# Convolutional Ring *C*_{α}

Chapter

## Abstract

In the 1950-th J. Mikusinski proposed a approach to the operational calculus of the operator of differentiation which is based on interpretation of the Laplace convolution as a multiplication in the ring of functions continuous on the half-axis and it is well known that many of the problems in mathematics such as ordinary differential equations, integral equations, partial differential equations and special functions were solved by using the Mikusinski’s operational calculus. It is worth mentioning that the Mikusinski’s scheme was used by several mathematicians for the development of operational calculi for differential operators with variable coefficients (see, for example, V.A. Ditkin (1957), V.A. Ditkin and A.P. Prudnikov (1963), N.A. Meller (1960), J. Rodrigues (1989)). These operators are particular cases of the hyper-Bessel differential operator and the case of the operator
was considered by I.H. Dimovski (1966). These calculi were used, in particular, to solve some linear differential equations with variable coefficients by V.A. Ditkin and A.P. Prudnikov (1963), N.A. Meller (1960), J. Rodrigues (1989). New result has been obtained in Yu. F. Luchko (1993), Yu. F. Luchko and S.B. Yakubovich (1993), (1994), where the operational calculus for the multiple Erdelyi-Kober fractional differentiation operator, which, among particular cases, contains both the operator (18.1) and the Riemann-Liouville fractional differentiation operator, was developed. This operational calculus is based on the interpretation of the convolution (11.34) of the generalized Obrechkoff transform (3.105) as an operation of multiplication in some space of functions. In this and the following chapters, we present these results as well as some new results concerning the operational calculus based on the convolution of the Kontorovich-Lebedev transform and their applications.

$$
\left( {Bf} \right)\left( x \right) = x^{ - \beta } \prod\limits_{i = 1}^n {\left( {\gamma i + \frac{1}
{\beta }x\frac{d}
{{dx}}} \right)f\left( x \right)}
$$

(1)

## Keywords

Differential Operator Linear Differential Equation Fractional Integration Operational Calculus Fractional Integration Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media Dordrecht 1994