Comparative Analysis of Wavelet Bases for Solving First-Kind Fredholm Integral Equations
Temirbekov N. Tamabay D. Tleulesova A. Mukhanova T.
August 2025Multidisciplinary Digital Publishing Institute (MDPI)
Computation
2025#13Issue 8
This research presents a comparative analysis of numerical methods for solving first-kind Fredholm integral equations using the Bubnov–Galerkin method with various wavelet and orthogonal polynomial bases. The bases considered are constructed from Legendre, Laguerre, Chebyshev, and Hermite wavelets, as well as Alpert multiwavelets and CAS wavelets. The effectiveness of these bases is evaluated by measuring errors relative to known analytical solutions at different discretization levels. Results show that global orthogonal systems—particularly the Chebyshev and Hermite—achieve the lowest error norms for smooth target functions. CAS wavelets, due to their localized and oscillatory nature, produce higher errors, though their accuracy improves with finer discretization. The analysis has been extended to incorporate perturbations in the form of additive noise, enabling a rigorous assessment of the method’s stability with respect to different wavelet bases. This approach provides insight into the robustness of the numerical scheme under data uncertainty and highlights the sensitivity of each basis to noise-induced errors.
approximate solution , Bubnov–Galerkin method , first-kind Fredholm integral equation , orthonormality , projection method , wavelet basis
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Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
National Engineering Academy of the Republic of Kazakhstan, Almaty, 050060, Kazakhstan
Faculty of Mechanics and Mathematics
National Engineering Academy of the Republic of Kazakhstan
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