Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method

Abstract

In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method.

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

“…LetX(t) be the approximation of X(t), using Equations (20), (37), and (38), this error for approximate of X ∈ H 2 (0, 1) is as follows:…”

“…Given the advantages of these polynomials, the researchers have used them for solving some of the equations, such as the generalized pantograph equations, 37 space fractional diffusion equation, 38 and linear complex differential equations. 39 Now, given that some physical phenomena lead to mathematical models with piecewise smooth solutions, and considering the advantages of Fibonacci polynomials, we introduce Fibonacci wavelets, which are the proper tools for solving equations with piecewise smooth solutions, also.…”

“…Given the advantages of these polynomials, the researchers have used them for solving some of the equations, such as the generalized pantograph equations, space fractional diffusion equation, and linear complex differential equations …”

Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

“…But Azizi and American Journal of Computational Mathematics Loghmani applied collocation technique with Gauss-Lobatto nodes whereas Xie et al used tau method to determine expansion coefficients. For numerical approximations of 1D fractional diffusion equation Bahsi and Yalcinbas [10] chosen Fibonacci polynomials to express the trial solution in both space and time domain and then used evenly spaced collocation points. Pirim and Ayaz [11] also chosen evenly spaced collocation points but expressed the trial solution in terms of Hermite polynomials for approximations of fractional order system of differential equations.…”

Comparison of Numerical Approximations of One-Dimensional Space Fractional Diffusion Equation Using Different Types of Collocation Points in Spectral Method Based on Lagrange’s Basis Polynomials

“…Lin and Xu [6] introduced a method where they applied Legendre spectral scheme in space and finite difference in time to approximate the solution of time fractional diffusion equation. Bahsi and Yalcinbas [7] reduced the fractional order diffusion equation into a system of linear algebraic equations by expanding trial solution in terms of Fibonacci polynomials in both space and time and then using collocation technique with evenly spaced collocation points. Pirim and Ayaz [8] introduced Hermite collocation method with evenly spaced nodes for numerical approximations of fractional order system of differential equations.…”