A modified Laplace transform for certain generalized fractional operators

Year 2018, Volume: 1 Issue: 2, 88 - 98, 31.08.2018

Fahd Jarad , Thabet Abdeljawad

Abstract

It is known that Laplace transform converges for functions of exponential order. In order to extend the possibility of working in a large class of functions, we present a modified Laplace transform that we call \rho-Laplace transform, study its properties and prove its own convolution theorem. Then, we apply it to solve some ordinary differential equations in the frame of a certain type generalized fractional derivatives. This modified transform acts as a powerful tool in handling the kernels of these generalized fractional operators.

Keywords

Generalized fractional derivatives, Generalized Caputo, convolution theorem

References

• [1] R. Hilfer, Applications of Fractional Calculus in Physics, Word Scientic, Singapore, (2000).[2] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional DierentialEquations, North Holland Mathematics Studies 204, (2006).[3] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, (2006).[4] I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego CA, (1999).[5] S. G. Samko, A. A. Kilbas, O. I.Marichev, Fractional Integrals and Derivatives: Theory andApplications, Gordon and Breach, Yverdon, (1993).[6] A. Atangana, D. Baleanu, New fractional derivative with non-local and non-singular kernel,Thermal Sci., 20 (2016), 757{763.[7] M. Caputo, M. Fabrizio, A new denition of fractional derivative without singular kernel, Progr.Fract. Dier. Appl.,1 (2015), 73{85[8] F. Gao, X. J. Yang, Fractional Maxwell uid with fractional derivative without singular kernel,Thermal Sci., 20 (2016), Suppl. 3, S873-S879.[9] J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr.Fract. Dier. Appl.,1(2015), 87{92.[10] X. J. Yang, F. Gao, J. A. T. Machado, D. Baleanu, A new fractional derivative involving thenormalized sinc function without singular kernel, arXiv:1701.05590 (2017).[11] T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractionalderivative with Mittag-Leer nonsingular kernel, J. Nonlinear Sci. Appl. (2017) 10 (3), 1098{1107.[12] T. Abdeljawad, D. Baleanu, Monotonicity results for fractional dierence operators with discreteexponential kernels, Advances in Dierence Equations (2017) 2017:78[13] T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discreteversions, Journal of Reports in Mathematical Physics, 2017.[14] U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput.,218 (2011), 860-865.[15] U. N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math.Anal.Appl., 6 (2014), 1{15.[16] A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc.,38 (2001), 1191{1204.[17] Y. Y. Gambo, F. Jarad, T. Abdeljawad, D. Baleanu, On Caputo modication of the Hadamardfractional derivative, Adv. Dierence Equ.,2014 (2014), 12 pages.[18] F. Jarad, T. Abdeljawad,D. Baleanu, Caputo-type modication of the Hadamard fractionalderivative, Adv. Dierence Equ., 2012 (2012), 8 pages.[19] Y. Adjabi, F. Jarad , D. Baleanu, T. Abdeljawad, On Cauchy problems with Caputo Hadamardfractional derivatives, Journal of Computational Analysis and Applications, Vol. 21, issue 1(2016), pages 661-681.[20] F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their Caputomodication, to appear in Journal of Nonlinear Science and Applictions.[21] T. Abdeljawad, On conformable fractional calculus , J. Comput. Appl. Math., 279 (2013),57{66.[22] R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional dierential equations with dependenceon the Caputo-Katugampola derivative, J. Comput. Nonlinear Dynam., 11(6) (2016), 11 pages.10