Searcharxiv⌕ Search

arXiv subjects

Ahmed Aberqi

Publications and source records attributed to Ahmed Aberqi.

8 recordsLinked to original sources
arXiv2026

Solving the Fisher nonlinear differential equations via Physics-Informed Neural Networks: A Comprehensive Retraining Study and Comparative Analysis with the Finite Difference Method

Physics-Informed Neural Networks (PINNs) represent a groundbreaking paradigm in scientific computing, seamlessly integrating the robust framework of deep learning with fundamental physical laws. This paper meticulously applies the standard PINN framework to solve the challenging one-dimensional nonlinear Fisher-KPP equation, a critical model in reaction-diffusion dynamics describing phenomena such as population spread and flame propagation. We detail a comprehensive methodology, encompassing the neural network architecture, the physics-informed loss function, and an in-depth investigation into retraining strategies aimed at optimizing model performance. Our approach is rigorously validated through a direct comparison of the PINN solution against both the known analytical solution and a numerical solution derived from the Finite Difference Method (FDM). Through this work, we elucidate the intricate balance between model complexity, training efficiency, and accuracy. Results highlight the PINN's remarkable capability in accurately approximating the solution to this complex PDE, while also shedding light on the critical aspects and challenges of model retraining, particularly concerning the optimizer's state. This study provides a thorough quantitative error analysis, demonstrating the efficacy of PINNs as a viable and competitive alternative to traditional numerical methods for solving nonlinear differential equations, and discusses their broader applications across various scientific domains.

math.NA↗
arXiv2024

Approximate Controllability of Fractional Differential Systems with Nonlocal Conditions of Order $q\in ]1,2[$

This manuscript is concerned with the approximate controllability of fractional nonlinear differential equations with nonlocal conditions of order $1<q<2$ in Banach spaces. As far as we know, few articles have investigated this issue. The idea is to see under which sufficient conditions the proposed control problem is approximately controllable. The discussion is based on the theory of resolvent operator, fractional calculus techniques and Krasnoselskii's fixed point theorem under the assumption that the associated linear system is approximately controllable. The obtained results improve some existing analogous ones on this topic. Finally, an example is provided to illustrate the applications of the obtained results.

math.OC↗
arXiv2023

Homogeneous incompressible Bingham viscoplastic as a limit of bi-viscosity fluids

In this paper, the existence of a weak solution for homogeneous incompressible Bingham fluid is investigated. The rheology of such a fluid is defined by a yield stress $τ_y$ and a discontinuous stress-strain law. This non-Newtonian fluid behaves like a solid at low stresses and like a non-linear fluid above the yield stress. In this work we propose to build a weak solution for Navier stokes Bingham equations using a bi-viscosity fluid as an approximation, in particular, we proved that the bi-viscosity tensor converges weakly to the Bingham tensor.

math.AP↗
arXiv2022

Existence of solutions for a singular double phase in Sobolev-Orlicz spaces with variable exponents in a complete manifold

The purpose of this paper is to study a class of double phase problems, with a singular term and a superlinear parametric term on the right-hand side. Using the method of Nehari manifold combined with the fibering maps, we prove that for all small values of the parameter λ > 0, there exist at least two non-trivial positive solutions. Our results extend the previous works Papageorgiou, Repovus, and Vetro [24] and Liu, Dai, Papageorgiou, and Winkert [21], from the case of Musielak-Orlicz Sobolev space, when exponents p and q are constant, to the case of Sobolev-Orlicz spaces with variable exponents in a complete manifold.

math.AP↗
arXiv2021

Existence Results for double phase problem in Sobolev-Orlicz spaces with variable exponents in Complete Manifold

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti-Rabinowitz type condition in the framework of Sobolev-Orlicz spaces with variable exponents in complete compact Riemannian n-manifolds. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder inequality, continuous and compact embedding results are proved.

math.AP↗
arXiv2021

Nonlinear Elliptic Equations With Variable Exponents Satisfying Cerami Condition

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems involving the (p(x), q(x))-equation and the nonlinearity is superlinear but does not satisfy the usual Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in Complete manifolds. The main results are established by means of the mountain pass theorem and Fountain theorem with Cerami condition. Moreover, we are giving an example of a (p(x), q(x)) equation that verifies all our demonstrated results.

math.AP↗