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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
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
2.2.2. Basic Results The generalized Mellin integral transform of a function f(x) in 0
∞−1 ∞−1 Where L is the line Re p=-c and M + [ xf ),( 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 = a = r 1 − , where Re(p)< γ (1) M − [ xf ),( r] F− r)( ∫ (x r+1 )() dxxf 0 x
The corresponding inverse transform is −1 −1 1 − M a [ xf ),( r] = F a r)( =f(x )= x r M a )( drr (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 + [ xf ),( r] F+ r)( ∫ (x r+1 )() dxxf 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 [ xf ),( r] = F a r)( =f(x)= x r M )( drr (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 a a a M − [α xf )( + βg x),( r] =α M − xf ),([ r],+ β M − g x),([ r] (5a) a a a Similarly M + [α xf )( + βg x),( r] =α M + xf ),([ r],+ β M + g x),([ r] (5b) 2.2.3.2. Scaling Property The GFMIT of a function f(x) in 0 a ab Similarly M + ([ bxf ), r]= M + [ tf ),( r] (6b) 2.2.3.3. Power Property The GFMIT of a function f(x) in 0 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 − [ (xf ), r]= (t ) − ) tf )( t dt ∫ 1 n 0 (t n ) r+1 an r 2r 1 −1 a = n − ∫ (t r ) )( dttf n +1 0 t n 1 an r = M − tf ),([ ] n n a n 1 an r M − [ (xf ), r]= M − tf ),([ ] (7a) n n a n 1 an r Similarly M + [ (xf ), r]= M + tf ),([ ] (7b) n n 2.2.4. Theorems 2.2.4.1. Convolution Theorem The GFMIT of a function f(x) in 0 +iac 1 − Similarly M a [ ()( tgxf − x), r]= x r M a [ xf ),( r]M a ([ tg − x), ]drr (8b) + π ∫ + + 2 i −iac 2.2.4.2. Orthogonality ( Parsevals Theorem) The GFMIT of a function f(x) in 0 +iac 1 − Similarly M a [ gxf x),()( r]= x r M a xf ),([ r]M a g x),([ ]drr (9b) + π ∫ + + 2 i −iac 2.2.4.3 .Shifting Theorem The GFMIT of a function f(x) in 0 a n = a + M − [ xfx ),( r] M − [ xf ),( r n] a 2r − a Proof If a = a = r 1 − , then M − [ xf ),( r] F− r)( ∫ (x r+1 )() dxxf 0 x 1120 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a 2r − a a n = r 1 − n M − [ xfx ),( r] ∫ (x r+1 ) )( dxxfx 0 x a 2r −+ a = nr 1 − ∫ (x nr +− 1 )() dxxf 0 x = a + M − [ xf ),( r n] a n a M − [ xfx ),( r] = M − [ xf ),( r + n] (10a) Similarly a n = a + (10b) M + [ xfx ),( r] M + [ xf ),( r n] 2.2.5. Propositions 1 1 2.2.5.1. Proposition for the function f ( ) x x The GFMIT in [0,a] is a 2r − a a = a = r 1 − , where Re(p)< γ , then M − [ xf ),( r] F− r)( ∫ (x r+1 ) )( dxxf 0 x a 2r 1 1 − a 1 1 a = r 1 − M − [ f ( ), r] ∫ (x r+1 ) f ( )dx x x 0 x x x 1 1 − dt 1 substitute =t ,x= , dx= . if x=0 the t= ∞ and if x=a then t= x t t 2 a 1 a 2r 1 1 1 − a − dt a = r 1 − M − [ f ( ), r] ∫ (( ) )tf t)(( 2 ) x x t 1 r+1 t ∞ ( ) t ∞ 2r +− a dt = r 1 − ∫ (t r−− 1 tf )(() ) 1 t t a ∞ 2r − a = r − ∫ (t −r )() dttf 1 t a ∞ 2r −+− a = r 11 − ∫ (t r +−− 11 )() dttf 1 t a ∞ 2r 1 1 −−− a a = r 1)1( − M − [ f ( ), r] ∫ (t r ++− 1)1( ) )( dttf x x 1 t a ∞ 2r a 1 1 ∞ − + r −−− 1)1( − a M − [ f ( ), r] = M 1 [ tf ),( r ]1 = (t ++− ) )( dttf (11a) −, ∫ r 1)1( x x a 1 t a 1121 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ ∞ 2r a 1 1 ∞ − + r −−− 1)1( + a Similarly M + [ f ( ), r] = M 1 tf ),([ r ]1 = (t ++− ) )( dttf (11b) +, ∫ r 1)1( x x a 1 t a 1 Here, 11a and 11b are the new integral transforms in to ∞ a 2.2.5.2. Proposition for the function (logx)f(x) The GFMIT in [0, a] is a 2r − a a = a = r 1 − , where Re(p)< γ , M − [ xf ),( r] F− r)( ∫ (x r+1 ) )( dxxf 0 x For a=1 , we have the particular case of (1) 1 − 1 1 = 1 = r 1 − , then M − xf ),([ r] F− r)( ∫ (x r+1 ) )( dxxf 0 x 1 − 1 1 = r 1 − M − [(log ) xfx ),( r] ∫ (x r+1 )(log )() dxxfx 0 x 1 2r − a = r 1 − , ∫ [(log )xx (log x) r+1 ) )( dxxf 0 x 1 d − d −− = ∫ [ (x r 1 ) + (x r 1 )() dxxf 0 dr dr 1 d − 1 = r 1 + ∫ (x r+1 ) )( dxxf dr 0 x d 1 d 1 = M + xf ),([ r] = F+ r)( dr dr 1 d 1 d 1 M − [(log xfx ),() r]= M + xf ),([ r] = F+ r)( (12a) dr dr 1 1 d 1 1 d 1 Similarly M + [(log xfx ),() r] = M + xf ),([ r] = F+ r)( M − xf ),([ r] = F− r)( (12b) dr dr 2.2.6. GFMIT of Differential Operators The GFMIT in [0, a] is a 2r − a a = a = r 1 − , where Re(p)< γ ,then M − [ xf ),( r] F− r)( ∫ (x r+1 ) )( dxxf 0 x a 2r d − a d a = r 1 − M − [( x ) xf ),( r] ∫ (x r+1 )( x )() dxxf dx 0 x dx a 2r − a = r 1 − ∫ (x r+1 )xf )(' dxx 0 x a a 2r = r − ∫ (x r f )(') dxx 0 x 1122 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ 2r a 2r a − a = r − a - r 1 − − [( x r xf )]() 0 ∫ (rx r+1 ( r)) )( dxxf x 0 x 2r a 2r a − a = r − - r 1 + [( a af )]() r∫ (x r+1 ) )( dxxf a 0 x a =0-r M + [ xf ),( r] a d a M − [( x ) xf ),( r] =- r M + [ xf ),( r] (13a) dx a d r a Similarly M + [( x ) xf ),( r] = 2a af )( - rM − xf ),([ r] (13B) dx a 2r d − a d a 2 = r 1 − 2 M − [( x ) xf ),( r] ∫ (x r+1 )( x ) )( dxxf dx 0 x dx a 2r − a = r 1 − 2 + ∫ (x r+1 )( fx x)('' xf x))(' dx 0 x a 2r a 2r − a − a = r 1 − 2 + r 1 − ∫ (x r+1 ) fx )('' dxx ∫ (x r+1 )xf )(' dxx 0 x 0 x a 2r + a = r 1 − - r a ( from 1 3a) ∫ (x r−1 f )('') dxx M + [ xf ),( r] 0 x 2r a 2r + a a = r 1 − a - + r − − + - r a [( x r−1 f x)](') 0 ∫ (( r )1 x r ( r ))1 f )(' dxx M + [ xf ),( r] x 0 x 2r a 2r 2r + a a a = r 1 − - (rx r + x r + r − f )(') dxx - r a (a r−1 f a)](') ∫ r r M + [ xf ),( r] a 0 x x a 2r a 2r + + a a = r 1 − r 1 - r + + r − - r a (a a f a)](') r∫ (x r f )(') dxx ∫ (x r f )(') dxx M + [ xf ),( r] 0 x 0 x 2r a 2r a − a =0- r + a + r (rx r 1 + (−r)) )( dxxf + r a r[( x r xf )]() 0 ∫ r+1 M + [xf x),(' r] x 0 x a -r M + [ xf ),( r] a 2r − a =0- r + r + 2 r 1 − r[( x a ) af )( r ∫ (x r+1 ) )( dxxf 0 x r 2 a =- 2ra f(a)+ r M − xf ),([ r] 2 a r = r M − xf ),([ r] - 2ra f(a) a d 2 2 a r M − [( x ) xf ),( r] = r M − xf ),([ r] - 2ra f(a) (14a) dx a d 2 2 a r−1 Similarly M + [( x ) xf ),( r] = r M + xf ),([ r] + 2a f a)(' (14b) dx 1123 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ 2.2.7. GFMIT of Integral Operators a − 2.2.7.1. The Integral Expression is ∫ a r (xtf )() dttg 0 The GFMIT in [0, a] is a 2r − a a = r 1 − , where Re(p)< γ ,then M − [ xf ),( r] ∫ (x r+1 )() dxxf 0 x a a 2r a − − a − a r = r 1 − r M − [∫ a (xtf ), )( dttg r], ∫ (x r+1 )[ ∫ a (xtf )() dttg ]dx 0 0 x 0 a a 2r − − a = r r 1 − ∫ a )( dttg ∫ (x r+1 () xtf )dx 0 0 x z dz substitute xt=z then x= , dx= ,if x=0 then z=0 and x=a then z=at t t a at 2r − z − a dz = ∫ a r )( dttg ∫ (( )r 1 − zf )() t z r+1 t 0 0 )( t a at 2r − − − (at ) = rr r 1 − ∫ a t )( dttg ∫ (z r+1 )() dzzf 0 0 z a at 2r − −− − (at ) = r 1 r 1 r 1 − ∫ a t )( dttg ∫ (z r+1 )() dzzf 0 0 z a − at = M 0 tg 1),([ r] M − zf ),([ r] a a −r a − at M − [∫ a (xtf ), )( dttg r], = M 0 tg 1),([ r] M − zf ),([ r] (15a) 0 a a −r a − at Similarly M + [∫ a (xtf ), )( dttg r], = M 0 tg 1),([ r] M + zf ),([ r] (15b) 0 a a −r µ a − + µ at M − [∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M − zf ),([ r] (15c) 0 a a −r µ a − + µ at M + [∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M + zf ),([ r] (15d) 0 a a λ −r µ λ a − + µ at + λ M − [x ∫ a xt (xtf ), )( dttg r], = M 0 tg 1),([ r ] M − [ zf ),( r ] (15e) 0 a a λ −r µ λ a − + µ at + λ M + [x ∫ a xt (xtf ), )( dttg r], = M 0 tg 1),([ r ] M + [ zf ),( r ] (15e) 0 a − 2.2.7.2. The Integral Expression is ∫ a r (xtf )() dttg 0 The GFMIT in [0, a] is 1124 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a 2r − a a = r 1 + , where Re(p)< γ ,then M + [ xf ),( r] ∫ (x r+1 )() dxxf 0 x a a 2r a − − a − a r = r 1 + r M + [∫ a (xtf ), )( dttg r], ∫ (x r+1 )[ ∫ a (xtf )() dttg ]dx 0 0 x 0 a a 2r − − a = r r 1 + ∫ a )( dttg ∫ (x r+1 () xrf )dx 0 0 x z dz substitute xt=z then x= , dx= ,if x=0 then z=0 and x=a then z=at t t a − = ∫ a r )( dttg 0 a at 2r − − − (ar ) = rr r 1 + ∫ a t )( dttg ∫ (z r+1 ) )( dzzf 0 0 z a at 2r − −− − (ar ) = r 1 r 1 r 1 + ∫ a t )( dttg ∫ (z r+1 ) )( dzzf 0 0 z a − ar = M 0 tg 1),([ r] M + [ zf ),( r] a a −r a − ar M + [∫ a (xtf ), )( dttg r], = M 0 tg 1),([ r] M + [ zf ),( r] (16a) 0 a a −r a − ar M − [∫ a (xtf ), )( dttg r], = M 0 tg 1),([ r] M − [ zf ),( r] (16b) 0 a a −r µ a − + µ ar Similarly M = [∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M − [ zf ),( r] ;’ (16c) 0 a a −r µ a − + µ ar M + [∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M + [ zf ),( r] (16d) 0 a a λ −r µ a − + µ ar + λ M = [x ∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M − [ zf ),( r ] (16e) 0 a a λ −r µ a − + µ ar + λ M + [x ∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M + [ zf ),( r ] (16f) 0 a − x 2.2.7.3. The Integral Expression is ∫ a r f )()( dttg 0 t The GFMIT in [0, a] is a 2r − a a = r 1 − , where Re(p)< γ ,then M − [ xf ),( r] ∫ (x r+1 )() dxxf 0 x a a 2r a − x − a − x a r = r 1 − r M − [∫ a f ),( )( dttg r], ∫ (x r+1 )[ ∫ a f )()( dttg ]dx 0 t 0 x 0 t 1125 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a a 2r − − a x = r r 1 − ∫ a )( dttg ∫ (x r+1 f )() dx 0 0 x t x a substitute =z then x= tz , dx= tdz ,if x=0 then z=0 and x=a then z= t t a a t 2r − − a = r r 1 − ∫ a )( dttg ∫ (( tz ) r+1 )tf z)( tdz 0 0 tz )( a a 2r a t )( − − = rr r 1 − t ∫ a )( dttgt ∫ (z r+1 )() dzzf 0 0 z a a 2r a t )( − −+ − = rr 11 r 1 − t ∫ a t )( dttg ∫ (z r+1 ) )( dzzf 0 0 z a a + t = M 0 tg ),([ r ]1 M − [ zf ),( r] a a − x a r a + t M − [∫ a f ),( )( dttg r], = M 0 tg ),([ r ]1 M − zf ),([ r] (17a) 0 t a a − x a r a + t Similarly M + [∫ a f ),( )( dttg r], = M 0 tg ),([ r ]1 M + zf ),([ r] (17b) 0 t a a −r µ a + µ + at M − [∫ a (xtft ), )( dttg r], = M 0 tg ),([ r ]1 M − zf ),([ r] (17c) 0 a a −r µ a + µ + at M + [∫ a (xtft ), )( dttg r], = M 0 tg ),([ r ]1 M + zf ),([ r] (17d) 0 a a λ −r µ a + + µ at + λ M − [x ∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M − [ zf ),( r ] (17e) 0 a a λ −r µ a + + µ at + λ M + [x ∫ a (xtft ), )( dttg r], = M 0 tg 1),([ r ] M + [ zf ),( r ] (17f) 0 a − x 2.2.7.4. The GFMIT Of Integral Expression is ∫ a r f )()( dttg 0 t The GFMIT in [0, a] is a 2r − a a = r 1 + , where Re(p)< γ ,then M + [ xf ),( r] ∫ (x r+1 )() dxxf 0 x a a 2r a − x − a − x a r = r 1 + r M + [∫ a f ),( )( dttg r], ∫ (x r+1 )[ ∫ a f )()( dttg ]dx 0 t 0 x 0 t 1126 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a a 2r − − a x = r r 1 + ∫ a )( dttg ∫ (x r+1 f )() dx 0 0 x t x a substitute =z then x= tz , dx= tdz ,if x=0 then z=0 and x=a then z= t t a a t 2r − − a = r r 1 + ∫ a )( dttg ∫ (( tz ) r+1 ) zf )( tdz 0 0 tz )( a a 2r a t )( − − = rr r 1 + t ∫ a )( dttgt ∫ (z r+1 )() dzzf 0 0 z a a 2r a t )( − −+ − = rr 11 r 1 + t ∫ a t )( dttg ∫ (z r+1 )() dzzf 0 0 z a a + t = M 0 tg ),([ r ]1 M − [ zf ),( r] a a − x a r a + t M − [∫ a f ),( )( dttg r], = M 0 tg ),([ r ]1 M + zf ),([ r] (18a) 0 t a a − x a r a + t Similarly M + [∫ a f ),( )( dttg r], = M 0 tg ),([ r ]1 M − zf ),([ r] (18b) 0 t a a a −r µ a + µ + t M + [∫ a (xtft ), )( dttg r], = M 0 tg ),([ r ]1 M − zf ),([ r] (18c) 0 a a a −r µ a + µ + t M − [∫ a (xtft ), )( dttg r], = M 0 tg ),([ r ]1 M + zf ),([ r] (18d) 0 a a a λ −r µ λ a + + µ t + λ M + [x ∫ a xt (xtf ), )( dttg r], = M 0 tg 1),([ r ] M − zf ),([ r ] (18e) 0 a a a λ −r µ λ a + + µ t + λ M − [x ∫ a xt (xtf ), )( dttg r], = M 0 tg 1),([ r ] M + zf ),([ r ] (18f) 0 2.2.8. GFMIT of Integral Equations 2.2.8.1. The GFMIT in [0,a] is a 2r − a a = r 1 − M − [ xf ),( r] ∫ (x r+1 )() dxxf 0 x a − The integral equation is ∫ a r tf )( K(xt )dt = tg )( , 0 a a 2r a − − a − a r = r 1 − [ r dx = a M − [∫ a tf )( K(xt )dt r], ∫ (x r+1 ) ∫ a tf )( K(xt )dt ] M + tg ),([ r] 0 0 x 0 1127 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a 2r a − a − r 1 − r dx= a ∫ (x r+1 ) ∫ a tf )( K(xt )dt ] M − tg ),([ r] 0 x 0 a a 2r − − a r r 1 − dx= ∫ a )( dttf ] ∫ (x r+1 )K(xt ) rg )( 0 0 x z dz substitute xt=z , x= ,dx= , if x=0 then z=0 and if x=a then z=at t t a at 2r − z − a dz ∫ a r )( dttf ] ∫ )( r 1 − )K z)( = rg )( t z r+1 t 0 0 )( t a at 2r − − − (at ) rr r 1 − = ∫ a t )( dttf ] ∫ (z r+1 )K )( dzz rg )( 0 0 z a at 2r − −− − (at ) r 1 r 1 r 1 − = ∫ a t )( dttf ] ∫ (z r+1 )K )( dzz rg )( 0 0 z at at f 1( − r) K − r)( = rg )( , similarly f 1( − r) K + r)( = rg )( replace ( r) by (1-r) at rf )( K − 1( − r) = g 1( − r) g 1( − r) 1 1 rf )( = , where rL )( = and rL )( = at at at K − 1( − r) K − 1( − r) K + 1( − r) rf )( = g 1( − r) rL )( , then the solution is a − f(x)= ∫ a r Ltg ()( xt )dt (19) 0 2.2.8.2. The GFMIT in [0,a] is a 2r − a a = r 1 − M − [ xf ),( r] ∫ (x r+1 ) )( dxxf 0 x 1 2 If K(x)= x Yv x)( , where Yv x)( is the Bessel function of second kind of order v. 1 2 and L(x)= x H v x)( , where H v x)( is the Struvi’s function a 1 −r 2 = The integral equation is ∫ a (xt ) )( Ytf v (xt )dt tg )( 0 The GFMIT in [0,a] of this integral equation is a 1 a −r 2 M − [∫ a (xt ) )( Ytf v (, xt )dt r], 0 a 2r a 1 − a − = r 1 − r 2 = a ∫ (x r+1 )[ ∫ a (xt ) )( Ytf v (xt )dt ]dx M − g x),([ r] 0 x 0 1128 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a a 2r 1 − − a r r 1 − 2 = ∫ a )( dttf ∫ (x r+1 )( xt ) Yv (xt )dx rg )( 0 0 x z dz substitute xt=z then x= , dx= , if x=0 then z=0 and if x=a then z=at t t a at 2r 1 − z − a dz r r 1 − 2 = ∫ a )( dttf ∫ (( ) )( z) Yv z)( rg )( t z r+1 t 0 0 )( t a at 2r 1 − − − (at ) rr r 1 − 2 = ∫ a t )( dttf ∫ (z r+1 )( z) Yv )( dzz rg )( 0 0 z a at 2r − − − (at ) rr r 1 − = ∫ a t )( dttf ∫ (z r+1 )K v )( dzz rg )( 0 0 z at at f 1( − r) K − r)( = rg )( , similarly f 1( − r) K + r)( = rg )( replace (r ) by (1-r) at rf )( K − 1( − r) = g 1( − r) g 1( − r) 1 1 = , where = and = rf )( at H r)( at H r)( at K − 1( − r) K − 1( − r) K + 1( − r) rf )( = g 1( − r) H r)( , then its solution is a 1 −r 2 f(x) = ∫ a (xt ) tg )( H v (xt )dt (20) 0 2.2.8.3. The GFMIT in [0,a] is a 2r − a a = r 1 − M − [ xf ),( r] ∫ (x r+1 ) )( dxxf 0 x a 1 −r 2 = The integral equation defined by g(x)= ∫ a (xt ) )( Ytf v (xt )dt tg )( is called 0 the Y-transform of f(x) , then we write g(t)= Yv [ xf t]),( a 1 −r 2 Similarly the integral equation defined by f(x)= ∫ a (xt ) tg )( H v (xt )dt , is called the H-transform of 0 * −= * g(t) , we write f(x)= H v tg ),([ x] . Here H v H v . a −r x dy = The integral equation is ∫ a K1 ( ) f1 y)( g1 x)( .\Using GFMIT in [0,a] is 0 y y a a −r x dy a M − [∫ a K1 f ( () 1 yxf )( ), r]= M − [g1 x),( r] 0 y y a 2r a − a − x dy r 1 − r = ∫ (x r+1 )dx ∫ a K1 ( ) f1 y)( g1 r)( 0 x 0 y y 1129 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ a a 2r − dy − a x r r 1 − = ∫ a f1 y)( ∫ (x r+1 )K1 ( )dx g1 r)( 0 y 0 x y x a substitute = z , x=yz , dx=ydz , if x=0 then z=0 and if x=a then z= y y a a y 2r − dy − a r r 1 − = ∫ a f1 y)( ∫ (( yz ) r+1 )K1 z)( ydz g1 r)( 0 y 0 (yz ) a a y 2r − − dy − (ay 0 rr 1 r 1 − = ∫ a y f1 y)( ∫ (z r+1 )K1 )( dzz g1 r)( 0 y 0 z a t f1 r)( K1 r)( = = g1 r)( , 1 1 f r)( = g r)( , where L r)( = 1 1 a 1 a t t K1− r)( K1− r)( f1 r)( = g1 r)( L1 r)( , its solution is a −r x f1 (x)= ∫ a g1 )( Lt 1 ( )dt (21) 0 y 2.2.9. GFMIT of Weyl Fractional Transform The Weyl fractional transform of the function f(t) is denoted by −α W [ tf )]( = F x α),( and defined as ∞ −α 1 α − W [ tf )]( = F x α),( = (t − x) 1 )( dttf ,0 The GFMIT in [0, a] is a 2r − a a = a = r 1 − , where Re(p)< γ , then M − [ xf ),( r] F− r)( ∫ (x r+1 ) )( dxxf 0 x a 2r ∞ − a 1 α − M a [F x α),,( r] = (x r 1 − )[ (t − x) 1 )( dttf ]dx − ∫ r+1 ∫ Γ α 0 x x )( ∞ a 2r 1 α − − a = (t − x) 1 )( dttf (x r 1 − )dx Γ α ∫ ∫ r+1 )( x 0 x ∞ r 2r 1 α − x a = (t − x) 1 )( dttf [ − ]a Γ α ∫ − r 0 )( x r ( )xr ∞ r 2r 1 α − a a = (t − x) 1 )( dttf [ − ] Γ α ∫ − r )( x r ( )ar 1130 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ ∞ 1 α − = (t − x) 1 )( dttf [a r + a r ] Γ α ∫ r )( x ∞ 1 1 α − = 2a r (t − x) 1 )( dttf Γ α ∫ r )( x r 2a a = F− x α),{ r r a 2a a M − [F x α),,( r]= F− x α),{ (22) r 2.2.10. Application of The GFMIT to the Summation of the Series 2.2.10.1. The GFMIT in [0, a] is a 2r − a a = a = r 1 − , where Re(p)< γ , M − [ xf ),( r] F− r)( ∫ (x r+1 ) )( dxxf 0 x then its inversion is +iac 1 − f(x)= x r M a xf ),([ ]drr , exists for r>0 π ∫ − 2 i −iac +iac 1 − f(nx)= (nx ) r M a xf ),([ ]drr , π ∫ − 2 i −iac +iac 1 − − = n r x r M a xf ),([ ]drr π ∫ − 2 i −iac ∞ ∞ +iac 1 − − (nxf ) = n r x r M a [ xf ),( ]drr ∑ π ∑ ∫ − n=0 2 i n=0 −iac +iac ∞ 1 − − = ( n r )x r M a [ xf ),( ]drr π ∫ ∑ − 2 i −iac n=0 +iac 1 −r a = x M − xf ),([ r ξ )(] drr 2πi ∫ −iac ∞ +iac 1 − (nxf ) = x r M a xf ),([ r ξ )(] drr (23a) ∑ π ∫ − n=0 2 i −iac ∞ 1 where is the Riemann Zeta function is denoted by ξ and defined as ∑ r r)( n=0 n ∞ 1 ξ = , Re(r)>1 r)( ∑ r n=0 n ∞ +iac 1 − Similarly (nxf ) = x r M a xf ),([ r ξ )(] drr (23b) ∑ π ∫ + n=0 2 i −iac 1131 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______2.2.10.2. The GFMIT in[0, a] is a 2r − a a = a = r 1 − , where Re(p)< γ , M − [ xf ),( r] F− r)( ∫ (x r+1 ) )( dxxf 0 x then its inversion is +iac 1 − f(x) = x r M a xf ),([ ]drr , exists for r>0 π ∫ − 2 i −iac The Hurvitz Zeta function is denoted by ξ ap ),( and defined as ∞ 1 ξ = ,0 1132 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. Res., 2012, 4 (2):1117-1134 ______ ∞ +iac − 1 − − (− )1 n 1 (nxf ) = 1[ − 21 r ξ )(] xr r M a xf ),([ ]drr (25a) ∑ π ∫ − n=1 2 i −iac ∞ +iac − 1 − − Similarly (− )1 n 1 (nxf ) = 1[ − 21 r ξ )(] xr r M a xf ),( ]drr (25b) ∑ π ∫ + n=1 2 i −iac 2.2.11. Application of the GFMIT to the Cauchy’s Linear Differential Equation The Cauchy’s linear differential equation is 2 fx x)('' + xf x)(' + xf )( = 0 By using GFMIT in [0,a) , we have a 2 M − [ fx x)('' + xf x)(' + xf ),( r] =0 − a rf a)( af a)(' M − [ xf ),( r]= (26a) r 2 − a rf a)( af a)(' Similarly M + [ xf ),( r]= (26b) r 2 +iac 1 − f(x)= x r M a xf ),([ ]drr π ∫ − 2 i −iac +iac 1 − rf a)( − af a)(' f(x)= x r M a [ ]dr (27a) π ∫ − 2 2 i −iac r +iac 1 − rf a)( − af a)(' Similarly f(x)= x r M a [ ]dr (27b) π ∫ + 2 2 i −iac r 2.2.12. Remarks 1. Properties and Theorems of Mellin Integral Transform in [0, ∞ ] are exists for the GFMIT in{0,a] 2.G FMITin[0,a] of derivatives , differential operators ,integrals , integral expressions and integral equations are obtained 1 1 3. Propositions for the functions f ( ) and (logx)f(x) gives new results for the GFMIT in [0, a] x x 4. New result is obtained of Weyl fractional transform by using GFMIT in [0,a] 5.Application of the GFMIT in [0,a] is in summation of the series is discussed. . 6. Using GFMIT in [0,a] solution of the Cauchy’s linear differential Equation is obtained CONCLUSION All the properties and theorems of Mellin integral transform are exists to GFMIT in [0,a]..Results for the derivatives , differential operators, integrals , integral expressions and integral equations are exists for GFMIT in [0,a].This integral transform is applies to obtain the results of summation of series and solution of the Caychy’s linear differential equation. In this paper we get new result of Weyl fractional transform 1 by using GFMIT in [0,a] and new integral transform named by Mellin type integral transform in [ ,∞ ] or a 1 it is also named as Pise-Khairnar integral transform in [ ,∞ ] a 1133 Scholars Research Library S. M. Khairnar et al Arch. Appl. Sci. 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