An introduction to the fractional calculus and fractional differential equations.

*(English)*Zbl 0789.26002
New York: John Wiley & Sons, Inc.. xiii, 366 p. (1993).

The reader will find here a systematic treatment of the theory of fractional calculus and its applications in the solution of fractional differential equations and fractional difference equations.

The following are the main features of the book:

There are eight chapters. The historical development of the fractional calculus from 1790 to the present is given in Chapter I. Several interesting mathematical arguments concerning the definition of fractional calculus are discussed in Chapter II which lead to the present definition of fractional integrals and derivatives. Chapter III is devoted mainly in developing the theory of Riemann-Liouville integral. Certain new techniques are investigated in finding the fractional integrals of more complicated functions. The theory of Riemann-Liouville fractional calculus is discussed in Chapter IV. The integral and differential representations of the ordinary special functions occurring in applied mathematics are derived as fractional integrals and derivatives which enhance the utility of fractional calculus. Chapter V deals with certain properties of fractional differential equations with constant coefficients. Chapter VI gives a method for deriving the solution of fractional integral equations, fractional differential equations with non-constant coefficients, sequential fractional differential equations, and vector fractional differential equations. The theory of Weyl fractional calculus is dealt with in Chapter VII. Certain selected physical problems are discussed in the last chapter which lead to fractional integral or differential equations.

There are four appendices which have further enhanced the utility of the book. Appendix A deals with some identities associated with partial fraction expansions. Appendix B contains elementary properties of certain higher transcendental functions. Laplace transforms as applied to the functions \(E_ t(\nu,a)\), \(C_ t(\nu,a)\), and \(S_ t(\nu,a)\), are discussed in Appendix C including short tables of these functions. Appendix D contains a brief table of fractional integrals and derivatives.

The book is well written which may be used as a text or a reference book. It contains many results from research papers published during the last decade, hence it is useful to research workers in the field of fractional calculus, special functions, integral equations and integral transforms.

The following are the main features of the book:

There are eight chapters. The historical development of the fractional calculus from 1790 to the present is given in Chapter I. Several interesting mathematical arguments concerning the definition of fractional calculus are discussed in Chapter II which lead to the present definition of fractional integrals and derivatives. Chapter III is devoted mainly in developing the theory of Riemann-Liouville integral. Certain new techniques are investigated in finding the fractional integrals of more complicated functions. The theory of Riemann-Liouville fractional calculus is discussed in Chapter IV. The integral and differential representations of the ordinary special functions occurring in applied mathematics are derived as fractional integrals and derivatives which enhance the utility of fractional calculus. Chapter V deals with certain properties of fractional differential equations with constant coefficients. Chapter VI gives a method for deriving the solution of fractional integral equations, fractional differential equations with non-constant coefficients, sequential fractional differential equations, and vector fractional differential equations. The theory of Weyl fractional calculus is dealt with in Chapter VII. Certain selected physical problems are discussed in the last chapter which lead to fractional integral or differential equations.

There are four appendices which have further enhanced the utility of the book. Appendix A deals with some identities associated with partial fraction expansions. Appendix B contains elementary properties of certain higher transcendental functions. Laplace transforms as applied to the functions \(E_ t(\nu,a)\), \(C_ t(\nu,a)\), and \(S_ t(\nu,a)\), are discussed in Appendix C including short tables of these functions. Appendix D contains a brief table of fractional integrals and derivatives.

The book is well written which may be used as a text or a reference book. It contains many results from research papers published during the last decade, hence it is useful to research workers in the field of fractional calculus, special functions, integral equations and integral transforms.

Reviewer: Ram Kishore Saxena (Jodhpur)

##### MSC:

26A33 | Fractional derivatives and integrals |