On Generalized Composite Fractional Derivative
Keywords:
Adomian decomposition method, free electron laser equation, generalized composite fractional derivative, Laplace transform, Riemann-Liouville fractional integralAbstract
In the present paper, we define a generalized composite fractional derivative and obtain results, which include the image of power function, Laplace transform and composition of Riemann-Liouville fractional integral with the generalized composite fractional derivative. We also obtain the closed form solution of a generalized fractional free electron laser equation with this fractional derivative by using the Adomian decomposition method.doi:10.14456/WJST.2014.61
Downloads
Metrics
References
J Liouville. Memoire sur quelques questions de geometrie et de mecanique et surun nouveau genre de calcul pour resoudre ces questions. J. Ecole Polytech. 1832; 13, 1-69.
AK Grunwald. Ueber ‘begrenzte’ derivationen und deren anwendung. Zeitschrift fur Mathematik und Physik 1867; 12, 441-80.
AV Letnikov. Theory of differentiation of arbitrary order. Matematiceskij Sbornik 1868; 3, l-68.
GFB Riemann. Versuch Einer Allgemeinen Auffasung Der Integration und Differentiation. Gesammelte Werke, 1876.
M Riesz. L'integral de Rimann-Liouville et le problem de Cauchy. Acta Math. 1949; 81, 1-223.
W Feller. On a generalization of Marcel Riesz' potentials and the semi-groups generated by them. Comm. Sem. Math. Univ. Lund. 1952; 72; 73-81.
M Caputo. Elasticita e Dissipazione. Zanichelli, Bologna, 1969.
T Osler. Fractional derivatives and Leibniz rule. Am. Math. Month. 1971; 78, 645-9.
KS Miller and B Ross. Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus, and Their Applications, Nihon University, Koriyama, Japan, 1989, p. 139-52.
K Nishimoto. An Essence of Nishimoto's Fractional Calculus. Descartes Press, 1991.
J Hadamard. Essai sur l'etude des fonctions donnees par leur development de Taylor. J. Pure Appl. Math. 1892; 4, 101-86.
AA Kilbas and AA Titjura. Hadamard-type fractional integrals and derivatives. Tr. Inst. Mat. Minsk 2002; 11, 79-87.
K Kolwankar and AD Gangal. Local Fractional derivatives and fractal functions of several variables. In: Proceedings of the Conference on Fractals in Engineering, Archanon, 1997.
R Hilfer. Application of Fractional Calculus in Physics. World Scientific Publishing, Singapore, 2000.
G Jumarie. Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions: Further results. Comput. Math. Appl. 2006; 51, 1367-76.
SG Samko, AA Kilbas and OI Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, New York, 1993.
AM Wazwaz. Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing, 2009.
IN Sneddon. The Use of Integral Transforms. Tata McGraw Hill, 1979.
R Hilfer, Y Luchko and Ž Tomovski. Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal. 2009; 12, 299-318.
HM Srivastava and Z Tomovski. Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 2009; 211, 198-210.
A Erdelyi, W Magnus, F Oberhettinger and FG Tricomi. Higher Transcendental Functions. Vol III. McGraw Hill, New York, 1955.
RK Saxena and SL Kalla. On a fractional generalization of the free electron laser equation. Appl. Math. Comput. 2003; 143, 89-97.
M Garg and A Sharma. Fractional free electron laser equation in single mode case with Caputo fractional derivative using Adomian decomposition method. J. Rajasthan Acad. Phys. Sci. 2012; 11, 125-32.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2014 Walailak University
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.