arXiv subjects

Mondher Benjemaa

Publications and source records attributed to Mondher Benjemaa.

5 recordsLinked to original sources

arXiv2023

Rescaling method for blow-up solutions of nonlinear wave equations

We develop a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. In order to numerically reproduce the blow-up phenomena, we propose a rule of scaling transformation, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A careful study of the convergence of the proposed scheme is carried out and several numerical examples are performed in illustration.

math.NA↗

arXiv2021

Discontinuous Galerkin method for blow-up solutions of nonlinear 1D wave equations

We develop and study a time-space discrete discontinuous Galerkin finite elements method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several numerical examples and benchmarks.

math.AP↗

arXiv2021

On the fractional operators with respect to another function

This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove that these results are valid at once for several standard fractional operators such as the Riemann-Liouville and Caputo operators, the Hadamard operators, the Erdélyi-Kober operators, etc., depending on the choice of the scaling function. We also show that our technique can be useful to solve a wide range of Volterra integral equations. The numerical approximation of solutions of systems involving the fractional derivatives with respect to another function is also investigated and the optimal convergence rate of the schemes is reached in graded meshes, even in the case of singular solutions. Various examples and numerical tests, with an application to the Erdélyi-Kober operators, are performed at the end to illustrate the efficiency of the proposed approach.

math.NA↗

arXiv2020

New results on the uniform exponential stability of non-autonomous perturbed dynamical systems

In this paper, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic systems nearby an equilibrium point, when the nominal parts are subject to non necessarily small perturbations. We show that, under some estimates on the perturbation terms, the equilibrium point remains (globally) uniformly exponentially stable. The results we obtained can easily be applied in practice since they are based on the Gronwall-Bellman inequalities rather than the classical Lyapunov methods that require the knowledge of a Lyapunov function. Several numerical examples are presented in order to illustrate the validity of our study, especially when the standard Lyapunov approaches are useless.

math.DS↗

arXiv2017

Generalized Taylor formulas involving generalized fractional derivatives

In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{α,ρ}\left(x^ρ-a^ρ\right)^{jα} + e_m(x)}$ \noindent with $m\in \mathbb{N}$, $c_j^{α,ρ}\in \mathbb{R}$, $x>a$ and $0< α\leq 1$. In case $ρ= α= 1$, this expression coincides with the classical Taylor formula. The coefficients $c_j^{α,ρ}$, $j=0,\dots,m$ as well as an estimation of $e_m(x)$ are given in terms of the generalized Caputo-type fractional derivatives. Some applications of these results for approximation of functions and for solving some fractional differential equations in series form are given in illustration.

math.CA↗