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A fractional integro-differential equation of Volterra type. (English) Zbl 0993.65153

Summary: An analytical and numerical treatment of a fractional integro-differential equation was considered recently by L. Boyadjiev, S. L. Kalla and H. G. Khajah [ibid. 25, No. 12, 1-9 (1997; Zbl 0932.45012)]. The present paper deals with a fractional generalization of the free electron laser equation, and the solution is obtained by a method that combines the variation of parameters and successive approximations. The numerical values have been obtained by employing the algebra system MAPLE V.

MSC:

65R20 Numerical methods for integral equations
78A60 Lasers, masers, optical bistability, nonlinear optics
45J05 Integro-ordinary differential equations
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 0932.45012

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boyadjiev, L.; Kalla, S. L.; Khajah, H. G., Analytical and numerical treatment of a fractional integro-differential equation of Volterra type, Mathl. Comput. Modelling, 25, 12, 1-9 (1997) · Zbl 0932.45012
[2] Dattoli, G.; Lorenzutta, S.; Maino, G.; Torre, A., Analytical treatment of the high-gain free electron laser equation, Rad. Phys. Chem., 48, 1, 29-40 (1996)
[3] Dattoli, G.; Reniere, A.; Torre, A., (Lectures on the free electron laser theory and related topics (1993), World Scientific: World Scientific Singapore)
[4] Brunner, H.; van der Houwen, P. I., The Numerical Solution of Volterra Equations (1986), North-Holland: North-Holland Amsterdam · Zbl 0611.65092
[5] (Erdèlyi, A.; etal., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York) · Zbl 0051.30303
[6] Oldham, K. B.; Spanier, J., The Fractinal Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[7] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley & Sons: John Wiley & Sons New York · Zbl 0789.26002
[8] Samko, S.; Kilbas, A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach London · Zbl 0818.26003
[9] Kress, R., Linear Integral Equations (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0671.45001
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