High-accuracy solution of pantograph differential equations subject to mixed boundary conditions via shifted Vieta–Lucas polynomials

• Published in: Boundary value problems Vol. 2025; no. 1; pp. 114 - 20
• Main Authors: Hafez, R. M., Ahmed, H. M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Hindawi Limited; SpringerOpen
• Subjects: Accuracy; Analysis; Approximations and Expansions; Basis functions; Boundary conditions; Collocation methods; Difference and Functional Equations; Differential equations; Galerkin method; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Pantograph-type equations; Pantographs; Partial Differential Equations; Polynomials; Shifted Vieta–Lucas polynomials; Spectral methods; Pantograph-type equations; 65L05; Galerkin method; 45J05; 65M70; Shifted Vieta–Lucas polynomials
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-025-02102-x

We propose two precise spectral techniques based on shifted Vieta–Lucas polynomials (SVLPs) for the solution of pantograph-type differential equations (PTDEs) subject to general mixed-type boundary conditions (MTBCs). A Galerkin method (GM) is formulated for constant coefficient-type equations, and a spectral collocation method (SCM) is given for variable coefficient cases. The basis functions are developed to fulfill the given MTBCs, which reduces implementation and enhances accuracy. Derivatives are calculated analytically through recurrence relations, whereas functional computations are carried out directly at the collocation nodes to circumvent the necessity of integral operation matrices. The methods are applied to several benchmark problems, covering both linear and nonlinear cases, with numerical results validating exponential convergence at minimal computational expense. Comparisons with other spectral methods (SPMs) show that the new SVLP-based approach possesses enhanced performance regarding stability, accuracy, and efficiency. The method is general and flexible, yielding an effective technique for solving a broad class of delay-type differential equations (DEs).