The construction of operational matrix of fractional derivatives using B-spline functions

Numerical Inversion Technique for the One and Two-Dimensional L2-Transform Using the Fourier Series and Its Application to Fractional Partial Differential Equations

“…To control of the truncation error E t , we can use the epsilon algorithm (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) to accelerate the convergence of the series and evaluate the valuesf p+1 (t 1 , t 2 ), f p+ p 4 (t 1 , t 2 ) in (3-9) until the difference between them be small.…”

“…This methods can be considered as a promising technique beside the existing methods for solving fractional partial differential equations, see [6,7] and [9,13,14]. …”

An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation

“…Moreover, a number of local versions of fractional derivatives also presented for the analysis of local behavior of fractional models such as Jumarie's modified Riemann-Liouville derivative [6], Cresson's derivative [7], and Kolwankar-Gangal local derivative [8]. On the other hand, the recent appearance of fractional differential equations (FDEs) as adequate models in science and engineering made it necessary to develop methods of solutions (both analytical and numerical) These methods include finite difference method [9], finite element method [10], differential transform method [11], Adomian decomposition method [12], variational iteration method [13], homotopy perturbation method [14], first integral method [15], fractional sub-equation method [16], B-spline function method [17], Tau method [18], homotopy analysis method [19,20], and collocation method [21]. Although these methods lead to exact solutions in some special cases, exact solutions are much needed in engineering applications.…”

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

“…Hence, finding exact/approximate solutions to FDEs has become an important task. There are various analytical methods for fractional calculus, among them are the homotopy analysis method [10], the discrete method [11], the tau approach [12], the finite difference method [13], the B-spline functions method [14], the (G'/G)-expansion method [15], the operational matrix method [16], the finite element method [17], the variational methods [18], the homotopy perturbation method [19], the fractional sub-equation method [20], the first integral method [21], and the others [22][23][24][25].…”

“…where G D G. n / satisfies Equation (14), while a 0 and a 1 are arbitrary constants to be specified.…”