Hyperbolic interface problems are widely applied to model wave propagation and shock transmission across discontinuous media, such as acoustic waves in layered materials, seismic waves in the Earth’s crust, and stress or electromagnetic waves in composite structures. This study introduces a novel computational framework for hyperbolic interface problems, specifically designed to unify and extend the treatment of regular interfaces within partial differential equations. The proposed hybrid approach combined Haar wavelet-based spatial discretization with finite difference schemes for temporal integration. By employing truncated Haar series to approximate spatial derivatives and leveraging finite difference techniques for time evolution, the method delivers accurate solutions for both linear and nonlinear systems regardless of whether the governing coefficients are constant or spatially variable. In addressing linear problems, the resulting algebraic equations are solved efficiently using Gaussian elimination. For nonlinear formulations, the method incorporates a quasi-Newton linearization strategy, effectively transforming the system into a linear one. Extensive validation is performed through a suite of benchmark problems, with performance assessed via metrics including maximum absolute errors (MAEs), root mean square errors (RMSEs), and convergence behavior as a function of collocation point (CP) density. Numerical experiments highlight the method’s superior stability and accuracy, particularly in scenarios marked by discontinuities or sharp gradients in the solution. The approach proves especially effective in bridging inconsistencies between boundary and initial conditions, offering a robust alternative to existing techniques. Theoretical soundness, strong convergence properties, and comprehensive numerical validation collectively underscore the method’s reliability and adaptability across a broad spectrum of applications.

Background: Accurate classification of interstitial lung disease (ILD) patterns in HRCT images remains challenging due to overlapping textures across disease types. We propose a computer-aided diagnosis pipeline for classifying five lung tissue classes from HRCT images: ground glass, micronodule, emphysema, fibrosis, and normal. Methods: The method applies a multi-stage texture-enhancement pre-processing step (separating coarse/fine, high/low-contrast, and directional image components) followed by extraction of nine texture, gradient and gradient-orientation descriptors (LBP, RGLBP, BDIP, bilateral, CDF, SDLGFFO, SSGSM, HOG, and gradient orientation). All nine descriptors were evaluated individually, together with one fused configuration, a combination of four of the nine (LBP, RGLBP, BDIP, and HOG), which gave the highest accuracy. Feature dimensionality is reduced using PCA guided by an eigenvalue-based estimator, and classification is performed with an extreme learning machine (ELM), with SVM (RBF kernel) used as a comparison baseline. Results: On a dataset of 5000 HRCT images (1000 per class; 500 train/500 test per class), the fused four-dimensional representation, classified with ELM (without kernel), achieved 92% accuracy—the best result among 46 configurations compared directly on the held-out test set and therefore a test-set-optimized figure rather than an independently validated estimate of generalization performance—outperforming individual descriptors and alternative dimensionality-reduction methods (LLE, Kernel LDA, Laplacian Eigenmaps, Nyström, Random Fourier). Conclusions: The proposed pipeline combines a multi-stage texture-enhancement step with multi-descriptor fusion; the enhancement step is intended to improve the separability of subtle ILD texture patterns, but its individual contribution has not been verified, as an enhanced-versus-unenhanced ablation was not performed.

Delay differential equations form a distinct class of differential equations in which the derivative of the dependent variable depends on its value at an earlier time. This unique structure makes them particularly suitable for modeling phenomena in areas such as population dynamics, epidemiology, and the spread or control of diseases. In this study, a high-order Haar wavelet collocation method (HoHWCM) is proposed for the numerical solution of second-order delay differential equations (SoDDEs). The delay term is approximated using a Taylor expansion, transforming the SoDDE into a standard second-order differential equation (SoDE). The nonlinear terms are linearized through an innovative Taylor series-based approach, which also serves as an efficient iterative scheme. The resulting SoDE is then discretized using Haar wavelet basis functions, yielding a system of linear algebraic equations that is solved iteratively. This strategy eliminates the need for Newton’s or Broyden’s methods, thereby reducing computational cost and improving time efficiency. A variety of linear and nonlinear benchmark problems are solved to evaluate the accuracy and efficiency of the proposed method. Comparative results with established approaches from the literature demonstrate that the HoHWCM achieves higher accuracy, faster convergence, and reduced computational time, making it a highly effective alternative for solving such problems.