Solving Differential Equations using Fractional Calculus

Question

Are there any integer order differential equations or integral equations that can be solved using fractional calculus? For example, the Abel's Integral (like from the Tautochrone Problem): $$\int_{0}^{h}\dfrac{u(y)}{\sqrt{h-y}}dy=C$$ where $C$ is a constant. Given the Riemann-Liouville Integral: $$I^{\alpha}f(x)=\dfrac{1}{\Gamma(\alpha)}\int f(t)(x-t)^{\alpha-1}dt$$ This problem can easily be solved by representing the integral by the half integral of $u(y)$ and taing the half derivative on both sides.

Are there any other such differential equations that can be solved using fractional calculus?

You showed an example of an integral, not differential, equation. — Gonçalo, Nov 15 '23 at 19:24
Fractional differential equations are differential equations which, among other things, can be solved using fractional calculus, e.g. $\left[D^{1.5}-D^{0}\right]\cdot y\left(x\right)=0$.You can also solve differential equations with derivatives that do not have an explicit number as order but an unknown constant, e.g. $\left[D^{2v}+\alpha D^{v}+\beta D^{0}\right]\cdot y\left(x\right)=0$ with fractional calculus. Do you mean something like that? — The Art Of Repetition, Nov 19 '23 at 11:07
@KevinDietrich I am more interested in whether there any such results using fractional calculus to solve ordinary order differential or integral equations. Are there any other examples of where this can be used? — Anirudh Yamunan Govindarajan, Nov 19 '23 at 17:43

Solving Differential Equations using Fractional Calculus

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