Computational scheme for the numerical solution of fractional order pantograph delay-integro-differential equations via the Bernstein approach

The main objective of this research is to provide an efficient computational algorithm for solving a class of fractional order pantograph delay-integro-differential equations. The Bernstein approximation technique is employed for approximating the solutions of such equations by taking into account the properties of Bernstein polynomials, which transform the original problem into a simplified linear system of algebraic equations. The formulation of the suggested scheme is carefully described, and its convergence performance is examined. Both existence and uniqueness theorems are provided to establish the theoretical reliability of the methodology. Additionally, an estimation of the error bounds is reported to evaluate the precision of the approximation. The reliability, stability and computational effectiveness of the algorithm developed are confirmed by a series of numerical experiments. A comparative analysis with other available methodologies and analytical solutions further highlights the practicality and robustness of the new technique, offering reduced computational and memory costs.