## An extension of the finite Hankel transform.(English)Zbl 1055.44005

Integral transforms have been used extensively to solve certain class of initial and boundary value problems. I. Ali and the reviewer [J. Aust. Math. Soc., Ser. B 41, 105–117 (1999; Zbl 0982.44003)] introduced a generalized form of the Hankel transform and applied it to the problem of a heavy pollutant from a ground level aerial source within the framework of diffusion theory. Recently, H. G. Khajah [Integral Transforms and Spec. Funct. 14, No. 5, 403–412 (2003; Zbl 1049.44005)] considered a modified form of a finite Hankel transform.
In this paper, the authors consider a generalized finite Hankel transform involving Bessel functions as kernel. Several properties including an inversion formula are given. This transform is suitable for solving certain class of mixed boundary value problems. A problem involving the flow through concentric cylinders when one of the cylinders starts rotating impulsively with a uniform angular velocity, while the other is kept fixed is solved by the use of this transform.

### MSC:

 44A15 Special integral transforms (Legendre, Hilbert, etc.) 44A20 Integral transforms of special functions 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 35A22 Transform methods (e.g., integral transforms) applied to PDEs

### Citations:

Zbl 0982.44003; Zbl 1049.44005
Full Text:

### References:

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