Richard Olu Awonusika

arXiv2026

Numerical Solution of Pantograph Delay Integrodifferential Equation of Volterra Type: Collocation Method Based on Shifted Jacobi Polynomials

Pantograph arises in electric trains, material modelling, and the modelling of quantum dot lasers. Pantograph integrodifferential equations are integrodifferential equations involving proportional delays; and they appear in fields such as electrodynamics, epidemiology, control theory, astrophysics, economics, and engineering. This research presents an efficient collocation method based on shifted Jacobi polynomials for obtaining numerical solutions of a class of first order pantograph delay integrodifferential equation of Volterra type with an initial condition. The proposed method expresses the solution of the governing equation as a shifted Jacobi polynomial series with expansion coefficients which are to be determined. Collocating at the roots of the shifted Jacobi polynomials, the underlying problem is reduced to a system of algebraic equations in the unknown expansion coefficients of the shifted Jacobi polynomial series solution. Newton's method is subsequently used to solve the resulting system of algebraic equations and numerical values of the expansion coefficients are obtained. The obtained coefficients are substituted into the assumed series solution to obtain the required numerical solutions. The applicability, reliability, efficiency, and accuracy of the shifted Jacobi collocation method are demonstrated through illustrative examples. Results obtained using the proposed method are compared with exact solutions and other published results. Comparisons of errors which are presented in tables reveal that our method approximates the solution better than the methods under comparison.

math.AP↗

arXiv2026

Shifted Horadam Collocation Method for Solution of Nonlinear Fourth-Order Boundary Value Problem in Ordinary Differential Equation

In this paper, numerical solutions of a class of nonlinear ordinary differential equations are obtained using a collocation method based on the shifted Horadam polynomials. The proposed problem, which is of the fourth-order, satisfies a class of two-point boundary conditions. We first discuss definitions and basic properties of the Horadam polynomials before presenting new and useful differentiation formulae for them. Novel interesting identities are deduced from the differentiation properties. The collocation method under consideration assumes that the solution of the proposed problem can be expressed as a shifted Horadam polynomial series. To determine the expansion coefficients of the series solution, one collocates at the zeros of the shifted Horadam polynomials, and the proposed boundary value problem is subsequently reduced to a set of nonlinear algebraic equations. These algebraic equations are then solved using Newton's iterative method to obtain the numerical values of the expansion coefficients of the shifted Horadam polynomial series solution. Several examples of the proposed nonlinear boundary value problem are considered to demonstrate the accuracy, efficiency, and reliability of the proposed method. Numerical solutions and errors obtained are compared with existing solutions. Comparisons of results, which are shown in tables and graphs, clearly reveal that the shifted Horadam collocation method outperforms the existing methods.

math.AP↗