Mohamed Haiour

This paper develops a unified framework of fractional product inequalities for special functions, extending classical Cauchy–Schwarz type results to nonclassical convexity settings. The approach exploits the structural properties of generalized harmonic [ ]-[ ]-convex functions in combination with local fractional integral operators to derive sharp and explicitly computable bounds. The analysis employs auxiliary identities together with refined Hölder and power-mean techniques adapted to the fractional context. Within this framework, both direct and reverse fractional product inequalities are established, with the deformation mechanisms and coefficient functions characterized in terms of fractional order and convexity parameters. Applications are presented for several important classes of special functions, including Gamma-type, Bessel-type, and hypergeometric-type functions. Moreover, many existing inequalities are recovered as particular or limiting cases, demonstrating the flexibility and generality of the approach. These results provide a broad and practical extension of classical inequality techniques, with potential relevance to fractional calculus, analysis, and related mathematical applications.

In this paper, we develop a unified analytical and computational framework that integrates fractal geometry, scaling laws, and artificial intelligence into the study of parabolic quasi-variational inequalities associated with Hamilton–Jacobi–Bellman equations. The analysis is done for irregular domains with boundaries having a Hausdorff dimension [ ], which enables the influence of the geometric complexity of the fractal interface on the numerical approximation. We develop a generalized overlapping domain decomposition method in a fractal Sobolev space setting. The geometric convergence rate is dictated by a scaling law, with the contraction factor given by [ ] where [ ] denotes the overlap width. This result reveals that interface roughness directly controls the decay rate of the iterative error. Additionally, we prove a multifractal maximum norm error estimate of the form [ ] which shows that the spatial convergence rates depend on the boundary’s fractal dimension. In order to improve the computational efficiency, we have proposed an AI-assisted adaptive overlap method, which relies on the prediction of local spectral radii, and we have shown that it accelerates the convergence significantly. The above results link, in a rigorous way, the fractal geometric structure, the scaling properties of domain decomposition algorithms, and data-driven computational optimization, which provides a novel interdisciplinary tool for the analysis of nonlinear Partial Differantial Equations (PDEs).

This study focuses on a nonlinear viscoelastic wave equation involving logarithmic nonlinearity. It considers a nonlinear distributed delay influencing the boundary feedback, which is coupled with acoustic and fractional boundary conditions. Following the proof of global existence, we demonstrate the exponential growth and blow-up of solutions with positive initial energy under appropriate assumptions and for a general case of the kernel. This finding broadens and enhances earlier results.

This research presents advanced spectral algorithms based on Romanovski–Jacobi polynomials for solving various types of differential equations. The study explores the integration of these polynomials into pseudospectral methods, enabling efficient handling of higher-order ordinary differential equations (ODEs) and second-order linear partial differential equations (PDEs). The proposed methods leverage the operational matrices of derivatives to transform differential equations into systems of algebraic equations, ensuring high accuracy and rapid convergence. Numerical experiments validate the effectiveness of the algorithms in solving problems with complex geometries and boundary conditions. The results demonstrate the superiority of Romanovski–Jacobi-based techniques compared to traditional spectral methods, offering a robust and versatile framework for scientific and engineering applications.

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term.

This study aims to investigate numerically and theoretically a thermoelastic Bresse system where the heat conduction given by the Green and Naghdi theories. Theoretically, we establish the existence and uniqueness of solutions as well as the exponential stability regardless of the system's parameters. Numerically, we present a finite element approximation, and we show the discrete energy decay. At the end, we provide some numerical results with an error estimate based on additional regularity of the solution.

In this paper, Newton-multigrid scheme on adaptive finite element discretisation is employed for solving elliptic quasi-variational inequalities with nonlinear source terms. We use Newton’s method as the outer iteration for the standard linearization, and using standard multigrid as the inner iteration for the solution of the Jacobian system at each step. The uniform convergence of Newton-multigrid methods is shown in the sense that the multigrid methods have a contraction number with respect to the maximum norm.