A New Way to Define Fractional Derivatives

TLDRA new method for computing fractional derivatives using a combination of fractional integrals and ordinary derivatives has been discovered. This method avoids the shortcomings of the traditional fractional integral formula and successfully computes fractional derivatives with positive order values.

Key insights

🔑Fractional derivatives can be defined using a combination of fractional integrals and ordinary derivatives.

🔑The traditional formula for fractional integrals does not work for negative order values or non-positive values of p.

🔑Computing a half derivative can be achieved by first computing a half integral and then taking the ordinary derivative.

🔑The new method successfully computes fractional derivatives with positive order values.

🔑The method follows the property that derivatives cancel out integrals.

Q&A

What is the traditional method for computing fractional derivatives?

—The traditional method involves using a fractional integral formula, but it has limitations for negative order values and non-positive values of p.

What is the new method for computing fractional derivatives?

—The new method involves computing a fractional integral and then taking the ordinary derivative to achieve the desired fractional derivative.

Does the new method work for all fractional order values?

—No, the new method only works for positive order values.

What is the significance of the square root of pi in the fractional integral formula?

—The square root of pi is the result of evaluating the gamma function for negative order values of p.

How does the new method compare to the traditional method?

—The new method avoids the shortcomings of the traditional method and successfully computes fractional derivatives with positive order values.

Timestamped Summary

00:01A new method for computing fractional derivatives using a combination of fractional integrals and ordinary derivatives has been discovered.

02:56The traditional formula for fractional integrals does not work for negative order values or non-positive values of p.

05:57Computing a half derivative can be achieved by first computing a half integral and then taking the ordinary derivative.

09:13The new method successfully computes fractional derivatives with positive order values.

11:47The method follows the property that derivatives cancel out integrals.

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