This book contains a brief historical introduction and state of the art in fractional calculus. The author introduces some of the so-called special functions, in particular, those which will be directly involved in calculations. The concepts of fractional integral and fractional derivative are also presented. Each chapter, except for the first one, contains a list of exercises containing suggestions for solving them and at last the resolution itself. At the end of those chapters there is a list of complementary exercises. The last chapter presents several applications of fractional calculus.

Autorentext

Edmundo Capelas de Oliveira is a Professor and Researcher at the Department of Applied Mathematics at IMECC, Unicamp, Brazil. He has won the First Award for Undergraduate Teaching Incentive for outstanding teaching skills at undergraduate level in 2012.

Inhalt

Preface; 1. A bit of history; 2. Special functions-2.1 Functions and Pochhammer symbol. 2.2 Hypergeometric function and particular cases. 2.3 Confluent hypergeometric function and particular cases. 2.4 Generalized hypergeometric functions-2.4.1 Wright function. 2.4.2 Meijer's G-function. 2.4.3 Fox's H-function. 2.5 Exercises. 2.5.1 Exercise list. 2.5.2 Suggestions. 2.5.3. Solutions. 2.5.4 Proposed exercises.

3. Mittag-Leffler functions. 3.1 The Mittag-Leffler function. 3.2 Wright and Mainardi functions. 3.3 Exercises. 3.3.1 Exercise list. 3.3.2 Suggestions. 3.3.3. Solutions. 3.3.4 Proposed exercises.

   4. Integral transforms. 4.1 Methodology. 4.2 Fourier transform. 4.3 Laplace transform. 4.4. Mellin transform. 4.5 Exercises. 4.5.1 Exercise list. 4.5.2 Suggestions. 4.5.3. Solutions. 4.5.4 Proposed exercises.
   5. Fractional derivatives. 5.1 Grunwald-Letnikov derivative. 5.2 Integer order integral. 5.3 Riemann-Liouville and Hadamard integrals. 5.4 Riemann-Liouville, Caputo and Hadamard derivatives. 5.5 Exercises. 5.5.1 Exercise list. 5.5.2 Suggestions. 5.5.3. Solutions. 5.5.4 Proposed exercises.
   6. Applications and add-ons. Forty discussed applications and complements.

   Appendix. Mellin-Barnes integrals.

   References. More than one hundred references

   Remissive index. More than three hundred entries