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A Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function |
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PP: 463-478 |
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doi:10.18576/pfda/040402
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Author(s) |
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Evan Camrud,
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Abstract |
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| While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative
has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their
results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and
shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta function and fractional Laplace and
Fourier transforms. A new class of functions, known as Zero Functions and closely related to the Dirac delta function, are necessary for
one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra
integral equation, and provides a method of solving for Riemann’s complimentary function introduced during his research on fractional
derivatives. |
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