Fractional Vieta-Fibonacci wavelets: application for systems of fractionaldelay differential equations

In this study, the fractional Vieta-Fibonacci wavelets are introduced. These wavelets are applied to construct a numerical method to solve a class of fractional delay systems of differential equations. To this end, some relationships regarding fractional integration and derivative of these wavelets are extracted at the first. Then, all of the unknown functions in the system under consideration are approximated by these wavelets. Next, by substituting these approximations into the system and applying the collocation method, a system of algebraic equations is obtained. Finally, by solving the extracted system and can determine the unknown coefficients, a solution is obtained for the main system. Moreover, the upper bound of error for approximation with the fractional Vieta-Fibonacci wavelets and the convergence analysis of presented approach is derived. The proposed method is evaluated with solving several examples.

Automated summary

In this study, the fractional Vieta-Fibonacci wavelets are introduced to construct a numerical method for solving a class of fractional delay systems of differential equations. The relationships regarding fractional integration and derivative of these wavelets are extracted, and all unknown functions in the system are then approximated by these wavelets. By substituting these approximations into the system and applying the collocation method, a system of algebraic equations is obtained and solved to determine the unknown coefficients and obtain a solution for the main system. The proposed method is evaluated with solving several examples, and the convergence analysis and error bound for approximation with the fractional Vieta-Fibonacci wavelets are derived.