A Chebyshev Polynomial-Based Comparison of Collocation and Least-Squares Methods for High-Order Volterra and Fredholm Integro-Differential Equations

This study presents an approximate computational framework for solving high-order linear Volterra and Fredholm integro-differential equations (IDEs). The proposed methodology formulates a comparative appraisal between the Standard Collocation Method (SCM) and the Least Squares Method (LSM), utilising third-kind Chebyshev polynomials as basis functions. The solution framework employs an assumed trial polynomial expansion substituted directly into the governing integro-differential formulation. In the SCM scheme, the resulting continuum residual system is collocated at strategically selected equidistant nodes across the domain of interest, yielding a deterministic system of linear algebraic equations resolved computationally using Maple 18. Conversely, in the LSM formulation, the continuous residual function is squared and integrated over the unit interval [0, 1 ] to construct an integral error functional; minimization is achieved by setting the partial derivatives with respect to the undetermined expansion coefficients to zero, resulting in a closed normal system. The precision, numerical stability, and computational rate of convergence of both methodologies were appraised across fourth-order benchmark initial- value integro-differential problems for approximation orders N = 5 and N = 6. Numerical experiments confirm that while both spectral formulations demonstrate monotonic convergence towards the exact analytical solution, the Standard Collocation Method exhibited superior numerical stability and precision over the interval, attaining absolute error thresholds on the order of 10_8, compared with the Least Squares formulation which demonstrated localized sensitivity at higher approximation degrees. The practical implications of these solvers across viscoelastic mechanics, epidemiology, and structural engineering alongside direct alignments with the United Nations Sustainable Development Goals (SDGs 3, 4, 9, and 17) are systematically discussed.