We solve the FIBONACCI FUNCTIONAL EQUATION, f(x) = f(x-1)+f(x-2), and prove its Hyers-Ulam stability in the class of functions f: R ®X, where X is a real Banach space.

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 the current study, we design a new computational method to solve a class of Li$\acute{e}$nard's EQUATIONs (LEs). This EQUATION originates from advancements in radio and vacuum tube technology. To attain the proposed goal, we develop a method using a three-layer artificial neural network, consisting of an input layer, a hidden layer, and an output layer. We use the Morgan-Voyce even FIBONACCI polynomials (MVEFPs) and $\sinh$ function as activation functions for the hidden layer and the output layer, respectively. Then, the neural network is trained using a classical optimization method. Finally, we analyze four examples using graphs and tables to demonstrate the accuracy and effectiveness of the numerical approach.

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At the Fourty-forth International Symposium on FUNCTIONAL EQUATIONs in 2006, Louisville, Kentucky, USA, T.M.K. Davison presented a talk under the title "D’Alembert’s FUNCTIONAL EQUATION on compact groups" in which he offered a solution method for this EQUATION on any compact group. In this paper we offer another approach based on some very recent results on spectral synthesis, which method seems to be useful in the case of other EQUATIONs.

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.