Linear Fractional Integro-differential Equations Research Articles

A Legendre collocation method for fractional integro-differential equations

A numerical method for solving the linear and non-linear fractional integro-differential equations of Volterra type is presented. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations. The properties of Legendre polynomials together with the Gaussian integration method are utilized to reduce the fractional integro-differential equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.

Operational Tau approximation for a general class of fractional integro-differential equations

In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, errorestimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method. Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

On an approach to complex analysis via differintegrations

The ordinary concept of derivative (primitive) of order n has been extended from (v = n) (v = —n) a positive (negative) integer n IN to fractional or complex v differintegration operators applied to real [1, 2], complex [3, 4] and generalized [5, 6] functions. We recall (sec. 1) the definition of Liouville (Riemann) differintegration Dv (dv ) of an analytic function (function with one branch-point), and use the corresponding rules of derivation with complex order to solve (sec. 2) a class of linear fractional integro-differential equations with constant (power) coefficients: this leads to some results on the number of roots of a class of relations between Gamma functions (sec. 7). We proceed to compare the rules of differintegration in the Liouville and Riemann systems, and to introduce (sec. 3) a branch-point derivative δ P–v which annihilates the Liouville derivative Dv for all non-integer v ∊ C — Z and integer p ∊ Z. We review (sec. 4) the generalization of Taylor's, Laurent's, Lagrange's and Teixeira'...