This paper proposes a unified operator-theoretic framework for analyzing the Finite Difference Method (FDM) and the Variational Iteration Method (VIM) for solving nonlinear second-order boundary value problems (BVPs). The VIM iteration is interpreted as a Picard-type fixed-point iteration in a Banach space, and a computable a priori error bound is established under a Lipschitz condition on the nonlinearity. Concurrently, FDM is treated as a finite-rank projection method, and its convergence is demonstrated via consistency and stability. The main contribution of this work is a novel hybrid a posteriori error estimator that uses the continuous VIM iterate to bound the global discretization error of the FDM solution without requiring additional PDE solves or mesh-doubling, requiring only the VIM iteration already computed as part of the error estimation process. Unlike traditional residual-based estimators, the proposed estimator relies solely on the discrepancy between the smooth semi-analytical approximation and the discrete numerical solution. Theoretical convergence rates and the efficiency of the estimator are validated through numerical experiments on a nonlinear BVP with a known exact solution and a non-solvable linear BVP. The results indicate that the hybrid estimator provides a reliable and computationally efficient indicator for adaptive mesh refinement.
Edem, Inemesit DavidAfrican Journal of Mathematics, Statistics and Computer Science · 2026
Image denoising is a classical inverse problem in which the recovery of a clean image from its noisy observation is complicated by ill-posedness. This paper develops a functional-analytic framework in which a broad class of denoising methods is expressed as the iteration of firmly nonexpansive or quasi-nonexpansive operators on a Hilbert space. Total variation regularization, wavelet shrinkage, projection onto convex sets (POCS), projection onto epigraph sets (PESC), and regularization by denoising via fixed-point projection (RED-PRO) are each shown to fit this structure, and their convergence is obtained from a single Krasnosel'skii–Mann argument. The proximal nature of wavelet shrinkage is distinguished from metric projection onto the \(\ell^1\)-ball, and the demicontractive hypothesis underlying RED-PRO is stated explicitly. Numerical experiments on standard test images show that RED-PRO with a BM3D denoiser attains the highest PSNR at every noise level tested, that Chambolle's TV projection follows within \(1.2\)–\(1.7\) dB, and that POCS and wavelet shrinkage trade accuracy for speed. The framework clarifies the operator-theoretic content of each method and provides a common language for their analysis and comparison.
Inemesit D. Edem, Donald Goodhead, Utomobong M. OkonKtrend - International Journal of Computational Mathematics and Scientific Computing · 2026
A feasible manufacturing adjustment can preserve mandatory commitments while losing most of the contribution attainable through reoptimization. This paper studies nominal-surplus scaling under simultaneous capacity losses and increasing production floors. For arbitrary finite product and resource counts with nonnegative resource use, it derives the exact feasible stress interval and a finite breakpoint audit of the largest scaled-plan regret. A two-product family with strictly positive resource coefficients and positive commitments proves that the optimum-to-scaled-value ratio can grow without bound. A dual-certificate tracing procedure supplies an interval-wide regret bracket without enumerating every dual vertex. The generic tracing principle is established parametric optimization; its specialization here couples commitment-adjusted scaling with a complete regret audit. An explicit two-resource model also gives exact product-mix transitions, shadow prices and coefficient-uncertainty limits. Experiments cover 90 manufacturing instances and compare oracle tracing, direct optimizer grids and independently enumerated dual envelopes. The trace requires a mean of 3.58 LP queries on these instances; 62 have an interior regret maximum, and a 101-point grid understates the largest regret by up to 0.4425 accounting units. These synthetic results support complete-path auditing rather than extrapolation from a nominal shadow price.
Okeke Ikenna StephenKtrend - International Journal of Computational Mathematics and Scientific Computing · 2026
