Theory for general fractional differintegral equations?

Question

I am aware there exist ways to construct fractional calculus, fractional differential operators and integral operators, for example by using Cauchy integral theorem in complex analysis or by Fourier analysis.

But do there exist any theory for differential equations involving such fractional differential and integral operators?

For context, a simple example of equation is $f^{(1/2)}(t) = 2f(t)$ ( which I don't know solution to ).

If we half-differentiate both sides

(would this make sense? would such an operation be equivalence relation?)

we get $f'(t) = 2f^{(1/2)}(t)$ which implies $f'(t) = 4f(t)$ and now we have something we can solve using normal differential equation theory.

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So maybe we can solve easy special cases like this one using ordinary theory of DE, but for more complicated ones, does there exist any theory for how to approach those?

Answer 1

Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, 2010:
Theory of Fractional Differential Equations

• Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations
• Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results
• Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases
• Multi-Term Caputo Fractional Differential Equations