Analytical and numerical treatment of a fractional integro-differential equation of Volterra-type
The object of the paper is the study of the following fractional integro-differential equation \[ D^\alpha_ta(t)=\lambda\int^t_0sa(t-s)\exp(ics) ds,\quad t\in[0,1],\tag{1} \] where \(\lambda\) is a complex number, \(c\) is a real constant and \(\alpha\) is a positive real constant. The equations of the type (1) describe the behaviour of free electron laser. The derivative \(D^\alpha\) is defined as the so-called fractional derivative. The solution of (1) is obtained in terms of Kummer functions while the tau method is used in order to obtain approximations of solutions of (1) via the numerical estimates.
fractional integro-differential equation
scientific article; zbMATH DE number 1326449
scientific article