Theory And Numerical Approximations Of Fractional Integrals And Derivatives ~upd~ Jun 2026

Integrate a fractional order of an integer derivative (more common in engineering). $$ a^CD^\alpha t f(t) = aI^n-\alpha t \left( \fracd^nf(t)dt^n \right), \quad n-1 < \alpha \le n$$ Advantage: Uses standard integer-order initial conditions $f(a), f'(a), \dots$

The reverses the order of operations—it first differentiates integer-order, then integrates fractionally: Integrate a fractional order of an integer derivative

Fractional calculus, a branch of mathematics that deals with derivatives and integrals of arbitrary order, has gained significant attention in recent years due to its applications in various fields, including physics, engineering, economics, and computer science. In this content, we will discuss the theory and numerical approximations of fractional integrals and derivatives. Unlike integer calculus, where the derivative is unique,

Unlike integer calculus, where the derivative is unique, several definitions of fractional derivatives exist. The choice depends on the problem's initial/boundary conditions and desired properties. Unlike integer calculus

$$t^-\alpha \approx \sum_j=1^N_\textexp w_j e^-s_j t$$

Modeling materials like polymers and polymers that exhibit both liquid and solid characteristics (viscoelasticity).