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Posted on Authorea 29 May 2020 — The copyright holder is the author/funder. All rights reserved. No reuse without permission. — https://doi.org/10.22541/au.159078077.78858403 — This a preprint and has not been peer reviewed. Data may be preliminary. $\psi$–Katugampola Fractional Derivatives and Integrals-Application to Mass–Spring Damper System Ramazan OZARSLAN 1 , Yadigar Sekerci 2 , and Erdal BAS 1 1 Firat Universitesi 2 Amasya University May 29, 2020 Abstract We propose a new type of generalized fractional derivatives with respect to (wrt) another function. These new generalized fractional derivatives generalize $\psi$–Caputo, Riemann–Liouville (R–L) wrt another function, Caputo Hadamard wrt another function, R–L Hadamard wrt another function, Caputo, R–L, Caputo Hadamard and R–L Hadamard fractional derivatives. We propose a newly modified Laplace transform for linear $\psi$–Katugampola fractional differential equations (FDEs). Properties of this newly generalized Laplace transform are analyzed. Cauchy problems and mass-spring damper system with $\psi$– Katugampola fractional derivative are solved analytically by means of modified Laplace transform. Finally, a new numerical method is proposed for nonlinear $\psi$–Katugampola FDEs. Hosted file $psi$--Katugampola Fractional Derivatives and Integrals-Application to Mass--Spring Damper System.pdf available at https://authorea.com/users/327906/articles/455299--psi-katugampola-fractional- derivatives-and-integrals-application-to-mass-spring-damper-system 1

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    $\psi$–Katugampola Fractional Derivatives andIntegrals-Application to Mass–Spring Damper System

    Ramazan OZARSLAN1, Yadigar Sekerci2, and Erdal BAS1

    1Firat Universitesi2Amasya University

    May 29, 2020

    Abstract

    We propose a new type of generalized fractional derivatives with respect to (wrt) another function. These new generalized

    fractional derivatives generalize $\psi$–Caputo, Riemann–Liouville (R–L) wrt another function, Caputo Hadamard wrt anotherfunction, R–L Hadamard wrt another function, Caputo, R–L, Caputo Hadamard and R–L Hadamard fractional derivatives. We

    propose a newly modified Laplace transform for linear $\psi$–Katugampola fractional differential equations (FDEs). Propertiesof this newly generalized Laplace transform are analyzed. Cauchy problems and mass-spring damper system with $\psi$–Katugampola fractional derivative are solved analytically by means of modified Laplace transform. Finally, a new numerical

    method is proposed for nonlinear $\psi$–Katugampola FDEs.

    Hosted file

    $psi$--Katugampola Fractional Derivatives and Integrals-Application to Mass--Spring Damper System.pdf

    available at https://authorea.com/users/327906/articles/455299--psi-katugampola-fractional-derivatives-and-integrals-application-to-mass-spring-damper-system

    1

    https://authorea.com/users/327906/articles/455299--psi-katugampola-fractional-derivatives-and-integrals-application-to-mass-spring-damper-systemhttps://authorea.com/users/327906/articles/455299--psi-katugampola-fractional-derivatives-and-integrals-application-to-mass-spring-damper-system

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