The Approximate Solution of High-Order Linear Fractional Differential Equations with Variable Coefficients in Terms of Generalized Taylor Polynomials

Abstract

In this paper, we have developed a new method called Generalized Taylor collocation method (GTCM), which is based on the Taylor collocation method, to give approximate solution of linear fractional differential equations with variable coefficients. Using the collocation points, this method transforms fractional differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Generalized Taylor coefficients. Generally, the method is based on computing the Generalized Taylor coefficients by means of the collocation points. This method does not require any intensive computation. Moreover, It is easy to write computer codes in any symbolic language. Hence, the proposed method can be used as effective alternative method for obtaining analytic and approximate solutions for fractional differential equations. The effectiveness of the proposed method is illustrated with some examples. The results show that the method is very effective and convenient in predicting the solutions of such problems.

Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics

“…Moreover, some numerical methods have been proposed for approximate solutions of this type equations such as the operational matrix method [10][11][12], Adomian decomposition method [13], homotopy-perturbation method [14], collocation method [15][16] and others [17][18][19][20].…”

Numerical Solutions for Linear Fractional Differential Equations of Order 1 <Alpha<2 Using Finite Difference Method (ffdm)

“…Regarding to [13], the author's develop a new method called Generalized Taylor Collocation Method (GTCM) to give approximate solutions for linear fractional differential equations with variable coefficients. They found that this method is easy to write in computer codes and can be used as an effective method for obtaining analytic and approximate solutions for fractional differential equations.…”

The Maple Program Procedures at Solution Systems of Differential Equation with Taylor Collocation Method

“…Firstly, this method is based on taking the truncated Taylor series of the function in equations and then substituting their matrix forms in the given equation. Hence, the result of the matrix equation can be solved and the unknown Taylor coefficients can be found approximately [1,[3][4][5][6].…”