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On a fractional generalization of the free electron laser equation. (English) Zbl 1110.45300

Summary: The following fractional generalization of the free electron laser equation is investigated: \[ D_\tau^\alpha h(\tau)= \lambda \int_0^\tau t^\delta h(t-t) \Phi(b;\delta+1;ivt)\, dt+\mu\tau^\gamma \Phi(\beta,\gamma+1;iv\tau), \quad 0\leq \tau\leq 1, \] where \(\beta,\gamma,\lambda\in C\); \(v,b,\beta\in R\), \(\alpha>0\), \(\gamma>-1\) and \(\delta>-1\). A closed form solution is derived in terms of Kummer’s function \(\Phi(\alpha,\beta;z)\) by the application of Riemann-Liouville fractional integration operators. Tau method approximation is used in the evaluation of the results in a suitable form for numerical computation. The results derived are of general character and provide extension of the work reported by L. Boyadjiev, S. L. Kalla and H. G. Khajah [Math. Comput. Model. 25, No. 12, 1–9 (1997; Zbl 0932.45012)] and A.H. Al-Shammery, S. L. Kalla and H. G. Khajah [Integral Transform. Spec. Funct. 9, No. 2, 81–90 (2000; Zbl 0962.45002); Fract. Cal. Appl. Anal. 2, No. 4, 501–508 (1999; Zbl 1033.65118)].

MSC:

45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
41A10 Approximation by polynomials
78A60 Lasers, masers, optical bistability, nonlinear optics
81V80 Quantum optics
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References:

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