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Note on Integral Transform of Fractional Calculus International Journal of Research and Review www.ijrrjournal.com E-ISSN: 2349-9788; P-ISSN: 2454-2237 Research Paper Note on Integral Transform of Fractional Calculus S. V. Nakade1, R. N. Ingle2 1Head & Assistant Professor, Dept. of Mathematics, Sharda Mahavidyalaya, Parbhani, (Affiliated to SRTMU Nanded) Maharashtra, India. 2Principal, Bahirji Smarak Mahavidyalaya, Basmatnagar, India. Corresponding Author: S. V. Nakade ABSTRACT In recent years Fractional Calculus is highly growing field in research because of its wide applicability and interdisciplinary approach. In this article we study various integral transform particularly Laplace Transform, Mellin Transform, of Fractional calculus i.e. Fractional derivative and Fractional Integral particularly of Riemann-Liouville Fractional derivative, Riemann-Liouville Fractional integral, Caputo’s Fractional derivative and their properties. AMS subject classification 2000: 26A33, 44A10 Keywords: Fractional Calculus, Riemann-Liouville Fractional derivative, Riemann-Liouville Fractional integral, Caputo’s Fractional derivative , Integral Transform, Laplace Transform, Mellin Transform. 1. INTRODUCTION end of 17th century but good work held’s Fractional Calculus: only since last three to four decades. It is observed that study of classical Oldham and Spanier (1974) [2] and Miller calculus i.e. differential calculus and and Ross (1993) [3] gives in brief historical integral calculus begins very close or development of Fractional Calculus. Special equally at same period as study of functions are very closely related with Fractional Calculus begins. Up to 30th Fractional Calculus and plays important role September 1695 no one knows that this is in development of fractional calculus. [7] the new field of mathematics and called as Different approaches of Fractional calculus Fractional Calculus. Answer given by great are given throughout the history such mathematician Leibnitz on 30th September Riemann approach, Liouville approach, 1695 to the question raised by another great Caupto approach, Wely approach, Lacorix mathematician L’Hospital as “What will be approach, Grunwald approach, Letnikov 푛 the meaning of 푑 푓 , if n=1/2?” as “This is approach, Riemann-Liouville approach, 푑푥 푛 Grunwald-Letnikov approach and many an apparent paradox from which one day a more. useful consequence will be drawn”. After th Integral Transform: 30 September 1695 mathematicians who An integral transform is a particular are working on classical calculus with type of mathematical operator. To each special case of differentiation and integral transform there is its associated integration of arbitrary order named the inverse integral transform. This inverse field as Fractional calculus. Though the integral transform associated to respective study of Fractional calculus begins at the integral transform since, integral transform International Journal of Research & Review (www.ijrrjournal.com) 106 Vol.6; Issue: 2; February 2019 S. V. Nakade et.al. Note on Integral Transform of Fractional Calculus maps its original domain into another ∞ 푥−1 −푡 Where, 훤(x) = 0 푡 . 푒 푑푡 , called as domain where it is solved very easily than in Euler’s Gamma function. its original domain than again mapped to If α 휖 N then these definitions acts as like original domain by using inverse integral classical derivative of order α. transform. There are numerous integral [8-10] In general Riemann-Liouville Fractional transform in study. Derivative of order α of the function f(x) Integral Transform of the function f (t) for with a < x < b is defined as t ≤ x≤ t is defined as n 1 2 α 1 d x n−α−1 t2 D f(x) = x − t f t dt , Γ(n−α) dx a 푇 푓 푡 = 푓 푡 퐾 푡, 푥 푑푡 If 0 < α < 1, we obtain α 1 d x f(t) t1 D f(x) = a α dt . Where K is the function of two variables t Γ(1−α) dx (x−t) and x called kernel of integral transform. Riemann-Liouville Fractional Integral: Choice of function K of two variables Let α > 0 and n-1 < α < n , n 휖 N, and a < x means kernel is different for different < b , Left hand and Right hand Riemann- integral transform. Liouville Fractional integral is defined as Inverse integral transform is of the form α 1 x α−1 I f(x) = x − t f t dt , a+ Γ(α) a x2 푓 푡 = 푇 푓 푥 퐾−1 푥, 푡 푑푥 α 1 b α−1 I f(x) = x − t f t dt , b− Γ(α) x x1 Where 퐾−1(푥, 푡) is the kernel of inverse In general Riemann-Liouville Fractional integral transform and is the inverse of integral of order α 휖 (-∞, ∞) of the function kernel K (t, x). f(x) with a < x < b is defined as 훼 −훼 1 x α−1 퐼 푓(푥) = 퐷 f(x) = x − t f t dt . This paper is divided in five sections which Γ(α) a are as follows In section 1, we introduced about fractional Caputo Fractional Derivative: calculus and integral transform, in section 2 Let α > 0 and n-1 < α < n, n 휖 N, and a < x we see some definitions of fractional < b, Left hand and Right hand Caputo calculus, in section 3 some definitions of Fractional derivative is defined as integral transform are given, in section 4 C 훼 1 푥 푛−훼−1 푛 main result means some integral transform 퐷 푓(푥) = 푥 − 푡 푓 (푡) 푑푡 , 푎+ 훤(푛−훼) 푎 of some fractional approaches are studied C (−1)푛 푏 퐷훼 푓(푥) = 푥 − 푡 푛−훼−1푓 푛 (푡) 푑푡 , with their properties and section 5 is the 푏− 훤(푛−훼) 푥 conclusion part. In general, Caputo Fractional derivative of order α 휖 (-∞, ∞) of the function f(x) with a 2. Some Definitions of Fractional < x < b is defined as 푐 훼 1 푥 푛−훼−1 푛 퐷 푓(푥) = 푥 − 푡 푓 (푡) 푑푡 , calculus: 푎 푥 훤(푛−훼) 푎 Riemann-Liouville Fractional Derivative: Where n-1 < α < n. [1-6] Let α > 0 and n-1 < α < n, n 휖 N, and a < x 3. Some Integral Transform: < b, Left hand and Right hand Riemann- Laplace Transform: Liouville Fractional Derivative is defined as Laplace Transform [8] is very useful in 1 푑 푛 푥 퐷훼 푓(푥) = 푥 − 푡 푛−훼−1푓 푡 푑푡 , solving differential equations especially 푎+ 훤(푛−훼) 푑푥 푎 when initial values are zero. The Laplace (−1)푛 푑 푛 푏 Transform of a function f(x) defined for all 퐷훼 푓(푥) = 푥 − 푡 푛−훼−1푓 푡 푑푡 , 푏− 훤(푛−훼) 푑푥 푥 real number x ≥ 0 is the function F(s) given respectively. by International Journal of Research & Review (www.ijrrjournal.com) 107 Vol.6; Issue: 2; February 2019 S. V. Nakade et.al. Note on Integral Transform of Fractional Calculus ∞ −푠푡 (f (t))= 퐹(푠) = 0 푓(푡)푒 dt, Where is the the Mellin transform operator and s is the Mellin transform Inverse Laplace transform of F(s) i.e. f (t) is variable given by which is complex number. 푐+푖∞ −1 푠 Inverse mellin transform is defined as ℒ 퐹 푠 = 푓 푡 = 푒 F(s) ds, 푐−푖∞ −1 1 푐+푖∞ −푠 ℳ 퐹 푠 = 푓 푥 = 푥 퐹 푠 푑푠 c=Re(s) > c0, 2휋푖 푐−푖∞ −1 Where ℳ is the inverse Mellin transform where c0 lies in right half plane of the operator. absolute convergence of Laplace integral. Some properties of Mellin Transform: Some properties of Laplace Transform: i) {a f(t)}=a ℳ{f(t)} ; where a is i) {a f(t)}=a {f(t)} ; where a is constant constant. ii) {a f(t) + b g(t)}= a {f(t)} + b ii) {a f(t) + b g(t)}= a {f(t)} + b {g(t)} ; Linearity {g(t)} ; Linearity iii) If {f(t)}= 퐹(푠) then {f(at)} = 푎−푠 iii) If {f(t)}= 퐹(푠) then £{f(at)} = 퐹 푠 ; change of scale 1 + 퐹(푠 ) ; change of scale iv) Convolution property : If F(s) and G(s) 푎 푎 are the Mellin Transform of f(x) and iv) Convolution property: If F(s) and G(s) g(x) respectively, then are the Laplace Transform of f(x) and g(x) respectively, then a. f(x) * g(x); s} = F(s) G(s)= 1 푐+푖∞ ℒ{ F(x) * g(x); s} = F(s) G(s) 푓 푠 − 푧 푔 푧 푑푧 푥 2휋푖 푐−푖∞ =ℒ 0 푓 푥 − 푧 푔 푧 푑푧 , b. Where convolution is given by Where convolution is given by 푥 푥 c. f * g = 푓 푥 − 푧 푔 푧 푑푧 . f * g = 푓 푥 − 푧 푔 푧 푑푧 . 0 0 v) The Mellin transform of the n-th (n 휖 N) v) The Laplace transform of the n-th (n 휖 derivative of f(x) is given by N) derivative of f(x) is given by 훤(푠) ℳ f n x : s = (−1)푘 퐹 푠 − 푘 푛−1 훤(푠 − 푘) ℒ f n x : s = 푠푛 퐹 푠 − 푠푛−푘−1푓 푘 0 푘=0 4. Integral transform of some fractional approaches: 푛 푛−1 푘 푛−푘−1 = 푠 퐹 푠 − 푘=0 푠 푓 0 5. Laplace Transform of the Riemann- Mellin Transform: Liouville Fractional Integral: Mellin Transform [10] is closely related The Riemann-Liouville Fractional Integral with Laplace transform and Fourier is given by transform. Hjalmar Mellin (1854–1933) x 1 gave his name to the Mellin transform 0f a D−α f(x) = x − t α−1f t dt . Γ(α) function f(x) defined over the positive real’s, a the complex function M [f (x); s]. Mellin Transform is the multiplicative version of Its Laplace transform is given by two sided Laplace Transform.
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