup
0
R
sr
xq
nu
u dxxuKxAue )(,
2sup
0
0
,
2sup
0 )( dxxuKxAue sr
xq
nu
u
Since
00
);,()( dxxucaexdxxuKx qxuss
0
1 )();,()( xudxucaxueu sqxus
0
1 );,( dttcateu sqts
)(;;,1).1( ,
111 IHqcasBqsu ss
)()1(;;,1)1).(1( ,
111 IHqcacsBqsu ss
21
311 )2.(
)()(
)(2.).2( s
scss q
aac
cqsu , or
2311
1
12.
)()(
2.)1).(2(
sscss
qaacqsu
provided 1Re,0Re,0)1Re( qqs and 11,1 qq by using the results 4.158 and 4.159
from Babister [1967,Ch.4,§4.17,p.120]:
i)
0
11 );;,()();,( kpcabBpbdzkzcaze bbpz,
),ReRe,0Re,0(Re kpkppb .
ii)
0
1 ;;,))(();,(pk
kcacbBkpbdzkzcaze bbpz ,
),ReRe,0Re,0(Re kkpkppb .
But
R
sq
n dxxuKx )(,, =
R
sr
xq
nu
u dxxuKxAue )(,
2sup
0
)(.. ,
sup2sup
0 xuKAuedxx r
xqxR
nu
uR
s
R
s dxxC.
where C is constant with respective to x and u. Since the integrand is finite and is independent of u,
vanishes as R . Hence
R
sq
n dxxuKx )(,, = 0 as R .
Therefore taking limit as R in (6.2) we have the required result.
This completes the proof of Lemma 6.2.
Lemma 6.3 Let )()(,, sFfF caq for fs Re0 , let )(ID and set
0
)()( dyyys s
where iws , and is fixed such that f )1,0max( . Then for any fixed real number r
with r0
r
r
r
r
ss dwsuufdwsuuf )(),()(),( 11
(6.8)
Proof. For 0)( y , the proof is trivial. So assume that 0)( y .
Let )(),( 1 suuf s
(6.9)
(6.9) is justified since )(,1 IHu s
for 1Re s .
It can be seen that )(s is analytic function in f )1( and )(s is analytic for all finite
values of s. Hence the integrals in (6.8) exist. We find that
r
r
sn
uq
nu
u dwsuAue )(1
,
2sup
0 , n = 0,1,2,3…
r
r
sn
uq
nu
u dwsuAue )(1
,
2sup
0
n
j
j
j
nsu
r
r
u dwssjn
suqCue
2
0
21sup
0 )()!12(
)!1(.])1[(.
Since
1sup
0
su
u ue for 0 and 1Re s and
dwssjn
suqC
r
r
n
j
j
j
n
2
0
2 )()!12(
)!1(.])1[( is finite for n = 0,1,2,3….Hence
r
r
sn
uq
nu
u dwsuAue )(1
,
2sup
0
proving that
r
r
s dwsu )(1 as a function of u belong to )(, IH , so that right hand side of (6.8) is
meaningful.
As in Zemanian [1968, §3.5] by partitioning the path of integration on the straight line from
irs to irs into m intervals each of length (2r/m) , we can write,
m
rsuufuVuf p
m
p
s
mp
2)(),()(),(
1
1
(6.10)
where m
rsuuV p
m
p
s
mp
2)()(
1
1
and pp iws
is a point in the pth interval. Since )()( pp ss is a continuous function pw we have as m , the
right summation of (6.10) tends to
r
r
s dwsuuf )(),( 1 .
Let be a suitably fixed real number and be a positive real number such that
fq )1/()1(0 . Since )(', IHf , our lemma will be proven when we show that
Vm(u) converges in )(, IH to
r
r
s dwsu )(1 as m . In other words we have to prove that for
each fixed r,
r
r
sm
nuq
nu dwsuuVAuemuB ])()([),( 1,,
2
converges uniformly to zero on u0 as m .
The proof can be carried on as in Zemanian [1968, pp.65-66].
Lemma 6.4 Let and be real numbers such that 0 , 0 and fix sRe such that
)1/()1( q . Also let )(ID . Then
dyyuu
yuryuy
)/log(.
))/log(.sin(.)/)((
1
0
converges to )(u in )(, IH as r .
Proof. Setting log (u/y) = t, we will prove that
dtt
rtueueAueu ttn
uq
nu
r
)sin()]()([)()( )1(
,
12
converges uniformly to zero in u0 as r , for n = 0,1,2,3….Since is smooth and is of
bounded support, we have by differentiating under the integral sign
)()()()( 321 uIuIuIur ,
Where dtt
rtuAuueAueueuI n
xq
ntn
uq
ntu )sin()()(.)()( ,
2
,
2)1(1
1
.
)(2 uI and )(3 uI are the same integrals with intervals of integration ),( and ),( respectively.
As in Zemanian [1968] we can prove now that )(1 uI , )(2 uI and )(3 uI tends to zero uniformly as
r , which proves the lemma.
Theorem 6.5 Inversion Theorem. Let )()(,, sFfF caq for fs Re0 . Then in the sense of
convergence in )(ID ,
dsssMyiyfiT
iT
s
T )()()2()( 1lim
,
(6.11)
where is any fixed real number such that f 0 ,
11)1( ;;,1)].1(./[)( qcasBsqsM s , 1q
11)1( )1(;;,1)].1(./)1([ qcacsBsq s , 11 q
(6.12)
And
0
)()( dxxFxs s .
(6.13)
Proof. Let )(ID , and choose real numbers and such that 0 , 0 and
fq )1/()1(0 .
Our object is now to show that
,)(,)()()2( 1lim fydsssMyi
iT
iT
s
T
(6.14)
Now the integral on s is a continuous function of y and therefore the left hand side without the limit
notation can be rewritten as:
dwdyssMyyT
T
s )()()()2(0
1
, )0,( Tiws
Since )(y is of bounded support and the integrand is a continuous function of (y,w), the order of
integration may be changed. This yields,
T
T
ss dydwyyxuKufxsM00
1 )()(),()()2(
which by Lemma 6.2, is equal to
T
T
ss dydwyyuuf0
11 )(),()2(
provided ,0)1Re( s Req>0, 11or 1 qq and 0)( ac .
By Lemma 6.3,
T
T
ss dydwyyuuf0
11 )(),()2( =
T
T
ss dydwyyuuf0
11 )()2(),( .
The order of integration for the repeated integral herein may be changed because again
)(y is of bounded support and the integrand is continuous function of (y,w).
Upon doing this, we obtain
0
11 )()2(),(T
T
ss dwdyyuyuf
0
1
)/log(.
))/log(sin(.)/)(()(),( dy
yuu
yuryuyuf .
The last expression tends to )(),( uuf as T because )(, IHf and according to Lemma
6.4, the testing function in the last expression converges to )(u in )(, IH . This completes the proof.
Theorem 6.6 Uniqueness Theorem.
Let )()( ,, fFsF caq for fs Re0 and )()( ,, gFsG caq
for gs Re0 and let
F’(s)=G’(s) for ).,min(Re0 gfs Then in the sense of equality in )(ID , f = g.
Proof. For any arbitrary )(ID ,
iT
iT
k
T dkykkMigf
,)()()2(, 1lim
where
0
0)]()([)( dssGsFsk k
since F’(s)=G’(s) for ).,min(Re0 gfs Hence 0, gf for )(ID .
Hence f = g in the sense of equality in )(ID . This completes the proof.
7. Application: (Distributional solution to a class of differential equations).
The distributional generalized modified Struve transform can be used to solve certain boundary value
problems. From Property 4.4, we see that the operator ...)3,2,1,0(, rAt r
tq
r is a continuous linear
mapping of )(, IH into itself. Its adjoint operator r
tqB , is a continuous linear mapping of )(', IH
into itself and is defined by
r
tq
rr
tq AtffB ,, ,,
(7.1)
So we see that
)(),(
)(
)())}(
)(
)({(
)2(
)2(
,
)2(
)2(
,
'
,, stKtfc
acBsf
c
acBF
r
rr
tq
r
rr
tqcaq, by (5.5.1)
)(
)(
)(),( ,
)2(
)2(stKAt
c
actf r
tq
r
r
r
);,(
)(
)(),( ,
)2(
)2(stcaeAt
c
actf qstr
tq
r
r
r
);2,(..).(
)(
)(),(
)2(
)2(strcaest
c
actf qstr
r
r
)()}()( 2,, stfFst rcaq
r
Thus ))(
)((
)2(
)2(
,
'
,, fc
acBF
r
rr
tqcaq
)('
2,, fF rcaq
(7.2)
We can exploit the relation (7.2) to solve a differential equation, with certain boundary conditions, of
course, of the type gfc
acB
r
rr
tq
)2(
)2(
,)(
)(
(7.3)
where g is a known generalized function belonging to )(', IH and is to be determined.
On applying distributional generalized modified Struve transform to (7.3) and using (7.2), we get
)(2,, fF rcaq = )()(,, sGgF caq , say. Hence )]([)( 1
2,, sGFf rcaq
which gives a solution of (7.3); where dsysMsi
sGF siT
iTTrcaq
)()(
2
1)]([)(
lim1
2,,
where 11)1( )];;,1()][1(/[)( qcasBsqs s
11)1( )1(;;,1)]1(/)1([ qcacsBsq s , ( 11 q )
and
0
,, )()( dyyFysM caq
s ; iTs ; be a real with f
and )(),()(,, stKtfsF caq .
Here we have not given the form fDttDtftB r
tq
221
, )2/7()2/3()( r
tqB , . However, these
can be calculated by the method of integration by parts. For r = 1, r
tqB , is given by
ftDDtfB r
tq
21
, )2/7()2/3()(
Or fDtDt )( 2/32/1
which is the adjoint operator of r
tqtA , and for r = 1 is given by
fDttDftAr
tq })2/1{()( 22
, fDDtt )( 2/12/3 .
-------------------------
Acknowledgement: Author is thankful to my Ph.D. guide Dr. Malgonde. S. P. for his useful suggestions
to improve the presentation of this paper. The author also wish to express his sincere thanks to U. G.
C.(WRO) Pune, India for it’s financial assistance during the preparation of this paper and the Principal,
Karmveer Bhaurao Patil Mahavidyalaya, Pandharpur for his full support to carry out this research
work.
---------
REFERENCES
1. Pathak, R.S. [1975],
Generalized modified Struve transform, Portugaliae Mathematica, vol.34, Fasc.4,
225-231.
2. Babister, A.W. [1967],
Transcendental Functions Satisfying N on-homogeneous Linear Differential
Equations, The MacMillan Company, New York.
3. Erdelyi, A. [1953],
Higher Transcendental Functions,Vol.II, McGraw-Hill Book Company Inc. N.Y.
4. Zemanian, A. H. [1968],
Generalized Integral Transformations, Interscience, Pub. N.Y., (Republished by
Dover, N. Y.).
*****
