Robust estimation and convergence analysis of multi-dimensional singular integral models employing advanced higher-order iterative schemes

Nonlinear fractional-order unsteady weakly singular integral equations are encountered in many scientific fields. The combination of weakly singular kernels, nonlinear terms, and fractional-order derivatives presents significant analytical challenges. To address these, we have developed and applied two computational schemes to approximate solutions for such equations. For steady problems, we introduce a linearized fully spectral method, that represents u ( x ) using shifted Gegenbauer polynomials (SGPs) in the vector Λ( x ), and use Picard's iterative method for the nonlinear terms. For unsteady problems, we extend a semi-discrete scheme that estimates temporal derivatives with a forward difference formula and spatial variables with the SGPs vector Λ( x ). We also establish new operational matrices to estimate singular integral terms, such as: ∫ 0 x ∫ 0 y ∫ 0 z K ( x , p ) u γ ( p , t ) / ( x ρ 1 − p ρ 1 ) α 1 ( y ρ 2 − q ρ 2 ) α 2 ( z ρ 3 − r ρ 3 ) α 3 d r d q d p ; with ρ 's > 1, 0 < α 's < 1. These methods convert the original nonlinear problem into a system of linear algebraic equations that are straightforward to solve. We implement both schemes in Maple 2015 and validate our results against existing literature. Various test problems are used to demonstrate the accuracy, stability, and reliability of the proposed schemes. Simulations across a range of nonlinear parameters γ, α, β, ρ, M , and N confirm that our methods are accurate, stable, and effective for these challenging problems.