A New Numerical Mechanism for Solving Two Models of Variable Order Delay Differential Equations

Abstract

Variable order delay differential equations with initial functions; Variable order Volterra delay integrodifferential equations; Operational matrix; Shifted fractional Gegenbauer polynomials; The method of steps; Collocation methodThe current paper offers an effective numerical mechanism for solving two models of variable order (VO) linear/nonlinear delay differential equations; the models represent the variable order delay differential equations (VODDEs) and the variable order Volterra delay integro-differential equations (VO-VDIDES) with initial functions. In the proposed mechanism, the method of steps is used to transfer the VO delay models (VO-DMs) into variable order non delay ones with initial conditions. After a novel operational matrix (OM) of VO derivative of the shifted fractional Gegenbauer polynomials (SFGPs) injunction with the spectral collocation method are utilized to transfer the aforementioned problem into a system of algebraic equations, which is simple to solve. The error estimation of the proposed technique is established through the article. The efficiency and accuracy of the proposed technique are verified by applying it to several numerical delay differential equations with constant or variable delay. The obtained numerical results are compared with other published data in the existing literature to expose the accuracy and efficiency of the proposed technique.