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Open AccessArticle

Several Results of Fractional Differential and Integral Equations in Distribution

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Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
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Department of Mathematics, Shanghai University, Shanghai 200444, China
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Author to whom correspondence should be addressed.
Mathematics 2018, 6(6), 97; https://doi.org/10.3390/math6060097
Received: 10 May 2018 / Revised: 3 June 2018 / Accepted: 6 June 2018 / Published: 8 June 2018
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
This paper is to study certain types of fractional differential and integral equations, such as θ ( x x 0 ) g ( x ) = 1 Γ ( α ) 0 x ( x ζ ) α 1 f ( ζ ) d ζ , y ( x ) + 0 x y ( τ ) x τ d τ = x + 2 + δ ( x ) , and x + k 0 x y ( τ ) ( x τ ) α 1 d τ = δ ( m ) ( x ) in the distributional sense by Babenko’s approach and fractional calculus. Applying convolutions and products of distributions in the Schwartz sense, we obtain generalized solutions for integral and differential equations of fractional order by using the Mittag-Leffler function, which cannot be achieved in the classical sense including numerical analysis methods, or by the Laplace transform. View Full-Text
Keywords: distribution; fractional calculus; convolution; Abel’s integral equation; product; Mittag-Leffler function distribution; fractional calculus; convolution; Abel’s integral equation; product; Mittag-Leffler function
MDPI and ACS Style

Li, C.; Li, C.; Clarkson, K. Several Results of Fractional Differential and Integral Equations in Distribution. Mathematics 2018, 6, 97.

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