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On Fractional Integrals And Derivatives Of A Function With Respect To Another Function

Author

Listed:
  • JUAN J. NIETO

    (CITMAga, Departamento de Estatística, Análise Matemática e Optimización, University of Santiago de Compostela, Santiago de Compostela 15782, Spain)

  • MADEAHA ALGHANMI

    (��Department of Mathematics, College of Sciences and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia‡Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia)

  • BASHIR AHMAD

    (��Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia)

  • AHMED ALSAEDI

    (��Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia)

  • BOSHRA ALHARBI

    (��Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia)

Abstract

In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem.

Suggested Citation

  • Juan J. Nieto & Madeaha Alghanmi & Bashir Ahmad & Ahmed Alsaedi & Boshra Alharbi, 2023. "On Fractional Integrals And Derivatives Of A Function With Respect To Another Function," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(04), pages 1-15.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:04:n:s0218348x23400662
    DOI: 10.1142/S0218348X23400662
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