fibonacci polynomials

In this study, we solve the nonlinear fractional differential equations with fractional integral boundary conditions. To solve the mentioned problems, we use an iterative method based on the reproducing kernel Hilbert spaces. In this method, the reproducing kernel of a finite-dimensional Hilbert space is constructed using Fibonacci polynomials. With the help of the obtained positive definite kernel, we produce bases that exactly satisfy the given integral boundary conditions. Then using the obtained bases, we construct fractional derivative operational matrices and obtain an approximation of the problem with the help of a simple iteration method. In fact, we construct an approximation of the solution in a finite-dimensional space. We have also shown the convergence of the method under certain conditions. To show the effectiveness of the proposed method, we have solved some examples, and the obtained results are ‎presented.‎

We suggest a convenient method based on the Fibonacci polynomials and the collocation points for solving approximately the Abel’s integral equation of second kind. Initially, the solution is supposed in the form of the Fibonacci polynomials truncated series with the unknown coefficients. Then, by placing this series into the main problem and collocating the resulting equation at some points, a system of algebraic equations is obtained. After solving it, the unknown coefficients and so the solution of main problem are determined. The error analysis is discussed elaborately. Also, the reliability of the method is quantified through numerical examples.