Collocation Method for Solving Problems of Generalized Fractional Bagley – Torvik Equation

Abstract

Collocation method for solving problems of Generalized fractional Bagley–Torvik equation is proposed: the boundary value problems of generalized Bagley–Torvik equation with constant coefficients is studied. Further, the fractional Cauchy-Euler equation which is a class of the initial value problem for the variable coefficients of the Bagley–Torvik equation is also considered. By using Mittag–Leffler (M-L) type function the existence and uniqueness of the solution of the boundary value problem of generalized Bagley–Torvik (BgT) equation with Caputo derivative of order 0 < α < 2 is proved. Further, a highly accurate numerical method is given by using the collocation-shooting method (C −SM). The collocation solution is constructed in the space S (1) m+1 as piecewise polynomials of degree at most m+1 and has continuous derivative on the considered domain. Furthermore, under some smoothness conditions of the exact solution, global convergence analysis are given. Numerical investigation is carried out on several examples and besides showing the justification of theoretical results comparisons with existing numerical methods are also given. Results indicate that C-SM gives highly accurate results and is efficient both in computational time and storage. In addition, we consider a model of the fractional Cauchy-Euler type equation, where the fractional derivative operator is the Caputo with order 0 < α < 2 . The problem also constitutes a class of examples of the Cauchy problem of the generalized fractional Bagley-Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the proposed problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is numerically shown that the proposed method and the algorithm are easy to implement on a computer and computational complexity is given. Keywords: Bagley–Torvik Equation, Caputo Fractional Derivative, Collocation-shooting Method, Convergence Analysis, Cauchy-Euler Equation, Existence and Uniqueness, Algorithm, Collocation Method.