A perturbed laguerre least-squares method for higher-order linear integro-differential equations

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We develop a perturbed Laguerre least-squares method (P-LLSM) for solving higher-order linear Fredholm and Volterra integro-differential equations subject to general linear side conditions. The approximation is constructed as UN,r(x)=∑ ᵢ₌₀N-1aᵢLᵢ(x)+∑ ⱼ₌₀ʳ⁻¹τ ⱼTN+j*(x), where the classical Laguerre basis provides the primary approximation space and shifted Chebyshev polynomials supply a non-overlapping enrichment. The unknown coefficients are obtained by minimizing the residual functional J(c)=∫ ₐᵇ(LUN,r-g)²\,dx under the constraint $$Bc=h$$ . Under explicit assumptions on regularity, feasibility, rank of the constraint matrix, and injectivity of the operator on the homogeneous feasible space, we prove existence and uniqueness of the minimizer, derive the associated Karush–Kuhn–Tucker system, establish residual orthogonality, and obtain a best-residual property with an error estimate. Numerical experiments for Fredholm and Volterra problems demonstrate that the proposed perturbation improves accuracy without altering the well-posed structure of the approximation space. In particular, the enrichment reduces the maximum nodal error and residual functional by several orders of magnitude for the non-polynomial Volterra example. The results show that the proposed method is an effective spectral framework for higher-order linear integro-differential problems.