System of Volterra Fredholm Integro-Fractional Differential Equations: Application of Fibonacci Polynomials

In this paper, we introduce the Fibonacci polynomials (FPs) and approximate functions using them. Furthermore, several lemmas and corollaries present the properties of FPs. Also, we derive the Fibonacci polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been undertaken before.  As applications of the Fibonacci polynomials operational matrix, we solve the system of Volterra Fredholm integro-fractional differential equations. In this scheme, we approximate one and two variable functions based on Fibonacci basis. Then by applying Fibonacci polynomials operational matrix, the system of Volterra Fredholm integro-fractional differential equations is reduced to a system of algebraic equations that is easily solvable with the help of a software (version ۱۳ of the Mathematica software). The obtained results are in good agreement with the exact solutions and with those in literature. As anticipated, the solutions converge to classical solutions as the fractional derivative order approaches integer values.

Keywords:

System of Fredholm Volterra integro-fractional differential equations , Fibonacci polynomials , operational matrix , Caputo derivative