Efficient spectral collocation algorithm for pantograph integro-differential equations involving delay and weakly singular kernels

We introduce a new algorithm to tackle Pantograph Volterra Integro-Differential Equations (PVIDEs) that have weakly singular kernels and delay terms. Our approach uses special polynomials called modified shifted Chebyshev polynomials within a collocation method to find solutions over a specific interval. We change the integro-delay equation into a system of algebraic equations and solve it using a Gaussian process. This helps deal with the problems modeled by delays and singular kernels. We also did a thorough analysis to ensure the method is reliable. To back this up, we ran several numerical tests, showing that our method converges quickly and is very accurate compared to other existing methods across five different cases. Overall, our modified shifted Chebyshev collocation methods prove to be an effective way to numerically solve tough integro-differential equations that have delays and singularities.