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.

In this paper, a class of new POLYNOMIALS based on FIBONACCI sequence using Newton interpolation is introduced. This target is performed once using Newton forward-divided-difference formula and another more using Newton backward-divided-difference formula. Some interesting results are obtained for forward and backward differences. The relationship between forward (and backward) differences and the Khayyam-Pascal’, s triangle are also examined.

Here, based on the FIBONACCI POLYNOMIALS, a new collocation method is presented in order to solve the system of linear fuzzy Volterra integral equations of the second kind. By using this method, these systems are reduced to a linear system of algebraic equations which are simply solvable. Also, the error analysis and existence of the solution of the suggested method are discussed. Finally, to show the importance and application of this method, we have used some rational examples. The method is computationally very attractive and gives very accurate results. Easy implementation and simple operations are the essential features of the 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.‎

In this paper, a new interconnection network structure called hierarchical extended FIBONACCI cubes interconnection networks HEFC1(n) is proposed, and its properties are evaluated. A set of the HEFC1(n)s constructed by the proposed method is a two level conventional hierarchical network. For each n (dimension of extended FIBONACCI cube-EFC1(n)), a HEFC1(n)   interconnection network can be constructed. All EFC1(n) for n>4 are recursively constructible, hence HEFC1(n)s for n>4 are also recursively constructible with some extra EFC1(n-1)s and EFC1(n-2)s. Furthermore, since a HEFC1(n) has a hierarchically structured character and the feature of uniformity, a wide variety of inter-cluster connections are possible. The comparisons of HEFC1(n)s with some of the traditional cubic and hierarchical cubic networks are presented in this study. The routing in HEFC1(n+2)s is as easy as routing in HCN(n,n) and the scalability of HEFC1(n+2) is better than the scalability of HCN(n,n) and H(2n). H(2n) and HCN(n,n) are symmetric interconnection networks, however, HEFC1(n+2) is not a symmetric interconnection network and HEFC1(n+2)s have a recurrent structure as H(2n) and HCN(n,n). The cost of HEFC1(n+2) is better than the costs of HCN(n,n) and H(2n).The edge connectivity of HEFC1(n+2) is worse than the edge connectivity of H(2n) and HCN(n,n). This makes the VLSI design of HEFC1(n+2) simpler than the VLSI designs of H(2n) and HCN(n,n).

The objective of this study was to develop a new optimal parallel algorithm for matrix multiplication which could run on a FIBONACCI Hypercube structure. Most of the popular algorithms for parallel matrix multiplication can not run on FIBONACCI Hypercube structure, therefore giving a method that can be run on all structures especially FIBONACCI Hypercube structure is necessary for parallel matrix multiplication. For creating this method, a new model for matrix multiplication with an algorithm for data distribution on FIBONACCI Hypercube structure was provided. Other than this, another optimized algorithm was designed on Mesh structure. By running the algorithms on a simulative parallel system and giving the results in graphical mode, it has been found that these two algorithms have optimized value in parallel matrix multiplication and they are more efficient than the previous algorithms.

We study two foremost Mahonian statistics, the major index and the inversion number for a class of binary words called restricted FIBONACCI words. The language of restricted FIBONACCI words satisfies recurrences which allow for the calculation of the generating functions in two different ways. These yield identities involving the $q$-binomial coefficients and provide non-standard $q$-analogues of the FIBONACCI numbers. The major index generating function for restricted FIBONACCI words turns out to be a $q$-power multiple of the inversion generating function.

This article deals with two new subclasses of analytic and bi-univalent functions in the open unit disk, which is defined  by applying subordination principle between analytic functions and the generalized Bivariate FIBONACCI POLYNOMIALS. Bounds for coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ of functions in these subclasses are estimated in terms of generalized Bivariate FIBONACCI POLYNOMIALS. In addition, the Fekete-Szeg\"{o} problem is handled for the members of these subclasses and several consequences and examples of the main results are presented. The results of article generalize some of the previously published papers in the literature.