"A Novel Approach to Fractional Calculus: Utilizing Fractional Integral" by Evan Camrud

Progress in Fractional Differentiation & Applications

Article Title

A Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function

Authors

Evan Camrud, Department of Mathematics, Concordia College, Moorhead, MN, USA\\ Department of Mathematics, Iowa State University, Ames, IA, USAFollow

Author Country (or Countries)

USA

Abstract

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.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/pfda/040402

Recommended Citation

Camrud, Evan (2018) "A Novel Approach to Fractional Calculus: Utilizing Fractional Integrals and Derivatives of the Dirac Delta Function," Progress in Fractional Differentiation & Applications: Vol. 4: Iss. 4, Article 2.
DOI: http://dx.doi.org/10.18576/pfda/040402
Available at: https://digitalcommons.aaru.edu.jo/pfda/vol4/iss4/2

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