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Babenko’s Approach to Abel’s Integral Equations

Author

Listed:
  • Chenkuan Li

    (Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada)

  • Kyle Clarkson

    (Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada)

Abstract

The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t ) + λ Γ ( α ) ∫ 0 t ( t − τ ) α − 1 y ( τ ) d τ = f ( t ) , ( t > 0 ) and its variants by fractional calculus. Applying Babenko’s approach and fractional integrals, we provide a general method for solving Abel’s integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary α ∈ R by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.

Suggested Citation

  • Chenkuan Li & Kyle Clarkson, 2018. "Babenko’s Approach to Abel’s Integral Equations," Mathematics, MDPI, vol. 6(3), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:3:p:32-:d:134174
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    References listed on IDEAS

    as
    1. Ming Li & Wei Zhao, 2013. "Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-4, May.
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    Cited by:

    1. 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.

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