Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.

The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach. [ABSTRACT FROM AUTHOR]