Fractional Calculus - Variable order derivatives and integrals

In summary, the conversation is about finding research on solving integral and differential equations where the unknown is the level of a fractional derivative in the equation. The topic is explored in the Wikipedia article on fractional calculus and in a paper from Reed College Physics Department. Further investigation into named theorems in this field is suggested. The initial conditions needed to solve such an equation are also questioned.
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TL;DR Summary
Looking for papers and research
Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the first term is F to the 1/2 derivative in fractional calculus and the second term is to an unknown power of derivative and the last term is a 1/3 fractional derivative. I am curious if anyone knows of any work on this kind of topic? What would be the initial conditions needed to solve such an equation?
 
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1. What is fractional calculus?

Fractional calculus is a branch of mathematics that deals with derivatives and integrals of non-integer orders. It extends the traditional calculus, which deals with integer orders, to fractional orders. This allows for a more precise and accurate representation of real-world phenomena.

2. What are variable order derivatives and integrals?

Variable order derivatives and integrals refer to the derivatives and integrals that are defined for non-integer orders. In traditional calculus, the order of a derivative or integral is always an integer. However, in fractional calculus, the order can be any real number, allowing for a more flexible and accurate description of systems.

3. What are the applications of fractional calculus?

Fractional calculus has various applications in physics, engineering, and other fields. It is used to model and analyze complex systems with memory, such as viscoelastic materials, electrical circuits, and biological systems. It also has applications in signal processing, control theory, and finance.

4. How is fractional calculus different from traditional calculus?

Fractional calculus differs from traditional calculus in that it deals with derivatives and integrals of non-integer orders. It also has a different set of rules and properties compared to traditional calculus. Additionally, fractional calculus allows for a more accurate and realistic representation of real-world systems.

5. What are some challenges in studying fractional calculus?

One of the main challenges in studying fractional calculus is the lack of a unified theory. There are several different approaches and definitions of fractional derivatives and integrals, making it difficult to compare and unify results. Additionally, there is a lack of standard techniques and tools for solving fractional calculus problems, making it a relatively new and developing field of study.

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