VKiryakova_fcaa113

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314 V. Kiryakova( x) m(γ= I k ),(1)I(eγ k ),(−η k )m,m−1 m,m−1 f(x) ,mthat is, D η I m f(x) = LD η f(x). For more details, see in [15], §3.5.Corollary 4. If ũ ∈ C (∞)m+max(α,−1)is a solution of the m-th orderdifferential equation ũ (m) (x) = λ ũ(x), then the P-D image u = D η ũ ∈C (∞)m+max(α,−1)is a solution of the hyper-Bessel differential equation Bu(x) =λ u(x) with parameters as in (48).Let us consider now some illustrations of the use of the P-D differentialtransformation. We use some examples of ODEs of hyper-Bessel type takenfrom the popular Kamke’s book [14].Example 2.Consider the ODEx y (m) (x) − n m y (m−1) (x) + a x y(x) = 0, 0 < x < ∞, (52)where m and n are natural numbers, m ≥ 2, a = const. It can be writtenalso in the form B y(x) = −y(x) with the m-th order hyper-Bessel operatorB = x −m Q m (x ddx ), where: β = m, γ 1 = 1 m − (n + 1); γ k = k m − 1 := ˜γ k, k =2, . . . , m. Therefore, ∆ 1 = η −1 = n > 0 - an integer, and ∆ k = η k = 0, k =2, . . . , m. Then the corresponding P-D transformation (50) has the form ofthe differential operatorn∏(D η f(x)= x m d )d(x m ) +γ 1+j f(x)=j=1([x (n+1)m−1 x 1−m d ) n ]x 1−m f(x),dxand, according to Corollary 4, if ỹ(x) denotes a solution of the simplerequation ỹ (m) (x) + a y(x) = 0, then the(P-D imagey(x) = D η ỹ(x) =[x (n+1)m−1 x 1−m d ) n ]x 1−m ỹ(x) (53)dxis a solution of eq. (52). The same result is given in Kamke [14], p. 483, forProblem 5.6.Example 3.Consider the 3-rd order ODE equationx 2 y ′′′ (x) − 3(p + q)xy ′′ (x) + 3p(3q + 1)y ′ (x) − x 2 y(x) = 0, 0 < x < ∞, (54)with integers p > 0, q > 0. It is a hyper-Bessel equation of the form By = y:B = x −3 Q 3 (x x dx ) with m = β = 3; γ 1 = q − 2 3 , γ 2 = −p − 1 3 , γ 3 = 0.Then the considered differences are: ∆ 1 = η 1 = q, ∆ 2 = η 2 = p, ∆ 3 = 0,and the P-D transformation (50) has the form
TRANSMUTATION METHOD FOR SOLVING . . . 315p−1∏= 3 (p+q)=D η = 3 (p+q)j 1 =0p−1∏j 1 =03∏η k∏k=1 j ′ =1(x 3( 13 x ddx − j 1 − 1 3(x ddx − 3j 1 − 1)dd(x 3 ) + γ k + j ′∏( 13 x x dx − j 2 −3)2) q−1j 2 =0) q−1∏j 2 =0(x x dx − 3j 2 − 2If ỹ(x) denotes a solution of the simpler 3-rd order differential equation∑ỹ ′′′ = ỹ, that is ỹ(x) = 3 c k exp(ω k x), ωk 3 = 1, then as it also shown ink=1Kamke [14], p. 466, Problem 3.49, we can find the solution of (54) in theformp−1∏(y(x) = D η ỹ(x) = x d ) q−1 ∏ (dx − 3j 1 − 1 x x )dx − 3j 2 − 2 ỹ(x).j 2 =0j 1 =0References[1] I. Ali, V. Kiryakova, S.L. Kalla, Solutions of fractional multi-orderintegral and differential equations using a Poisson-type transform. J.Math. Anal. and Appl. 269, No 1 (2002), 172-199.[2] P. Delerue, Sur le calcul symbolique à n variables et fonctions hyperbesséliennes,II. Annales Soc. Sci. Bruxelles, Ser. 1, 3 (1953), 229-274.[3] J. Delsarte, Oeuvres de Jean Delsarte, Vol-s 1,2. Gothier-Villar, Paris(1971).[4] J. Delsarte, J.L. Lions, Transmutations d’opérateurs différentiels dansle domaine complexe. Commentarii Helvetici 32 (1957), 113-128.[5] I. Dimovski, Operational calculus for a class of differential operators.C.R. Acad. Bulg. Sci. 19, No 12 (1966), 1111-1114.[6] I. Dimovski, Foundations of operational calculi for Bessel-type differentialoperators. Serdica (Bulg. Math. Publ.) 1 (1975), 51-63.[7] I. Dimovski, Isomorphism of the quotient fields generated by Besseltypedifferential operators. Math. Nachrichten, 67 (1975), 101-107.[8] I. Dimovski, Convolutional Calculus. Kluwer Acad. Publ., Dordrecht(1990).[9] I. Dimovski, V. Kiryakova, Transmutations, convolutions and fractionalpowers of Bessel-type operators via Meijer’s G-function. In:Proc.“Complex Anal. & Appl-s, Varna’1983” (1985), 45-66.).
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