A Novel Fractional Vieta-Lucas Wavelet-Based Projection Method for Solving Fractional Delay Differential Equations

The primary aim of this study is to present a novel approach that combines Fractional Shifted Vieta-Lucas Wavelets (FSV-LW) with a collocation technique to effectively address two types of delayed models involving both linear and nonlinear fractional-order differential equations, incorporating constant and proportional delays. Initially, we introduce an innovative formulation of (FSV-LW) defined over a given interval, and construct a closed-form expression for the fractional integral operator. The original models are then transformed into algebraic equations through the application of the collocation method and the fractional integral vector of these wavelets. In support of the method’s precision and convergence behavior, several mathematical theorems have been formulated and proven. Finally, we apply this method to solve specific examples from both models, highlighting its effectiveness and practical applicability.