A Unified Multifaceted Space-Time Wavelet Framework and Its Singularity Analysis for Weakly Singular Non-local Partial Integro-Differential Equations in High Dimensions

Abstract This work presents a comprehensive efficiency and convergence analysis of wavelet-based methods within a multi-dimensional framework for detecting singularities in nonlocal weakly singular integro-partial differential equations in one and two dimensions. The proposed approach incorporates the multi-resolution properties of wavelets to accurately identify and localize singularities in solutions to such equations. Combinations of space-time wavelets with their advantages are very limited for higher-dimensional problems, and their convergence analysis on collocation points is not fully clear till the present day. For problems having time singularity, the present work shows that multi-resolution analysis through 2D/3D Haar wavelets requires a lower regularity assumption for the convergence of the proposed procedure than several approaches on finite-difference setup or other wavelets like Hermite, Chebyshev, or Bernoulli’s wavelets. In particular, we produce a higher-order convergence result (second-order accurate) in the $L^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> norm, based on sufficient regularity assumptions on the solution. In addition to the higher-order estimate, we provide the wavelet-based truncation error estimate for several terms, such as the time-fractional derivative, Volterra &amp; Fredholm integral operators, classical derivatives, and their effects on the regularity of the function for future researchers in this domain. Numerical tests are performed in the $L^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and $L^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> norms to compare the efficiency of this method over existing approaches for weakly singular nonlocal integro-partial differential equations. These experiments show the efficiency of the proposed approach in several kinds of regularity assumptions of the solution. This also guarantees the convergence of approximations to the functions having weak singularities in time and the higher-order accuracy for sufficiently smooth solutions.