Fibonacci collocation pseudo-spectral method of variable-order

Abstract: In this article, we evaluated the approximate solutions of one-dimensional variableorder space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.


Introduction
As is known, the fractional order is a special case of ordinary and partial derivatives. Fractional diffusion equations (FDEs) (with linear and nonlinear forms) have attracted many scientists. They describe life phenomena by using the fractional order of differentiation and integration. There exist applications in different fields such as mathematics, chemistry, physics, biology, fluids, engineering and mechanics; all of them are described by ordinary and partial fractional differential equations. Fractional calculus investigates the rules and properties of the non-integer orders for the derivatives and integrals. Many researchers use numerical methods such as Adomain decomposition method (see [1][2][3]), variational iteration method (see [4][5][6]), the Haar wavelet method (see [7,8]), finite difference method (see [9,10]), finite element method (see [11,12]), the homotopy analysis method (see [13,14]) and the homotopy perturbation method (see [15,16]).
Recently, spectral methods have been used to evaluate the approximate solutions of fractional differential equations. These methods are distinguishable from the previous methods with a small error and small number of unknowns. The Chebyshev method is used for solving a class of linear and nonlinear Lane-Emden type equations (see [17]), but the second-order two-point boundary value problems are solved by using the Chebyshev wavelets method (see [18]). The methods Jacobi and shifted Jacobi are applied to solve second-and fourth-order fractional diffusion wave equations and fractional wave equations with damping, as well as linear multi-order fractional differential equations (see [19,20]). The system of high-order linear differential equations and multi-term fractional differential equations were solved by using Lucas and generalized Lucas methods (see [21][22][23]). Fibonacci and generalized Fibonacci methods are used to solve multi-term fractional differential equations (see [24,25]). The most common spectral methods are Galerkin, collocation and tau methods. Lane-Emden singular-type equations use a Galerkin operational matrix (see [26]); however, the third-and fifthorder differential equations use Petrov-Galerkin methods (see [27]). Tau methods are applied to solve multi-order fractional differential equations and variable coefficients fractional differential equations (see [28,29]). The fractional differential equations with constant and variable orders mandate the use of a collocation method (see [30,31]).
In this article, we solve FDEs with a variable-order and apply the generalized Fibonacci collocation (GFC) method. We obtain a system of nonlinear algebraic equations with unknown expansion coefficients; this system is studied by using Newton's method with the help of Mathematica. The obtained results are compared with the forward-substitution (FS) method and the fast divide and conquer (FDAC) method (see [32]). We show that our results are more efficient and yield higher accuracy.
The outline of this paper is as follows. In Section 2, the necessary definitions, properties of fractional calculus and generalized Fibonacci polynomials, which are used in the following sections, are introduced. In Section 3 the algorithm of this method is explored in details for a variable-order space-fractional diffusion equation (sFDE) with homogeneous boundary conditions. In Section 4 we investigate the convergence and error analysis of the proposed generalized Fibonacci expansion. We give examples, their numerical solutions and comparisons in Section 5 to prove the efficiency of the method. In the last section we present our conclusions.

Basic properties
In this section, we introduce the definition of a fractional derivative with variable order (see [33][34][35]). Then the important relations and the derivatives of the generalized Fibonacci polynomials, which are used in the following sections, are stated.

Overview and relations of generalized Fibonacci polynomials
If α and β are non zero real numbers, the recurrence relation for the generalized Fibonacci polynomials is [37] Υ α,β with the following initial values: Its Binet form is when there are no zeros in the denominator. The power form is:

Integer derivatives of Fibonacci polynomial
the first derivative of Φ(z) can be written as (see, [37]) Also, the integer derivatives of (Φ)(z) can be easily written in the form (see [37])

Fractional derivatives of Fibonacci polynomial
If λ is not an integer, then the fractional derivative of (Φ)(z) takes the form (see [37]) where χ (λ) = κ λ nm is the (M + 1) × (M + 1) Fibonacci operational matrix of fractional derivatives of order λ, which has the form κ λ nm can be written in the form So the fractional derivative has the relation

A variable-order sFDE and the algorithm for the method
In this section, we explain the method for solving a variable-order sFDE with homogeneous boundary conditions by using generalized Fibonacci polynomials.
Suppose that the function W(z) expanded in terms of generalized Fibonacci polynomials: If we approximate this function as: where The variable-order sFDE is given by [32] − with the boundary conditions and where f (z) and g(z) are continuous functions. Substituting Eq (2.1) with = 2 into Eq (3.4), we have Suppose that υ(z) = W (z); then, Substituting the expansion of Eq (3.1) into Eq (3.6), we have So Eq (3.7) can be written as: N j=0 c j h j (z) = g(z).
Collocating this equation at the generalized Fibonacci roots, we obtain the following system of equations N j=0 c j h j (z i ) = g(z i ). (3.8) The matrix form for this equation is: where H = h j (z i ) , i, j = 0, 1, ...., N and G = (g(z i )) = g(z 0 ), g(z 1 ), ..., g(z N ) T . With the boundary conditions given by Eq (3.5), we have Equations (3.8) and (3.9) give a linear system of equations with the coefficients c j , j = 0, 1, ...N. These coefficients must be determined. Lastly, we integrate υ(z) two times to evaluate W(z).

Investigation of convergence and error analysis
In this section, we explain the convergence and error analysis of generalized Fibonacci expansion. Theorem 1. If W(z) is defined on [0, ι] and W (i) (s) ≤ L i , i ≥ 0 where L is a positive constant, s is any point in [0, ι], and if W(z) has the expansion: then one has: 2) the series converges absolutely, where Li 6 (z) is the polylogarithmic function.
is the exact solution of Eq (3.6) and υ(z) is where f (z) and g(z) are continuous functions such that | f (z)| ≤ λ, where λ is a positive constant. Then we have the following global error estimate: By integrating and taking − 1 ≤ γ(z) ≤ , so we have From Theorems 1 and 2 By Lemma 1, we have Using some calculations, we obtain where Γ( ) and Γ( , ) are gamma and incomplete gamma functions respectively; therefore So the proof is completed is the exact solution of Eq (3.4) and it satisfies the hypotheses of Then we have the following global error estimate: From Theorems 1 and 2 By Lemma 1, we have N, ς)) .
In the same way as with the previous theorem, we obtain So the proof is completed
In Tables 1 and 2, we observe that the absolute error obtained via the GFC method is better than that obtained via the FS method and FDAC method [32]. Figure 1 illustrates the results of the present method at N = 5, 8 and 10. The figure shows that the convergence is exponential.
Tables 3-5 compare the methods at different values of γ 0 and γ 1 . We notice that the error calculated using this method is the best because it is the smallest. Also, we prove the accuracy and efficiency of the method. The absolute error at these values is plotted in Figures 2 and 3. We observe from the Figures that the convergence is exponential and the errors are better for different values of α and β.
With γ 0 = 1.2, γ 1 = 1.6   With γ 0 = 1, γ 1 = 1.5   Table 5 lists the numerical results obtained via the proposed method for N= 8, 12 and 16 and different values of a and b. The absolute errors of this method are plotted in Figure 2. We observe from the figure that the convergence is exponential.

Conclusions
This paper details an attempt to solve the variable-order sFDE by using a collocation method based on the operational matrix of fractional derivatives of the generalized Fibonacci polynomials. We modified the variable-order sFDE for compatibility with a system of linear algebraic equations which were solved by using Mathematica software. Then we evaluated the errors. The spectral results which were used throughout, indicate strong adequacy, viability and ease of application. Finally, we discussed the convergence and error analysis of our method.