Year 2018, Volume 1, Issue 2, Pages 88 - 98 2018-08-31

A modifi ed Laplace transform for certain generalized fractional operators

Fahd Jarad [1] , Thabet Abdeljawad [2]

35 353

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 modied Laplace

transform that we call \rho-Laplace transform, study its properties and prove its own convolution the-

orem. Then, we apply it to solve some ordinary differential equations in the frame of a certain type

generalized fractional derivatives. This modied transform acts as a powerful tool in handling the

kernels of these generalized fractional operators.

Generalized fractional derivatives, Generalized Caputo
  • [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 Di erentialEquations, North Holland Mathematics Studies 204, (2006).[3] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, (2006).[4] I. Podlubny, Fractional Di erential 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. Di er. 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. Di er. 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 di erence operators with discreteexponential kernels, Advances in Di erence 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. Di erence Equ.,2014 (2014), 12 pages.[18] F. Jarad, T. Abdeljawad,D. Baleanu, Caputo-type modi cation of the Hadamard fractionalderivative, Adv. Di erence 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 di erential equations with dependenceon the Caputo-Katugampola derivative, J. Comput. Nonlinear Dynam., 11(6) (2016), 11 pages.10
Primary Language en
Subjects Mathematics
Published Date 31 August 2018
Journal Section Articles
Authors

Author: Fahd Jarad (Primary Author)
Country: Turkey


Author: Thabet Abdeljawad
Country: Saudi Arabia


Bibtex @research article { rna452424, journal = {Results in Nonlinear Analysis}, issn = {}, eissn = {2636-7556}, address = {Erdal KARAPINAR}, year = {2018}, volume = {1}, pages = {88 - 98}, doi = {}, title = {A modifi ed Laplace transform for certain generalized fractional operators}, key = {cite}, author = {Jarad, Fahd and Abdeljawad, Thabet} }
APA Jarad, F , Abdeljawad, T . (2018). A modifi ed Laplace transform for certain generalized fractional operators. Results in Nonlinear Analysis, 1 (2), 88-98. Retrieved from http://dergipark.org.tr/rna/issue/37067/452424
MLA Jarad, F , Abdeljawad, T . "A modifi ed Laplace transform for certain generalized fractional operators". Results in Nonlinear Analysis 1 (2018): 88-98 <http://dergipark.org.tr/rna/issue/37067/452424>
Chicago Jarad, F , Abdeljawad, T . "A modifi ed Laplace transform for certain generalized fractional operators". Results in Nonlinear Analysis 1 (2018): 88-98
RIS TY - JOUR T1 - A modifi ed Laplace transform for certain generalized fractional operators AU - Fahd Jarad , Thabet Abdeljawad Y1 - 2018 PY - 2018 N1 - DO - T2 - Results in Nonlinear Analysis JF - Journal JO - JOR SP - 88 EP - 98 VL - 1 IS - 2 SN - -2636-7556 M3 - UR - Y2 - 2018 ER -
EndNote %0 Results in Nonlinear Analysis A modifi ed Laplace transform for certain generalized fractional operators %A Fahd Jarad , Thabet Abdeljawad %T A modifi ed Laplace transform for certain generalized fractional operators %D 2018 %J Results in Nonlinear Analysis %P -2636-7556 %V 1 %N 2 %R %U
ISNAD Jarad, Fahd , Abdeljawad, Thabet . "A modifi ed Laplace transform for certain generalized fractional operators". Results in Nonlinear Analysis 1 / 2 (August 2018): 88-98.