M a t h e m a t i c a B a l k a n i c a. On Some Generalizations of Classical Integral Transforms. Nina Virchenko

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1 M a t h e m a t i c a B a l k a n i c a New Series Vol. 26, 212, Fasc. 1-2 On Some Generalizations of Classical Integral Transforms Nina Virchenko Presented at 6 th International Conference TMSF 211 Using the generalized confluent hypergeometric function [6 some new integral transforms are introduced. They are generalizations of some classical integral transforms, such as the Laplace, Stieltjes, Widder-potential, Glasser etc. integral transforms. The basic properties of these generalized integral transforms and their inversion formulas are obtained. Some examples are also given. MSC 21: 44A15, 44A2, 33C6 Key Words: classical integral transforms, generalized integral transforms, generalized confluent hypergeometric function 1. Introduction The important role of the method of integral transforms is well known in applied mathematics [4, 5, 1, 3, in solving some boundary value problems of mathematical physics, astronomy, in the theory of the differential and integral equations, etc. Let us recall the definitions if some classical integral transforms [3: the Laplace transform: Lfx; y = the Widder-potential transform: P fx; y = the generalized Stieltjes transform: S p fx; y = e xy fx dx, 1 xfx x 2 dx, 2 + y2 fx dx. 3 x + y p

2 258 N. Virchenko We define the new generalized transforms by means of the τ, β generalized confluent hypergeometric function 1 Φ τ,β 1 a; c; z, see [6: [ 1Φ τ,β 1 a; c; z = 1 1 t a 1 1 t c a 1 c; τ 1Ψ 1 Ba, c a c; β ztτ dt, 4 where Re c > Re a >, τ, β R; τ > ; τ β < 1; B... is the classic beta-function, and 1 Ψ 1 [... is a special case of the generalized hypergeometric Wright function [4: [ ai ; α i 1,p pψ q b j ; β j 1,q z = n= with z C, a i, b j C, α i, β j R, + ; q p a i, b j ; i = 1, 2,..., p; j = 1, 2,..., q, 1 + β j α i. j=1 i=1 p i=1 Γa i + nα i z n q j=1 Γb j + nβ j n!, 5 2. The generalized integral transforms and their properties Some definitions: 1 The generalized Laplace integral transforms: L γ1,γ 2,γfx; y = L γ1,γ 2 fx; y = x γ 2 e xyγ1 fx dx, 6 x γ 2 e xy 1 Φ τ,β 1 α; c; bx, yγ fx dx = gy, 7 where x >, γ C, >, γ 2 >, b ; fx as x < ; x γ 2 fx < Me s x γ1, M > and s are consts as x >. Let us note that when γ 2 =, = 1, b =, the transform 7 coincides with the transform 1. 2 The generalized Stieltjes integral tranforms: P,γ 2,γ 3,γ 4 1 fu; x = P Γc 1 fu; x = Γa 1 Γa 2 u γ [ 2 fu a1 ; τ; a 2 ; γ x + u γ 3 c; β b u γ 1 γ4 x + u du = g 1 x, 8 P,γ 2,γ 3,γ 4 2 fu; x = P Γc 2 fu; x = Γa 1 Γa 2 u γ [ 2 fu a1 ; τ; a 2 ; γ x + u γ 3 c; β b x γ 1 γ4 x + u du = g 2 x, 9 where Re a 1 >, Re a 2 >, Re c >, >, i = 1, 4; τ, β R; τ > ; τ β < 1; b, 2 Ψ 1 is the function of the form 5.

3 On Some Generalizations of Classical Let us notice that as b =, = 1, γ 2 =, γ 3 = p, the transforms 8, 9 coincide with the Stieltjes integral transform 3. For the transform 7 the following properties are valid: i Linearity: n n L γ1,γ 2,γ c i f i x; y = c i g i y, 1 i=1 i=1 c i = const, i = 1, 4. ii Similarity: L γ1,γ 2,γfax; y = 1 a γ 2+1 L γ1,γ 2,γ fx; y, 11 a a = const >. iii If the functions fx L; +, gx L; +, then the following equality is valid: u γ 2 L γ1,γ 2,γft; ugu du = t γ 2 L γ1,γ 2,γgu; tft dt, 12 under the absolute convergence of integrals. iv Under conditions of existing and convergence of integrals 6-9, the relations are valid: L γ1,γ 2 L γ1,γ 2,γgu; x; y = 1 Γ γ2 + 1 L γ1,γ 2 L γ1,γ 2 gu; x; y = 1 Γ γ2 + 1 x γ 2 P 1 ft; xgx dx = x γ 2 L γ1,γ 2 hy; x Lgu; x dx = 1 γ2 + 1 Γ 15, 16 are equalities of Parseval type. Let us give some examples of the transform 7. P,γ 2, γ 2 +1,γ 1 gu; y, 13 P,γ 2, γ 2 +1,γ 2 gu; y, 14 x γ 2 P 2 gt; xfx dx, 15 y γ 2 hy P,γ 2, γ 2+1,γ 1 gu; y dy. 16

4 26 N. Virchenko Example 1. fx = ηx = L γ1,γ 2,γ 3 ηx; y = 1 Γc Γa y γ Ψ 1 1 as x >, as x <. [ a; τ; γ2 +1 ; γ c; β b. Example 2. fx = x k, L γ1,γ 2,γx k ; y = 1 Γc Γa y k γ Ψ 1 [ a; τ; γ2 +k+1 ; γ c; β b. Example 3. L γ1,γ 2,γ fx = e kxγ1, e kxγ1 ; y = 1 Γc Γa y + k γ2+1 [ a; τ; γ Ψ 1 ; γ c; β y γ2 γ b y γ. 1+k 3. The inversion formulae Theorem 1. Under the conditions of existing of the integral transform 1 fx; y the following inversion formula is valid: 1 fy = L,γ 2,γ L 1 Γa2,γ 2,a 2 1 g 1 z; x ; y, 17 P,γ 2,γ 3,γ 4 where g 1 z = P,γ 2,γ 3,γ 4 1 fu; z; γ 4 = γ, γ 3 = a 2. P r o o f. Let us consider the equality x a 2 1 y γ 2 e y+z x 1 Φ 1 a; c : bxy γ dx = 1 Γc y γ [ 2 a; τ; a2 ; γ Γa 1 y + z a 2 c; β b y γ 1 γ y + z. 18

5 On Some Generalizations of Classical Setting in 8 γ 4 = γ, γ 3 = a 2 and taking into account 18, we obtain: P,γ 2,a 2,γ 1 fy; z = g 1 z Γc y γ [ 2 fy a; τ; a2 ; γ = Γa 1 Γa 2 z + y a 2 c; β b y γ 1 γ y + z dy = 1 fy x a 2 1 y γ 2 e y+z x 1 Φ 1 a; c; bx, y γ dx dy Γa 2 from where we get 17. = γ 1 Γa 2 L,γ,a 2 1 L γ1,γ 2,γfy; x; z, Theorem 2. Under the conditions of existing of the integral transform L γ1,γ 2,γfx; y the following inversion formula is valid: where fu = Γa Γc u γ 2 gy = L γ1,γ 2,γfx; y, Kx = 1 σ+i x s 2πi σ i ζs ds, ux 1 gxkux dx, 19 [ a; τ; s ζs = 2 Ψ 1 c; β ; γ b. The proof of the theorem follows by using of the Mellin integral transform. Corollary. Under the conditions of existing and absolute convergence of integrals, the following equality is valid: x µ 1 L m,m 1,γ e tx gt; x dx = 1 [ Γc a; τ; µ m Γa 2 Ψ m ; γ 1 c; β b L t m 1 µ gt; u, 2 where L is the transform Some applications of the generalized integral transforms Let us give some applications of the generalized integral transforms for evaluation of some integrals, in the theory of differential equations. Example 1. Let gt = e at t ν+µ m.

6 262 N. Virchenko Then taking into account formula [3: L t m 1 µ gt; u = L t ν 1 e αt ; u = Γνu + α ν, we have: Re u > Re α x µ 1 L m,m 1,γ t ν+µ m e α+ut ; x dx = 1 ΓcΓν m Γa [ a; τ; µ m ; γ c; β b u + α ν. we get Example 2. Let gt = t µ m sin αt sin βt. Then taking into account formula [3: L t 1 sin αt sin βt; u = 2 2 ln u2 + α + β 2 u 2 + α β 2, Re u > Im ±α ± β x µ 1 L m,m 1,γ t µ m e tu sin αt sin βt; x [ a; τ; µ m ; γ 2 Γc = 2 mγa 2 Ψ 1 Remark. Noticing that c; β dx b L m fx; y = 1 m L f m z; y m, we obtain the inversion formula for L m fx; y: ln u2 + α + β 2 u 2 + α β 2. where fx = L 1 m F s; x = 1 2πi Example 3. F s = L m ft; s = c+i c i 2F m s e sxm ds, 21 t m 1 e xm t m ft dt. Let us consider the following problem: t 2 u + 1 m u t2 t t2m 1 2 u x 2 = tm x, x, t >, 22 ux, = ; u, t = 1, u xx, =. 23

7 On Some Generalizations of Classical Rewrite the differential equation 22: δ 2 mux, t 2 u x 2 = x, 24 tm 1 where δm 2 1 d = 1 m t 2m 1 dt + 1 d 2 t 2m 2 dt Let us apply to 24 the generalized integral Laplace transform L m. We obtain U xxx, s m 2 s 2m Ux, s = Γ 1 m x, 26 ms where Ux, s = L m ux, t, s. The solution of the differential equation is: Ux, s = C 1 e msm + C 2 e msm Γ 1 m m 3 s Taking into account 23 we have: C 1 =, C 2 = 1 ms m, from where Γ 1 m Ux, s = 1 ms m e msm = 2m+1 x. m 3 x. s2m+1 Applying the inversion formula for this integral transform 21 we get ux, t = 1 σ+i 1 Γ 1 2πi σ i s e ms m x e stm ds. 27 m 2 s 2+ 1 m References [1 M. Al-Hajri, S.L. Kalla, On a integral transform involving Bessel function, Bull. Soc. Math. Maced , [2 A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Fricomi, Higher Transcendental Functions, Vol. 1, Mc Graw-Hill, New York [3 A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Fricomi, Tables of Integral Transforms, Mc Graw-Hill, New York [4 A.A. Kilbas, M. Saigo, The H-Transforms, Chapman and Hall / CRC 24.

8 264 N. Virchenko [5 H.M. Srivastava, O. Yürekli, A theorem on Widder s potential transfo and its applications, J. Math. Anal. Appl , [6 N. Virchenko, On the generalized confluent hypergeometric function and its applications, Fract. Calc. Appl. Anal. 9, No 2 26, [7 N. Virchenko, S. L. Kalla, M. Al.-Zamel, Some results on a generalized hypergeometric function, Integr. Transf. Spec. Funct. 12, No 1 211, National Technical University of Ukraine Kyiv, UKRAINE nvirchenko@hotmail.com Received: October 21, 211