Remarks on Convolutions and Fractional Derivative of Distributions


  •  Chenkuan Li    
  •  Kyle Clarkson    

Abstract

This paper begins to present relations among the convolutional definitions given by Fisher and Li, and further shows that the following fractional Taylor's expansion holds based on convolution \[ \frac{d^\lambda}{d x^\lambda}  \theta (x) \phi(x) = \sum_{k = 0}^{\infty} \frac{\phi^{( k)}(0)\, x_+^{k - \lambda }}{\Gamma(k - \lambda  + 1)} \quad \mbox{if} \quad \lambda \geq 0, \] with demonstration of several examples.  As an application, we solve the Poisson's integral equation below  \[ \int_0^{\pi/2} f(x \cos \omega)\sin^{2 \lambda + 1} \omega d \omega = \theta(x) g(x) \] by fractional derivative of distributions and the Taylor's expansion obtained.


This work is licensed under a Creative Commons Attribution 4.0 License.
  • Issn(Print): 1916-9795
  • Issn(Onlne): 1916-9809
  • Started: 2009
  • Frequency: bimonthly

Journal Metrics

Google-based Impact Factor (2018): 3.1

  • h-index (August 2018): 16
  • i10-index (August 2018): 35
  • h5-index (August 2018): 9
  • h5-median (August 2018): 9

( The data was calculated based on Google Scholar Citations. Click Here to Learn More. )

Contact