Several Results of Fractional Differential and Integral Equations in Distribution

Chenkuan Li, Changpin Li and Kyle Clarkson
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Chenkuan Li: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Changpin Li: Department of Mathematics, Shanghai University, Shanghai 200444, China
Kyle Clarkson: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada

Mathematics, 2018, vol. 6, issue 6, 1-19

Abstract: 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.

Keywords: distribution; fractional calculus; convolution; Abel’s integral equation; product; Mittag-Leffler function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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