Derivatives of noninteger order and their applications.

*(English)*Zbl 0880.26007This interesting paper is devoted to the investigation of fractional calculus of functions of many variables and applications to solution of initial and boundary value problems for ordinary and partial differential equations of fractional order, to operational calculus and to multidimensional complex analysis. The main constructions of multidimensional fractional calculus considered on bounded domains in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) are the so-called mixed Riemann-Liouville fractional integrals and derivatives – see Section 24.2 in the book of S. G. Samko, the reviewer and O. I. Marichev [“Integrals and derivatives of fractional order and some of their applications” (Russian) (1987; Zbl 0617.26004); (English edition, 1993; Zbl 0818.26003)].

In Chapter I the problems of continuity and complete continuity of such fractional integro-differentiation operators are studied as well as the conditions for their existence, semigroup property, analogy of Taylor’s formula, solvability conditions for the multidimensional Abel integral equation, etc. These properties are similar to those for the classical one-dimensional Riemann-Liouville fractional integrals and derivatives – see Section 2 in the book above.

Chapter II deals with a characteristic boundary vlue problem for the \(n\)-dimensional nonlinear differential equation of fractional order which generalizes the Darboux problem for the Mangeron polyvibrating equation. When \(n=1\) such an ordinary fractional differential equation was examined by E. Pitcher and W. E. Sewell [Bull. Am. Math. Soc. 44, 100-107 (1938; Zbl 0019.40801)] and for \(n=2\) the corresponding two-dimensional partial differential equation of fractional order by J. Conlan [Appl. Anal. 14, 167-177 (1983; Zbl 0502.35062)]. The problem above is reduced to a nonlinear integral equation; sufficient conditions for the existence and uniqueness of its solution are given and continuous dependence of this solution on the initial data is proved.

In Chapter III a noncharacteristic boundary value problem which deals with the counterpart of Z. Szmydt’s problem [Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys. 5, 577-582 (1957; Zbl 0078.08701)] for a two-dimensional nonlinear partial fractional differential equation is considered and the conditions of its local and global solutions are given. The results obtained extend those of J. Conlan in the paper above.

Chapter IV is devoted to examine a solution of a multipoint problem for a certain nonlinear ordinary differential equation of fractional order. In particular, the exact solution of the Cauchy problem for the simplest linear fractional differential equation is constructed. This equation generalizes the equation which plays an important role in polarography – see, for example, the paper of K. B. Oldham and J. Spanier [J. Electroanal. Chem. 26, 331-341 (1970)] for chemical background.

In Chapter V the mixed Riemann-Liouville fractional integro-differentiation is used to construct some new examples of Mikusiński operators and to prove the Cauchy and Schwartz integral formulae for analytic functions of several complex variables defined in a polydisc. The latter extend to the case of arbitrary \(n\) the results of M. M. Dzhrbashyan [“Integral transforms and representations of functions in the complex domain” (Russian) (1966; Zbl 0157.37702), p. 594] obtained for \(n=1\).

In conclusion, we note that a number of the results presented have been obtained by the author earlier [Appl. Anal. 28, No. 2, 151-161 (1988; Zbl 0644.35087); ibid. 40, No. 2/3, 123-137 (1991; Zbl 0757.35039); Z. Anal. Anwendungen 8, No. 5, 479-483 (1989; Zbl 0681.34019); ibid. 10, No. 2, 205-210 (1991; Zbl 0761.34001)].

In Chapter I the problems of continuity and complete continuity of such fractional integro-differentiation operators are studied as well as the conditions for their existence, semigroup property, analogy of Taylor’s formula, solvability conditions for the multidimensional Abel integral equation, etc. These properties are similar to those for the classical one-dimensional Riemann-Liouville fractional integrals and derivatives – see Section 2 in the book above.

Chapter II deals with a characteristic boundary vlue problem for the \(n\)-dimensional nonlinear differential equation of fractional order which generalizes the Darboux problem for the Mangeron polyvibrating equation. When \(n=1\) such an ordinary fractional differential equation was examined by E. Pitcher and W. E. Sewell [Bull. Am. Math. Soc. 44, 100-107 (1938; Zbl 0019.40801)] and for \(n=2\) the corresponding two-dimensional partial differential equation of fractional order by J. Conlan [Appl. Anal. 14, 167-177 (1983; Zbl 0502.35062)]. The problem above is reduced to a nonlinear integral equation; sufficient conditions for the existence and uniqueness of its solution are given and continuous dependence of this solution on the initial data is proved.

In Chapter III a noncharacteristic boundary value problem which deals with the counterpart of Z. Szmydt’s problem [Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys. 5, 577-582 (1957; Zbl 0078.08701)] for a two-dimensional nonlinear partial fractional differential equation is considered and the conditions of its local and global solutions are given. The results obtained extend those of J. Conlan in the paper above.

Chapter IV is devoted to examine a solution of a multipoint problem for a certain nonlinear ordinary differential equation of fractional order. In particular, the exact solution of the Cauchy problem for the simplest linear fractional differential equation is constructed. This equation generalizes the equation which plays an important role in polarography – see, for example, the paper of K. B. Oldham and J. Spanier [J. Electroanal. Chem. 26, 331-341 (1970)] for chemical background.

In Chapter V the mixed Riemann-Liouville fractional integro-differentiation is used to construct some new examples of Mikusiński operators and to prove the Cauchy and Schwartz integral formulae for analytic functions of several complex variables defined in a polydisc. The latter extend to the case of arbitrary \(n\) the results of M. M. Dzhrbashyan [“Integral transforms and representations of functions in the complex domain” (Russian) (1966; Zbl 0157.37702), p. 594] obtained for \(n=1\).

In conclusion, we note that a number of the results presented have been obtained by the author earlier [Appl. Anal. 28, No. 2, 151-161 (1988; Zbl 0644.35087); ibid. 40, No. 2/3, 123-137 (1991; Zbl 0757.35039); Z. Anal. Anwendungen 8, No. 5, 479-483 (1989; Zbl 0681.34019); ibid. 10, No. 2, 205-210 (1991; Zbl 0761.34001)].

Reviewer: A.A.Kilbas (Minsk)

##### MSC:

26A33 | Fractional derivatives and integrals |

34A34 | Nonlinear ordinary differential equations and systems |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

45Gxx | Nonlinear integral equations |

44A40 | Calculus of Mikusiński and other operational calculi |