Fractional Calculus - Laplace Transform

Laplace Transform

We can also come at the question via the Laplace transform. Noting that

and

etc., we assert

.

For example

\begin{array}{lcr} J^\alpha\left(t^k\right) &= &\mathcal L^{-1}\left\{\dfrac{\Gamma(k+1)}{s^{\alpha+k+1}}\right\}\\ &= &\dfrac{\Gamma(k+1)}{\Gamma(\alpha+k+1)}t^{\alpha+k} \end{array}

as expected. Indeed, given the convolution rule (and shorthanding for clarity) we find that

\begin{array}{rcl} (J^\alpha f)(t) &= &\frac{1}{\Gamma(\alpha)}\mathcal L^{-1}\left\{\left(\mathcal L\{p\}\right)(\mathcal L\{f\})\right\}\\ &=&\frac{1}{\Gamma(\alpha)}(p*f)\\ &=&\frac{1}{\Gamma(\alpha)}\int_0^t p(t-\tau)f(\tau)\,d\tau\\ &=&\frac{1}{\Gamma(\alpha)}\int_0^t(t-\tau)^{\alpha-1}f(\tau)\,d\tau\\ \end{array}

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations.

Read more about this topic:  Fractional Calculus

Famous quotes containing the words laplace and/or transform:

    “Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
    —Pierre Simon De Laplace (1749–1827)

    “The inspired scribbler always has the gift for gossip in our common usage ... he or she can always inspire the commonplace with an uncommon flavor, and transform trivialities by some original grace or sympathy or humor or affection.”
    —Elizabeth Drew (1887–1965)