Mathematica B a l k a n i c a ||||||||| New Series Vol. 26, 2012, Fasc. 1-2 ®-Mellin Transform and One of Its Applications Yanka Nikolova Presented at 6th International Conference \TMSF' 2011" We consider a generalization of the classical Mellin transformation, called ®-Mellin transformation, with an arbitrary (fractional) parameter ® > 0. Here we continue the presen- tation from the paper [5], where we have introduced the de¯nition of the ®-Mellin transform and some of its basic properties. Some examples of special cases are provided. Its operational properties as Theorem 1, Theorem 2 (Convolution theorem) and Theorem 3 (®-Mellin trans- form of fractional R-L derivatives) are presented, and the proofs can be found in [5]. Now we prove some further properties of this integral transform, useful for its application to solving some fractional order di®erential equations. An example of such application is proposed for the fractional order Bessel di®erential equation of the form ¯+1 ¯+1 ¯ ¯ t 0Dt y(t) + t 0Dt y(t) = f(t) ; 0 < ¯ < 1: MSC 2010: 35R11, 44A10, 44A20, 26A33, 33C45 Key Words: integral transforms method, Mellin transformation, Riemann-Liouville fractional derivative, fractional Bessel di®erential equation 1. Introduction This paper deals with the theory and applications of the ®-Mellin trans- form. We derive the ®-Mellin transform and its inverse from the complex Fourier transformation. This is followed by several examples and basic operational prop- erties of the ®-Mellin transform. We discuss an application of the ®-Mellin transform for solving a fractional di®erential equation. Historically, Riemann (1876) ¯rst recognized the Mellin transform in his famous memoir on the prime numbers. Its explicit formulation was given by Cahen (1894). Almost simulta- neously Mellin (1896, 1902) gave an elaborate discussion of the Mellin transform and its inversion formula. 186 Y. Nikolova 2. De¯nition of the ®-Mellin transform In [2] Luchko, Martinez and Trujillo introduced the fractional Fourier transformation (FRFT). A substitution x = et in the FRFT leads to a general- ization of the Mellin transform, called further ®-Mellin transform. The reason for this name is explained by the following de¯nitions. De¯nition 2.1. Let 0 < ® · 1 and the variable be a complex number, such that ( Rep ¡ ij!j1=®;! · 0 p = : Rep + ij!j1=®; ! > 0 The integral transform of the form Z1 p sign!¡1 M®ff(t); pg = M(psign!) = t f(t)dt (1) 0 is called ®-Mellin transform of the function f(t). De¯nition 2.2. Let Re® > 0 and f 2 C. Then for a 2 R and x > a the integral operator Zx 1 D¡®f(x) = (x ¡ t)®¡1f(t)dt (2) a x ¡(®) a is called a Riemann-Liouville fractional integral operator of order ®. The symbol ¯ aDx with ¯ > 0 is interpreted as a corresponding di®erintegral operator, called Riemann-Lioville fractional derivative. For details and de¯nitions of these op- erators of Fractional Calculus (FC), see for example [7], [6]. 3. Basic operational properties of the ®-Mellin transforms Like for the FRFT in [8], [6], [3] our interest in the ®-Mellin transform is based on the possibilities of its applications for solving certain ordinary dif- ferential equations of fractional order. Using De¯nition 2.1, we can prove some basic properties of the ®-Mellin transforms. The following operational properties hold: Theorem 3.1. ([5]) If we denote M®ff(t); pg = M(p sign !), then: ¡p sign! a) M®ff(at); pg = a M(p sign!); a > 0 (Scaling Property); a b) M®ft f(t); pg = M(p sign! + a)(Shifting Property). Theorem 3.2. ([5]) (Convolution Theorem) If M®ff(t); pg = M(psign!) and M®fg(t); pg = G(psign!), then M®ff(t) ¤ g(t); pg = M(p sign!)G(1 ¡ p sign!); ®-Mellin Transform and One of Its Applications 187 where Z1 f(t) ¤ g(t) = f(t¿)g(¿)d¿: 0 Corollary 3.1. ([5]) If n > 0 and f(t) is a function such that lim tsign!Rep¡1jf(t)j = 0 ; lim tsign!Rep¡1jf(t)j = 0 ; t!1 t!0 then ¡(1 ¡ psign! + n) M ff (n)(t); pg = M(psign! ¡ n): ® ¡(1 ¡ psign!) Corollary 3.2. ([5]) If 0 < ¯ < 1 and f(t) is a function such that lim tsign!Rep¡1jf(t)j = 0 ; lim tsign!Rep¡1jf(t)j = 0 ; t!1 t!0 then ¡(1 ¡ p sign! + ¯) M f D¯f(t); pg = M(p sign! ¡ ¯) ; ® 0 t ¡(1 ¡ p sign!) ¯ where 0Dt is the Riemann-Liouville fractional derivative of order ¯. Theorem 3.3. If 0 < ¯ < 1 and M®ff(t); pg = M(psign!), then ½ ¾ n P ¯+k ¯+k M® akt 0Dt f(t); p k=0 Pn kQ¡1 = M(p sign!) ¡(1¡p sign!) (¡1)ka (p sign! + ¯ + j) ; ¡(1¡p !¡¯) k sign k=0 j=0 ¯ where 0Dt is the Riemann-Liouville fractional derivative operator of order ¯. P r o o f. Under (1) and Theorem 3.1 (b), ( ) Xn Xn n o ¯+k ¯+k ¯+k M® akt 0Dt f(t); p = akM® 0Dt f(t); psign! + ¯ + k : k=0 k=0 On the other hand, under Corollary 3.2, Xn n o ¯+k akM® 0Dt f(t); psign! + ¯ + k k=0 Xn ¡(1 ¡ p sign!) = a M(p sign!) k ¡(1 ¡ p sign! ¡ ¯ ¡ k) k=0 188 Y. Nikolova Xn a = M(p sign!)¡(1 ¡ psign!) k : ¡(1 ¡ p sign! ¡ ¯ ¡ k) k=0 Applying the property of the ¡¡function, we obtain k¡1 k Q n n (¡1) ak (p sign! + ¯ + j) X a X j=0 k = ¡(1 ¡ p sign! ¡ ¯ ¡ k) ¡(1 ¡ p sign! ¡ ¯) k=0 k=0 and this proves the theorem. Corollary 3.3. If 0 < ¯ < 1 and M®ff(t); pg = M(psign!), then (1¡p sign!¡¯)¡(1¡p sign!) M ft¯+1 D¯+1f(t)+t¯ D¯f(t); pg= M(p sign!) : ® 0 t 0 t ¡(1¡p sign!¡¯) 4. Generalized Bessel fractional equation The fractional order di®erential equation of the form ¯+1 ¯+1 ¯ ¯ t 0Dt y(t) + t 0Dt y(t) = f(t) ; 0 < ¯ < 1; (3) we call generalized Bessel fractional equation. Theorem 4.1. The solution of the boundary value problem for the Bessel fractional equation (3) with the conditions y(0) = y0(0) = 0 ; y(1) = y0(1) = 0; (4) has the form Z1 y(t) = f(t¿)g(¿)d¿; (5) 0 where ¡1 g(t) = M® fG(psign!); tg: P r o o f. Applying of the ®-Mellin transform (1) to both sides of (3) and condition (4), by Corollary 3.1 and Corollary 3.3, we obtain (1 ¡ psign! ¡ ¯)¡(1 ¡ psign!) Y (psign!) = M(psign!) ; ¡(1 ¡ psign! ¡ ¯) where M®fy(t); pg = Y (psign!) ;M®ff(t); pg = M(psign!) : ®-Mellin Transform and One of Its Applications 189 Rewriting the above equality in the following form: Y (psign!) = M(psign!)G(1 ¡ psign!) ; this leads to ¡(psign! ¡ ¯) G(psign!) = : (psign! ¡ ¯)¡(psign!) If the inverse ®-Mellin ransform for G(psign!) is ¡1 M® fG(psign!); tg = g(t) ; then, according to Theorem 3.2, we get the solution in the form Z1 y(t) = f(t¿)g(¿)d¿ : 0 This proves the theorem. R1 The solution is of the form y(t) = f(t¿)g(¿)d¿ ; if g(t) is zero for t > 1. 0 Acknowledgements: This paper is partially supported under Project D ID 02/25/2009: "Integral Transform Methods, Special Functions and Appli- cations", by National Science Fund - Ministry of Education, Youth and Science, Bulgaria. References [1] L. Debnath, D. Bhatta, Integral Transforms and Their Applications, Chap- man & Hall/ CRC, Boca Raton (2006). [2] Y. Luchko, H. Martinez, J. Trujillo, Fractional Fourier transform and some of its applications, Fract. Calc. Appl. Anal. 11, No 4 (2008), 457-470. [3] Y. Nikolova, L. Boyadjiev, Integral transforms method to solve time-space fractional di®usion equation, Fract. Calc. Appl. Anal. 13, No 1 (2010), 57-67. [4] Y. Nikolova, Basic properties of fractional Fourier transformation, In: Amer- ican Institute of Phys. Proc.: "Applications of Mathematics in Engineering and Economics' 2010 " (2010), 183-188. [5] Y. Nikolova, De¯nition of the ®-Mellin transform and some of its properties, In: American Institute of Phys. Proc.: "Applications of Mathematics in Engineering and Economics' 2011" (2011), 259-266. 190 Y. Nikolova [6] I. Podlubny, Fractional Di®erential Equations, New York-London, Academic Press (1999). [7] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), Gordon & Breach Sci. Publ., Switzerland (1993). [8] N. Wiener, Hermitian polynomials and Fourier analysis, J. Math. Phys. 8 (1929), 70-73. Faculty of Applied Mathematics and Informatics Technical University So¯a 1156, BULGARIA e-mail: [email protected] Received: November 8, 2011.