Operational-Matrix Methods for Caputo Fractional Delay Differential Equations; A Critical Comparison of Shifted Third-Kind Chebyshev Polynomials and Genocchi Wavelets; Review Article

Abstract

 Fractional delay differential equations (FDDEs) are a type of nonlocal memory fractional differential equations that have an explicit dependence on past states. This mixup is a source of confusion in the analysis and can make numerical solutions sensitive to the regularity of the solution, the delay geometry, the history data, and the conditioning of the basis chosen. There are reasons to trust operational-matrix methods, since they are based on a finite algebraic system instead of the governing equation, but comparisons published in the literature are sometimes made from different test problems, of different fractional order, of different truncation size, of different hardware, and of different error definitions. Such comparison does not demonstrate one basis is always more accurate and/or faster than another. This article reviews critically two representative methods: the shifted third-kind Chebyshev polynomial method, and the Genocchi wavelet method. A current mathematical framework is employed to allow identification of exact identities from projected and pseudo operational approximations. The analysis reveals that the shifted third kind Chebyshev polynomials are an orthogonal, globally supported set which is well adapted to smooth solutions and simple delay maps. While the Genocchi wavelets have compact support within dyadic subintervals and offer explicit operator constructions for a number of constant and proportional delays, the underlying Genocchi polynomials are not orthogonal, there is only approximate representation of the Caputo operator, and locality is restricted by the regularity and alignment of solution features. There is no general consensus in the literature about the relative ranking of the two methods due to the fact that no head-to-head benchmark was found that uses the same equations, reference solutions, tolerances, and computational environments. The review introduces a new selection framework based on evidence and a benchmark protocol for selecting a procedure that is reproducible. The conclusions that are reached are tentative, clear and appropriate for future comparative studies.