Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 4 (2017), pp. 1285–1302 © Research India Publications http://www.ripublication.com/gjpam.htm
Study of Generalized Integral Transforms their Properties and Relations
Bhausaheb R. Sontakke Department of Mathematics, Pratishthan College, Paithan, Dist: Aurangabad (M.S.), India.
Govind P. Kamble Department of Mathematics, P.E.S. College of Engineering Nagsenvana, Aurangabad (M.S.), India.
Abstract
In this article, we study the basic theoretical properties of Mellin-type and Weyl fractional integrals and fractional derivatives. Furthermore, we prove some prop- erties of Weyl fractional transform. Also, we study fractional Mellin transform and we prove relation between fractional Mellin transform and Fourier fractional Mellin transform.
AMS subject classification: Keywords: Caputo Fractional derivative, Integral transform, Fractional Mellin transform, Weyl transform, Fractional Fourier transform.
1. Introduction The differential equations have played a central role in every aspects of applied math- ematics for every long time and with the advent of the computer their importance has increased further. It is also well known that throughout science,engineering and far beyond, scientific computation is taking efforts to understand and control our natural environment in order to develop new technological processes.Thus investigation and 1286 Bhausaheb R. Sontakke and Govind P. Kamble analysis of differential equations arising in applications, led to develop different tech- niques to solve differential equations. In order to solve the differential equations,the integral transform were extensively used. Historically, the origin of the integral transform can be traced back to celebrated work of P.S. Laplace (1749–1827). In fact,Laplace classic book on La Theorie Analy- tique des Probabilities included some basic results of the Laplace transform, which was one of the oldest and most commonly used integral transforms available in the mathe- matical literature.In the papers [8, 9, 10], a broad study of fractional Mellin analysis was developed. In which the Hadamard-type integrals are considered and they represent the appropriate extensions of the classical Riemann–Liouville and Weyl fractional integral [11]. In the literature there are numerous integral transforms invented and widely used in physics, astronomy and engineering.Therefore large amount of research work was done on the theory and application of integral transform, such as development of the Laplace transform, Mellin transform, and Hankel transform.These integrals are also connected with the theory of moment operators [12, 13]. On the other hand,Fourier treatise provide the modern mathematical theory of heat conduction, Fourier series and Fourier integrals with applications. In an attempt to ex- tend his ideas Fourier discovered an integral transform known as Fourier transform and the inverse Fourier transform. There are many other integral transformation including the Mellin transform, the Hankel transform, the Hilbert transform, the Steiltjes trans- form and the Radon transform which are widely used to solve initial and boundary value ordinary differential equations and partial differential equations.Recently, fractional in- tegral transform is one of the flourishing fields of active research due to its wide range of applications. It is used in signal processing, image reconstruction,pattern recognition and acoustic signal processing etc. We organize this paper as follows: In second section, we study some definitions of integral transforms and fractional integral and fractional derivatives. Third section is devoted for Weyl fractional derivative and fractional in- tegrals and their integral transforms, also we prove some properties of Weyl fractional derivative and integrals. In section 4, we study fractional Mellin transform and Fourier fractional Mellin transform and their relation.
2. Technical Background
Definition 2.1. (Integral Transform) The integral transform of the function f(x) in a ≤ x ≤ b is defined by F {f(x)}=F(k) and is defined by b F {f(x)}=F(k) = K(x,k)f(x)dx (2.1) a where K(x,k) given function of two variables x and k, is called kernel of the transform. The operator F is usually called an integral transform operator or simply an integral Study of Generalized Integral Transforms their Properties and Relations 1287 transformation. The transform function F(k) is often referred to as the image of the given object function f(x)and k is called the transform variable. Similarly, the integral transform of a function of several variables is defined by F {f(X)}=F(K) = K(X,K)f(X)dx (2.2) S where x = (x1,x2,...,xn), K = (k1,k2,...,kn). Definition 2.2. (Fourier Transform) The Fourier transform of f(x) is denoted by F {f(x)}=F(k),k ∈ R, and is defined by the integral 1 ∞ F {f(x)}=F(k) = √ exp(−ikx)f(x)dx (2.3) 2π −∞ where F is called the Fourier transform operator or the Fourier transformation and k is the Fourier transform variable which is complex number. − The inverse Fourier transform, denoted by F 1{F(k)}=f(x), is defined by ∞ − 1 F 1{F(k)}=f(x)= √ exp(ikx)F(k)dk (2.4) 2π −∞ Definition 2.3. (MellinTransform) The Mellin transform of f(x)is defined by M{f(x)}= F(p)and is defined by ∞ − M{f(x)}=F(p) = xp 1f(x)dx (2.5) 0 Where M is the the Mellin transform operator and p is the Mellin transform variable which is complex number. − The inverse Mellin transform is denoted by M 1{F(p)}=f(x)and is defined by c+i∞ − 1 − M 1{F(p)}=f(x)= x pF(p)dp (2.6) 2πi c−i∞ − Where M 1 is the inverse Mellin transform operator. Definition 2.4. (Riemann–Liouville Fractional Integral) If f(t) ∈ C[a,b] and a< t Definition 2.5. (Riemann–Liouville Fractional Derivative) [5] If f(t)∈ C[a,b] and a Definition 2.6. (Caputo Fractional Derivative) If f(t)∈ C[a,b] and a Definition 2.7. (Grunwald–Letnikov Fractional Derivatives) The Grunwald-Letnikov definition of fractional derivative of a function generalize the notion of backward differ- ence quotient of integer order. In this case α = 1 if the limit exists the Grunwald-Letnikov fractional Derivative is the left derivative of the function. The Grunwald–Letnikov frac- tional Derivative of order α of the function f(t)is defined as t−a h α 1 jα D f(t)= lim → (−1) C f(t − jk) (2.10) a t h 0 hα j j=0 t − a where is integer and α ∈ C.Ifα =−1, we have a Riemann sum which is the first h integral. Definition 2.8. (Mittag–Leffler function) The Mittag–Leffler function of one param- eter is denoted by Eα(z) and is defined as ∞ zk E (z) = ,α ∈ C, Re(α) > 0. (2.11) α (αk + 1) k=0 Definition 2.9. (Mittag–Leffler function) The Mittag–Leffler function of two param- eter is denoted by Eα,β(z) and is defined as ∞ zk E (z) = ,α,β ∈ C, Re(α) > 0,Re(β)>0. (2.12) α,β (αk + β) k=0 The function Eα(z) was defined and studied by Mittag–Leffler in the year 1903. It is a direct generalization of the exponential series. For α = 1 we have the exponential series. The function defined by (2.12) gives generalization of (2.11). This generalization was studied by Wiman in 1905, Agrawal in 1953, Humbert and Agarwal in 1953, and others. Definition 2.10. (Fourier transform of Riemann–Liouville Fractional integral) The Fourier transform of Riemann–Liouville Fractional integral of order α is given as { α ; }= { −α ; }= −α { ; } F −∞It f(t) k F −∞Dt f(t) k (ik) F f(t) k (2.13) Study of Generalized Integral Transforms their Properties and Relations 1289 where k is real. Definition 2.11. (Fourier transform of fractional derivative) The Fourier transform of fractional derivative of order α is given as { α ; }= α { ; } F Dt f(t) k (ik) F f(t) k (2.14) α where the operator Dt represents any of the mentioned fractional derivative i.e. Riemann– α Liouville fractional derivative 0Dt f(t) defined by (2.8), Caputo fractional derivative c α α aDt f(t) defined by (2.9), Grunwald–Letnikov fractional derivative aDt f(t) defined by (2.10). Definition 2.12. (Mellin transform of Riemann–Liouville Fractional integral) The Mellin transform of Riemann–Liouville Fractional integral of order α is given as − (1 − p − α) M{ I αf(t); p}=M{ D αf(t); s}= F(p+ α) (2.15) 0 t 0 t (1 − p) where p is complex. Definition 2.13. (Mellin transform of Fractional derivative) The Mellin transform of Fractional derivative of order α is given as (1 − p + α) M{Dαf(t); p}= F(p− α) (2.16) t (1 − p) α where the operator Dt represents any of the mentioned fractional derivative i.e. Riemann– α Liouville fractional derivative 0Dt f(t) defined by (2.8), Caputo fractional derivative c α α aDt f(t) defined by (2.9), Grunwald–Letnikov fractional derivative aDt f(t) defined by (2.10). Example 2.14. The Mellin transform and Fourier transform of fractional derivative of following functions (i) f(t)= sint. we have pπ M{f(t)}=(p)sin 2 − + α (1 p α) M{D f(t); p}= [M{f(t)}] → − t (1 − p) p (p α) (1 − p + α) pπ = (p)sin − (1 p) 2 p→(p−α) (1 − p + α) (p − α)π = (p − α)sin (1 − p) 2 1290 Bhausaheb R. Sontakke and Govind P. Kamble (ii) f(t)= exp(−at2) we have 2 1 −( p ) p M{exp(−at )}= a 2 2 2 (1 − p + α) M{Dαexp(−at2); p}= M{exp(−at2)} t (1 − p) p→(p−α) (1 − p + α) 1 − p p = a ( 2 ) (1 − p) 2 2 p→(p −α) − + − − (1 p α) 1 −( p α ) p α = a 2 (1 − p) 2 2 Example 2.15. The Fourier transform of fractional derivative of the following function: f(t)= exp(−at2) we have 1 F {exp(−at2); k}=√ − k2 2aexp 4a F {Dαexp(−at2); k}=(ik)αF {exp(−at2); k} t ⎡ ⎤ α ⎣ 1 ⎦ = (ik) √ − k2 2aexp 4a (ik)α = √ − k2 2aexp 4a 3. Main Result 3.1. Weyl Fractional Derivative and Integrals and their Integral Transforms Definition 3.1. (Weyl Fractional Integral) The Weyl fractional integral of f(x) is −α denoted by xW∞ and is defined as ∞ −α 1 α−1 xW∞ [f(x)]= (t − x) f(t)dt, (3.1) (α) x where 0 (i) f(t)= exp(−at) ∞ −α 1 α−1 xW∞ [exp(−at)]= (t − x) exp(−at)dt (α) x by the change of variable t − x = y − ∞ −α exp( ax) α−1 xW∞ [exp(−at)]= y exp(−ay)dy (α) 0 by substituting ay = u − ∞ −α[ − ]=exp( ax) 1 α−1 − xW∞ exp( at) α u exp( u)du a (α) 0 − = exp( ax) aα − (ii) f(t)= t µ ∞ −α −µ 1 α−1 −µ xW∞ [t ]= (t − x) t dt (α) x by the change of variable t − x = y ∞ −α −µ 1 α−1 −µ W∞ [t ]= y (y + x) dy x (α) 0 ∞ α−1 = 1 y µ dy (α) 0 (y + x) xα−µ = β(α,µ − α) (α) (µ − α) = xα−µ (µ) (iii) f(t)= sinat ∞ −α 1 α−1 xW∞ [sinat]= (t − x) sinatdt (α) x by the change of variable t − x = y ∞ −α 1 α−1 W∞ [sinat]= y sin(ax + ay)dy x (α) x = 1 + − π α cos ax (α 1) a 2 1 απ = sin ax + aα 2 where 0 (iv) f(t)= cosat similar to above, we get −α 1 απ W∞ [cosat]= cos ax + x aα 2 where 0 Note: The modified form of the Dirichlet formula is a a a − + − (t − x)αdt (u − t)β 1f (u)du = β(α,β) (u − t)α β 1f (u)du (3.2) x t t Property 3.3. Weyl fractional integral obeys the semigroup property that is −α −β −(α+β) −β −α xW∞ xW∞ = xW∞ = xW∞ xW∞ (3.3) Proof. We have ∞ ∞ −α −β 1 α−1 1 β−1 xW∞ xW∞ = (t − x) f(t)dt (u − t) f (u)du (α) x (β) x 1 a = (u − t)α+β−1f (u)du (α + β) t −(α+β) = xW∞ by considering a →∞in formula (3.2). Property 3.4. Relation between nth derivative and Weyl fractional integral n −α −α n D [xW∞ f(x)]= xW∞ [D f(x)] (3.4) Proof. We prove this property by mathematical induction For n = 1 ∞ −α 1 α−1 D[xW∞ f(x)]=D (t − x) f(t)dt (α) x by the changing of variable t − x = y ∞ −α 1 α−1 D[xW∞ f(x)]=D y f(y+ t)dy (α) 0 1 ∞ = yα−1[Df (y + t)]dy (α) 0 −α = xW∞ [Df (x)] Study of Generalized Integral Transforms their Properties and Relations 1293 We assume that this property is true for n = k that is k −α −α k D [xW∞ f(x)]= xW∞ [D f(x)] Let us prove this property for n = k + 1 ∞ k+1 −α k+1 1 α−1 D [xW∞ f(x)]=D (t − x) f(t)dt (α) x by the changing of variable t − x = y ∞ k+1 −α k+1 1 α−1 D [xW∞ f(x)]=D y f(y+ t)dy (α) 0 1 ∞ = yk+1[Dk+1f(y+ t)]dy (α) 0 −α k+1 = xW∞ [D f(x)] Hence by induction, we prove n −α −α n D [xW∞ f(x)]= xW∞ [D f(x)] Definition 3.5. (Weyl Fractional Derivative) If β is a positive number and n is the smallest integer greater than β such that n − β = α>0, the Weyl fractional derivative of a function f(x) is defined by − n n ∞ β n −(n−β) ( 1) d (n−β−1) ∞[ ]= − ∞ [ ]= − xW f(x) ( D) xW f(x) n (t x) f(t)dt (n − β) dx x (3.5) Property 3.6. Weyl fractional derivatives obeys the law of exponents. β γ (β+γ) γ β xW∞ xW∞ = xW∞ = xW∞ xW∞ (3.6) Example 3.7. Weyl fractional derivative of following functions. β β (i) xW∞[exp(−ax)]=a exp(−ax) + β −µ (β µ) −(β+µ) (ii) W∞[x ]= x x (µ) β β βπ (iii) W∞[cosat]=a cos ax − x 2 β β βπ (iv) W∞[sinat]=a sin ax − . x 2 1294 Bhausaheb R. Sontakke and Govind P. Kamble Definition 3.8. Mellin transform of Weyl fractional Integral −α (p) M{ W∞ [f(x)]; p}= M{f(x)} → + (3.7) x (p + α) p (p α) Definition 3.9. Mellin transform of Weyl fractional derivative: β (p) M{ W∞[f(x)]; p}= M{f(x)} → − (3.8) x (p − β) p (p β) Example 3.10. We obtain the Mellin transform of Weyl fractional integrals and deriva- tives of some standard functions as follows. (i) Mellin transform of Weyl fractional integral We have − M{exp(−ax)}=n p (p),whereRe(p) > 0 −α (p) M{ W∞ [exp(−ax)]; p}= M{exp(−ax)} → + x (p + α) p (p α) (p) = n−p (p) (p + α) p→(p+α) (p) = n−(p+α) (p + α)) (p + α) Mellin transform of Weyl fractional derivative β (p) M{ W∞[exp(−ax)]; p}= M{exp(−ax)} → − x (p − β) p (p β) (p) = n−p (p) (p − β) p→(p−β) (p) = n−(p−β) (p − β) (p − β) (ii) Mellin transform of Weyl fractional integral We have πp M{sinax}=a−p (p)sin 2 where 0 −α (p) M{ W∞ [sinax]; p}= M {sinax} → + x (p + α) p (p α) (p) πp = a−p(p)sin (p + α) 2 p→(p+α) (p) π(p + α) = a−(p+α)(p + α)sin (p + α) 2 Study of Generalized Integral Transforms their Properties and Relations 1295 Mellin transform of Weyl fractional derivative β (p) M{ W∞[sinax]; p}= M {sinax} → − x (p − β) p (p β) (p) πp = a−p(p)sin (p − β) 2 p→(p−β) (p) π(p − β) = a−(p−β)(p − β)sin (p − β) 2 (iii) Mellin transform of Weyl fractional integral We have πp M{cosax}=a−p(p)cos 2 where 0 −α (p) M{ W∞ [cosax]; p}= M {cosax} → + x (p + α) p (p α) (p) πp = a−p(p)cos (p + α) 2 p→(p+α) (p) π(p + α) = a−(p+α)(p + α)cos (p + α) 2 Mellin transform of Weyl fractional derivative β (p) M{ W∞[cosax]; p}= M {cosax} → − x (p − β) p (p β) (p) πp = a−p(p)cos (p − β) 2 p→(p−β) (p) π(p − β) = a−(p−β)(p − β)cos (p − β) 2 Definition 3.11. The Fourier transform of Weyl fractional integral −α πiα −α F { W∞ f(x)}=exp − k F {f(x)} (3.9) x 2 α Theorem 3.12. The Riemann–Liouville fractional integral operator 0Ix f(x)defined in −α (2.8) and Weyl fractional integral operator xW∞ f(x)are adjoint of each other. i.e. α = −α 0Ix f (x), g(x) f(x), xW∞ g(x) (3.10) where < f (x), g(x) > is inner product notation. 1296 Bhausaheb R. Sontakke and Govind P. Kamble Proof. ∞ α = { α } 0Ix f (x), g(x) 0Ix f(x)g(x) dx 0 1 ∞ x = g(x)dx (x − t)α−1f(t)dt (α) 0 0 ∞ 1 ∞ = f(t)dt (x − t)α−1g(x)dx 0 (α) t ∞ 1 ∞ = f(t)dt (t − x)α−1g(t)dt 0 (α) x ∞ −α = f(t)xW∞ g(x) 0 −α =f(x), xW∞ g(x) Hence proved. 4. Fractional Mellin transform and Fourier fractional Mellin transform Definition 4.1. (Fractional Mellin Transform) Fractional Mellin transform in the range (0, ∞). The one dimensional fractional Mellin transform with parameter θ of f(x) denoted by Mfr{[f(x),θ,0, ∞]} performs a linear operation, given by integral transform ∞ fr 2πiu−1 πi 2 2 M [f(x),θ,0, ∞] = f(x)xsinθ exp (u + log x) dx (4.1) 0 tanθ π where 0 <θ≤ . 2 Property 4.2. Relation between Mellin transform and fractional Mellin transform. The fractional Mellin transform is given as πi πi Mfr [f(x),θ,0, ∞] = exp u2 M f(x)exp log2x ,θ,0, ∞ tanθ tanθ Study of Generalized Integral Transforms their Properties and Relations 1297 Proof. ∞ fr 2πiu−1 πi 2 2 M {f(x)}= f(x)xsinθ exp (u + log x) dx 0 tanθ ∞ πi 2 2πiu−1 πi 2 = exp u f(x)xsinθ exp log x dx tanθ 0 tanθ ∞ πi 2 2πiu−1 = exp u g(x)x sinθ dx tanθ 0 πi ∞ = exp u2 g(x)xp−1dx tanθ 0 πi = exp u2 M{[g(x),θ,0, ∞]} tanθ πi 2πiu where g(x) = f(x)exp log2x and p = . tanθ sinθ 1 Definition 4.3. Fractional Mellin transform in the range 0, Fractional Mellin a 1 1 transform with parameter of f(t) in the range 0, is denoted as Mfr f(t),p,0, a a and is defined as 1 1 a − Mfr f(t),p,0, = ap tp 1f(t)dt (4.2) a 0 1 where p>0, a>0 is a parameter and is fractional. a 4.1. Linearity Property 1 Mfr αf (t) + βg(t), p, 0, a 1 1 = αMfr f(t),p,0, + βMfr g(t),p,0, (4.3) a a 1298 Bhausaheb R. Sontakke and Govind P. Kamble Proof. 1 Mfr αf (t) + βg(t), p, 0, a 1 a = aptp−1[αf (t) + βg(t)]dt 0 1 1 a a = α aptp−1f(t)dt + β aptp−1g(t)dt 0 0 1 1 = αMfr f(t),p,0, + βMfr g(t),p,0, a a Hence proved. 4.2. Power Property 1 1 p 1 Mfr f(tb), p, 0, = Mfr f(tb), , 0, (4.4) a b b ab Proof. 1 1 a − Mfr f(tb), p, 0, = ap tp 1f(tb)dt a 0 1 − ab p p 1 1 1 −1 = a τ b f(τ)τb dτ 0 b 1 1 ab b p p −1 = a b f(τ)τ b dτ b 0 1 p 1 = Mfr f(tb), , 0, b b ab where τ = tb hence proved. 1 Example 4.4. Find fractional Mellin transform in the range 0, of following func- a tions (i) exp(t) and (ii) sinat Study of Generalized Integral Transforms their Properties and Relations 1299 Solution: (i) 1 1 a − Mfr exp(t), p, 0, = ap tp 1exp(t)dt a 0 1 ∞ k a t = ap tp−1 dt (k + 1) 0 k=0 ∞ p 1 a a = tk+p−1dt (k + 1) k=0 0 ∞ 1 p k+p a = a t (k + 1) (k + p) = k=0 k 0 ∞ ( 1 )k = a (k + p)(k + 1) k=0 where k is an integer. (ii) 1 1 a − Mfr sinat,p,0, = ap tp 1sinat dt a 0 1 ∞ 2k+1 a − (at) = ap tp 1 (−1)k dt (2k + 2) 0 k=0 ∞ p 1 a a + = (−1)k (at)2k pdt (2k + 2) k=0 0 ∞ k= 1 ap (at)2k+p+1 a = (−1)k (2k + 2) (2k + p + 1) = k=0 k 0 ∞ ( 1 )2k+1 = (−1)k a (2k + 2)(2k + p + 1) k=0 where k is an integer. 1 Definition 4.5. Fourier Fractional Mellin transform in the range 0, is denoted as a 1 FMfr f(x,t),s,0, ∞,p,0, and defined as a ∞ 1 1 a − FMfr f(x,t),s,0, ∞,p,0, = exp(−isx)ap tp 1f(x,t)dt (4.5) a 0 0 1300 Bhausaheb R. Sontakke and Govind P. Kamble 1 where a>0, p and s are parameters and is fractional. a 4.3. Linearity Property FMfr{αf (x, t) + βg(x, t)}=αFMfr{f(x,t)}+βFMfr{g(x,t)} (4.6) Proof. FMfr{αf (x, t) + βg(x, t)} ∞ 1 a = exp(−isx)ap tp−1[αf (x, t) + βg(x, t)]dxdt 0 0 ∞ 1 a = α exp(−isx)ap tp−1f(x,t)dxdt 0 0 ∞ 1 a + β exp(−isx)ap tp−1g(x,t)dxdt 0 0 = αFMfr{f(x,t)}+βFMfr{g(x,t)} Hence proved. 4.4. Power Property 1 FMfr f(x,tn), s, 0, ∞,p,0, a 1 p 1 n = FMfr f(x,t),s,0, ∞, , 0, (4.7) n n a Proof. 1 FMfr f(x,tn), s, 0, ∞,p,0, a ∞ 1 a = exp(−isx)ap tp−1f(x,tn)dxdt 0 0 ∞ 1 n ( ) − a p p 1 1 1 −1 = exp(−isx)a z n f(x,z) z n dxdz n 0 0 ∞ ( 1 )n 1 a n p p −1 = exp(−isx)(a ) n z n f(x,z)dxdz n 0 0 1 p 1 n = FMfr f(x,t),s,0, ∞, , 0, n n a where tn = z. Hence proved. Study of Generalized Integral Transforms their Properties and Relations 1301 5. Conclusions (i) We study various integral transforms and fractional integral transforms. (ii) We study Weyl fractional integral and fractional derivative. 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