# The Causal Fractional Derivatives

## Abstract

The fractional calculus is a 300 years old mathematical discipline. In fact and some time after the publication of the studies on Differential Calculus, where he introduced the notation \( {\frac{{{\text{d}}^{n} y}}{{{\text{d}}x^{n} }}} ,\) Leibnitz received a letter from Bernoulli putting him a question about the meaning of a non-integer derivative order. Also he received a similar enquiry from L’Hôpital: *What if n is* ½? Leibnitz’s replay was prophetic: *It will lead to a paradox*, *a paradox from which one day useful consequences will be drawn*, *because there are no useless paradoxes*. It was the beginning of a discussion about the theme that involved other mathematicians like Euler and Fourier. Euler suggested in 1730 a generalisation of the rule used for computing the derivative of the power function. He used it to obtain derivatives of order 1/2. Nevertheless, we can say that the XVIII century was not proficuous in which concerns the development of Fractional Calculus. Only in the early XIX, interesting developments started being published. Laplace proposed an integral formulation (1812), but it was Lacroix who used for the first time the designation “derivative of arbitrary order” (1819).

## Keywords

Fractional Derivative Fractional Calculus Laplace Transform Fractional Integration Caputo Derivative## References

- 1.Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives—theory and applications. Gordon and Breach Science Publishers, New YorkGoogle Scholar
- 2.Hoskins RF (1999) Delta functions. Horwood Series in Mathematics & Applications, Chichester, EnglandzbMATHGoogle Scholar
- 3.Dugowson S (1994) Les différentielles métaphysiques. PhD thesis. Université Paris NordGoogle Scholar
- 4.Diaz JB, Osler TJ (1974) Differences of fractional order. Math Comput 28(125):185–202CrossRefzbMATHMathSciNetGoogle Scholar
- 5.Ortigueira MD (2004) From differences to differintegrations. Fractional Calculus Appl Anal 7(4):459–471zbMATHMathSciNetGoogle Scholar
- 6.Ortigueira MD (2006) A coherent approach to non integer order derivatives. Signal processing special section: fractional calculus applications in signals and systems, vol 86(10). pp 2505–2515Google Scholar
- 7.Oldham KB, Spanier J (1974) The fractional calculus: theory and application of differentiation and integration to arbitrary order. Academic Press, New YorkGoogle Scholar
- 8.Krempl PW Solution of linear time invariant differential equations with ‘proper’ primitives. IEEE 32nd Annual Conference on Industrial Electronics, IECON 2006, 6-10 Nov, Paris, FranceGoogle Scholar
- 9.Ferreira JC (1997) Introduction to the theory of distributions. Pitman monographs and surveys in pure and applied mathematics. Longman, HarlowGoogle Scholar
- 10.Hoskins RF, Pinto JS (1994) Distributions, ultradistributions, and other generalised functions. Ellis Horwood Limited, ChichesterzbMATHGoogle Scholar
- 11.Silva JS (1989) The axiomatic theory of the distributions. Complete Works, INIC, LisbonGoogle Scholar
- 12.Zemanian AH (1987) Distribution theory and transform analysis. Dover Publications, New YorkzbMATHGoogle Scholar
- 13.Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
- 14.Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, San DiegoGoogle Scholar
- 15.Chaudhry MA, Zubair SM (2002) On a class of incomplete gamma functions with applications. Chapman & Hall, CRC, London, Boca RatonGoogle Scholar
- 16.Henrici P (1991) Applied and computational complex analysis, vol 2. Wiley, New York, pp 389–391Google Scholar
- 17.Ortigueira MD (2000) Introduction to fractional signal processing. Part 1: continuous-time systems. IEE Proc Vis Image Signal Process 1:62–70CrossRefGoogle Scholar
- 18.Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkzbMATHGoogle Scholar
- 19.Magin RL (2006) Fractional calculus in bioengineering. Begell House, ConnecticutGoogle Scholar
- 20.Ortigueira MD (2001) The comb signal and its Fourier transform. Signal Process 81(3):581–592CrossRefzbMATHMathSciNetGoogle Scholar
- 21.Podlubny I (2002) Geometric and physical interpretation of fractional integration and fractional differentiation. J Fract Calc Appl Anal 5(4):367–386zbMATHMathSciNetGoogle Scholar
- 22.Machado JAT (2003) A probabilistic interpretation of the fractional-order differentiation. Fractional Calculus Appl Anal 6(1):73–80zbMATHGoogle Scholar
- 23.Machado JAT (2009) Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun Nonlinear Sci Numer Simulat 14:3492–3497CrossRefGoogle Scholar