Quadratic spline solution for boundary value problem of fractional order

Abstract

Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.Keywords Boundary value problems of fractional order · Riemann-Liouville fractional derivative · Caputo fractional derivative · Quadratic polynomial spline · Error bound

An exponential spline approximation for fractional Bagley–Torvik equation

“…The maximum absolute errors in solutions of the methods are tabulated in tables. We compute the absolute error for examples and compare them with the methods in [4,9,17,21,28,29]. The convergence order (C.O.)…”

“…Method IV. In this section, we would like to develop a numerical method based on the methods in references [4,25,29], and [28]. Also we investigate the convergence analysis of this method.…”

“…Also Adomian decomposition method for solving the initial value problem of Bagley-Torvik equation is discussed in [19], fractional linear multistep method and a predictor-corrector method of Adams type based on finite difference methods for initial value problem of Bagley-Torvik equation are discussed in [8]; Legendre operational matrix method for fractional differential equation is applied in [20]; and a combination of collocation points and first-order Bessel functions, which is called Bessel-collocation method for boundary value problem of Bagley-Torvik equation, is discussed in [27]. Quadratic spline solution for boundary value problem of fractional order is applied in [28], and an exponential spline technique for solving fractional boundary value problem is employed in [4]. The first aim of the present work is to explore exponential spline interpolation with multiple parameters and to produce the error of approximate exponential spline.…”

“…In Sect. 4, the numerical examples are given to illustrate the applications of the method, and also the computed results are compared with another known method in [4,9,17,21,28,29].…”

An Efficient Approach of the Fractional Lagrange Interpolation

“…This called polynomial interpolation (Berrut and Trefethen, 2004;Dvornikov, 2008;Elsaid, 2010;Hamasalh and Muhammad, 2015;Zahra and Elkholy, 2012). Hamasalh and Muhammad (2015) presented a study of three interpolator fractional splines.…”