Available online a t www.scholarsresearchlibrary.com Scholars Research Library Archives of Applied Science Research, 2012, 4 (2):1117-1134 (http://scholarsresearchlibrary.com/archive.html) ISSN 0975-508X CODEN (USA) AASRC9 Generalizsd finite mellin integral transforms S. M. Khairnar, R. M. Pise*and J. N. Salunke** Deparment of Mathematics, Maharashtra Academy of Engineering, Alandi, Pune, India * R.G.I.T. Versova, Andheri (W), Mumbai, India **Department of Mathematics, North Maharastra University, Jalgaon-India ______________________________________________________________________________ ABSTRACT Aim of this paper is to see the generalized Finite Mellin Integral Transforms in [0,a] , which is mentioned in the paper , by Derek Naylor (1963) [1] and Lokenath Debnath [2]..For f(x) in 0<x< ∞ ,the Generalized ∞ ∞ ∞ ∞ Mellin Integral Transforms are denoted by M − [ f( x ), r] = F− ()r and M + [ f( x ), r] = F+ ()r .This is new idea is given by Lokenath Debonath in[2].and Derek Naylor [1].. The Generalized Finite Mellin Integral Transforms are extension of the Mellin Integral Transform in infinite region. We see for f(x) in a a a a 0<x<a, M − [ f( x ), r] = F− ()r and M + [ f( x ), r] = F+ ()r are the generalized Finite Mellin Integral Transforms . We see the properties like linearity property , scaling property ,power property and 1 1 propositions for the functions f ( ) and (logx)f(x), the theorems like inversion theorem ,convolution x x theorem , orthogonality and shifting theorem. for these integral transforms. The new results is obtained for Weyl Fractional Transform and we see the results for derivatives differential operators ,integrals ,integral expressions and integral equations for these integral transforms. Application of these transforms are for the summation of the series and solution of the Cauchy’s linear differential equation. Keywords: Integral transforms, Mellin integral transform, Finite Mellin integral transform. AMS MTHEMATICS CLASSIFICATION- 44A10 (200), 47D03(2001) ______________________________________________________________________________ INTRODUCTION In the theory of Integral Transform , Mellin Integral Transform has presented a direct and systematic technique, for resolution of certain types of classical boundary and initial value problems .To be successful the transform must be adopted to the form of differential operators to be eliminated as well as to the range of interest and associated boundary conditions. These transforms are the extension of the Mellin Integral Transform and have a similar inversion formulae. These transforms are suited to regions bounded by the natural coordinate surfaces of a cylindrical or spherical coordinate system and apply to finite or infinite regions bounded internally. 1117 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______________________________________________________________________________ Historically ,Riemann (1876) first recognized the Mellin transform in the famous memoir on prime numbers, Its explicit formulation was given by Cahen (1894) . Almost simultaneously Mellin (1896, 1902) gave an elaborate discussion of the Mellin transform and its inversion formula A method is prescribed for generating such transforms . .The method is adopted for the GMIT in [0, ∞ ] for the infinite interval x>0 and the result is modified for a finite interval 0<x<a. We see the properties for GFMIT in [0,a] like linearity property , scaling property ,power property and propositions for the 1 1 functions f ( ) and (logx)f(x) .Also we see the theorems for this integral transforms like inversion x x theorem ,convolution theorem , orthogonality and shifting theorem. The new result is obtained for Weyl transform. and other results are obtained for derivatives , differential operators ,integrals ,integral expressions and integral equations by using the GFMIT in [0,a]. Application of this transforms is for the summation of the series and solution of the Cauchy’s linear differential equation. 2.2.2. Basic Results The generalized Mellin integral transform of a function f(x) in 0<x< ∞ , introduced by the integral ∞ 2r ∞ ∞ r−1 a = = − , r>0 M − [ f( x ), r] F− ()r ∫ (x r+1 ) f() x dx 0 x The inverse of this transform is ∞1 ∞− 1 − ∞ M [ f( x ),r] = F 1 ()r =f(x )= x r M ()r dr − − π ∫ − 2 i L ∞ ∞ Where L is the line Re p=c and M − [ f( x ),r] = F− ()r is regular function on the strip Re( p) < γ with c< γ . On the other hand , if the derivative of f(x) is prescribed at r=a , it is convenient define the associated integral transform by ∞ 2r ∞ ∞ r−1 a = = + , r>0 M + [ f( x ),r] F+ ()r ∫ (x r+1 )()f x dx 0 x The inverse of this transform is ∞−1 ∞−1 1 − ∞ M [ f( x ),r] = F ()r =f(x )= x r M ()r dr + + π ∫ + 2 i L ∞−1 ∞−1 Where L is the line Re p=-c and M + [ f( x ),r] = F+ ()r is regular function on the strip Re( p) < γ with c< γ . If the range of the integral is finite, then we define the generalized Mellin integral transform by a 2r a a r−1 a = = − , where Re(p)< γ (1) M − [ f( x ),r] F− ()r ∫ (x r+1 )()f x dx 0 x The corresponding inverse transform is −1 −1 1 − M a [ f( x ),r] = F a ()r =f(x )= x r M a ()r dr (2) − − π ∫ − 2 i L Similarly we define the generalized finite Mellin integral transform pair by a 2r − a a = a = r 1 + ,r>0 (3) M + [ f( x ),r] F+ ()r ∫ (x r+1 )()f x dx 0 x The inverse of the corresponding transform is given by 1118 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______________________________________________________________________________ −1 −1 1 − ∞ M a [ f( x ),r] = F a ()r =f(x)= x r M ()r dr (4) + + π ∫ − 2 i L where line L is from c-i a to c +i a 2.2.3.Lemma 2.2.3.1. Linearity Property The GFMIT of a function f(x) in 0<x<a , introduced by the integral a 2r − a a = a = r 1 − , M − [ f( x ), r] F− ()r ∫ (x r+1 )()f x dx 0 x then for the constants α and β ,we have Linearity Property a a a M − [αf() x + βg(x ), r] =α M − [f ( x ),r],+ β M − [g (x ),r] (5a) a a a Similarly M + [αf() x + βg(x ), r] =α M + [f ( x ),r],+ β M + [g (x ),r] (5b) 2.2.3.2. Scaling Property The GFMIT of a function f(x) in 0<x<a , introduced by the integral a 2r − a a = a = r 1 − , then M − [ f( x ), r] F− ()r ∫ (x r+1 )()f x dx 0 x a 2r − a a = r 1 − M − [ f( x ), r] ∫ (x r+1 )(f bx)dx 0 x t dt substitute bx=t , x= ,dx= , if x=0 then t=0 and if x=1 then t= ab b b ab 2r a t r−1 − a dt M − [(f bx), r]= ∫ (( ) )()f t b t r+1 b 0 () b ab 2r 1 − (ab ) = (t r 1 − ) f t)( dt r ∫ r+1 b 0 t a ab M − [(f bx), r]= M − [ f( t ),r] (6a) a ab Similarly M + [(f bx), r]= M + [ f( t ),r] (6b) 2.2.3.3. Power Property The GFMIT of a function f(x) in 0<x<a , introduced by the integral a 2r − a a = a = r 1 − , then M − [ f( x ),r] F− ()r ∫ (x r+1 ) f() x dx 0 x a 2r a n r−1 a n = − M − [ f( x ), r] ∫ (x r+1 )(f x )dx 0 x 1 1 1 −1 substitute x n = t , x = t n ,dx= t n dt , if x=0 the t=0 and if x=a then t= a n n 1119 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______________________________________________________________________________ an 1 1 2r −1 a n n r−1 a 1 n M − [ f( x ), r]= (t ) − ) f() t t dt ∫ 1 n 0 (t n ) r+1 an r 2r −1 1 n a = − ∫ (t r ) f() t dt n +1 0 t n 1 an r = M − [f ( t ), ] n n a n 1 an r M − [ f( x ), r]= M − [f ( t ), ] (7a) n n a n 1 an r Similarly M + [ f( x ), r]= M + [f ( t ), ] (7b) n n 2.2.4. Theorems 2.2.4.1. Convolution Theorem The GFMIT of a function f(x) in 0<x<a , introduced by the integral a 2r − a a = a = r 1 − , then M − [ f( x ), r] F− ()r ∫ (x r+1 )()f x dx 0 x c+ ia 1 − M a [ f()( x g t − x),r]= x r M a [ f( x ),r]M a [(g t − x),r] dr (8a) − π ∫ − − 2 i c− ia c+ ia 1 − Similarly M a [ f()( x g t − x),r]= x r M a [ f( x ),r]M a [(g t − x),r] dr (8b) + π ∫ + + 2 i c− ia 2.2.4.2. Orthogonality ( Parsevals Theorem) The GFMIT of a function f(x) in 0<x<a , introduced by the integral a 2r − a a = a = r 1 − , then M − [ f( x ),r] F− ()r ∫ (x r+1 )()f x dx 0 x c+ ia 1 − M a [ f( x ) g (x ),r]= x r M a [f ( x ),r]M a [g (x ),r] dr (9a) − π ∫ − − 2 i c− ia c+ ia 1 − Similarly M a [ f( x ) g (x ),r]= x r M a [f ( x ),r]M a [g (x ),r] dr (9b) + π ∫ + + 2 i c− ia 2.2.4.3 .Shifting Theorem The GFMIT of a function f(x) in 0<x<a , introduced by the integral a 2r a a r−1 a = = − , then M − [f ( x ),r] F− ()r ∫ (x r+1 ) f() x dx 0 x a n a M − [x f( x ),r] = M − [ f( x ),r + n] a 2r − a Proof If a = a = r 1 − , then M − [ f( x ),r] F− ()r ∫ (x r+1 )()f x dx 0 x 1120 Scholars Research Library S.