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Fractional Derivative and Fractional Integral

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Introduction to Fractional Differential Equations

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 25))

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Abstract

For every α > 0 and a local integrable function f(t), the right FI of order α is defined:

$$\displaystyle{ }_aI_t^\alpha f(t) = \displaystyle\frac {1}{\Gamma (\alpha )}\displaystyle\int _a^t(t - u)^{\alpha - 1}f(u)du,\qquad-\infty \le a < t < \infty .$$

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Notes

  1. 1.

    G.F.B. Riemann (1826–1866).

  2. 2.

    J. Liouville (1809–1882).

  3. 3.

    J.P.G.L. Dirichlet (1805–1859).

  4. 4.

    A.L. Cauchy (1789–1857).

  5. 5.

    M. Caputo (1967–).

  6. 6.

    B. Taylor (1685–1731).

References

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  2. Ishteva, M., Scherer, R., & Boyadjiev, L. (2003). On the Caputo operator of fractional calculus and C-Laguerre functions. Technical report, Bulgarian Ministry of Education and Science, Grant MM 1305/2003.

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  3. Miller, K. S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. New York: John Wiley & Sons.

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  4. Valério, D., & Costa, J. S. (2013). An introduction to fractional control. London: CRC Press.

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  5. Podlubny, I. (1998). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in science and engineering. San Diego: Academic Press.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Polytechnic University of Timişoara, Timişoara, Romania

    Constantin Milici

  2. Research Center in Theoretical Physics, West University of Timişoara, Timişoara, Romania

    Gheorghe Drăgănescu

  3. Department of Electrical Engineering, Polytechnic of Porto, Porto, Portugal

    J. Tenreiro Machado

Authors
  1. Constantin Milici
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  2. Gheorghe Drăgănescu
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  3. J. Tenreiro Machado
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Milici, C., Drăgănescu, G., Tenreiro Machado, J. (2019). Fractional Derivative and Fractional Integral. In: Introduction to Fractional Differential Equations. Nonlinear Systems and Complexity, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-00895-6_2

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  • DOI: https://doi.org/10.1007/978-3-030-00895-6_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00894-9

  • Online ISBN: 978-3-030-00895-6

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