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Several Results of Fractional Differential and Integral Equations in Distribution

Author

Listed:
  • 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)

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.

Suggested Citation

  • Chenkuan Li & Changpin Li & Kyle Clarkson, 2018. "Several Results of Fractional Differential and Integral Equations in Distribution," Mathematics, MDPI, vol. 6(6), pages 1-19, June.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:6:p:97-:d:151390
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    References listed on IDEAS

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    1. Chenkuan Li & Kyle Clarkson, 2018. "Babenko’s Approach to Abel’s Integral Equations," Mathematics, MDPI, vol. 6(3), pages 1-15, March.
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