arXiv subjects

Nabil Nassif

Publications and source records attributed to Nabil Nassif.

4 recordsLinked to original sources

arXiv2026

A Two-Stage Learning PINN Approach for Solving the Inverse Problem of the 1D Porous Medium Equation

The Porous Medium Equation (PME), given by $u_t = \Delta(u^m)$ for $m > 1$, is a degenerate nonlinear parabolic partial differential equation that arises in various physical applications such as fluid flow in porous media, heat transfer in plasmas, and population dynamics. It is known for its nonlinear diffusion and finite propagation speed. In this paper, we study numerical solutions of the one-dimensional direct and inverse PME using Physics-Informed Neural Networks (PINNs), and compare them with classical numerical methods and available analytical and manufactured solutions. While PINNs provide a flexible framework for solving both forward and inverse problems, we show that the standard inverse formulation suffers from a strong sensitivity to the initial guess, leading to only local convergence. To address this issue, we propose a novel two-stage PINN training framework for the inverse problem, which significantly improves convergence stability and allows reliable recovery of the unknown parameter even for poor initial guesses. Overall, the proposed approach demonstrates that PINNs are a flexible and accurate alternative to classical methods for the 1D PME, and the introduced two-stage training strategy substantially improves their robustness in inverse problems, providing a solid basis for extensions to more complex geometries and higher-dimensional cases.

math.OC↗

arXiv2026

Direct Problem for Gas Diffusion in Polar Firn with Variable Coefficients

We consider the mathematical model of gas trapping in deep polar ice (firns), which consists of a parabolic partial differential equation, that can degenerate at one boundary extreme. In [1], we considered all the coefficients to be constants, except the diffusion coefficient D(z) that is to be reconstructed. In this paper, we assume both the diffusion coefficient D(z) and the volume fraction f(z) are functions. The difficulty in this problem, both theoretically and computationally, arises from the fact that D(z) and f(z) may be zero at bottom of the firn. To handle such degeneracy, we defined appropriate weighted Sobolev spaces and used Lion's theorem to prove existence and uniqueness of the semi-variational formulation of the Firn PDE. A full discrete system is obtained through a P1 Finite element Galerkin procedure in space and an Euler-Implicit scheme in time. Sufficient conditions for the existence and uniqueness of the solution for the discrete system are obtained.

math.NA↗

arXiv2022

Strong and Weak Solutions to the Hasegawa-Mima Equation with Periodic Boundary Conditions

The two dimensional Hasegawa-Mima (HM) equation $$ -Δu_t+u_t = \{u,Δu\} + ku_y$$ describes the time evolution of drift waves in magnetically-confined plasma. Several authors have treated the HM equation theoretically and numerically, with difficulties arising when handling the non-linear Poisson's bracket $\{u,Δu\}:=u_xΔu_u-u_yΔu_x $. In this paper, we introduce a new decoupling approach that avoids the Poisson's bracket term by reformulating the HM equation as a system of two linear PDEs, a solution of which is a pair $(u,w)$ such that $$(HM)\,\,\,\left\{\begin{array}{lll} w_t + \vec{V}(u) \cdot \nabla w = ku_y\\ -Δu+u=w, \\ \end{array}\right.$$ where $\vec{V}(u)= -u_y \vec{\textbf{i}} + u_x \vec{\textbf{j}}$ is a divergence-free vector field. Based on this coupled hyperbolic-elliptic system, we derive several variational frames, all propitious for finding weak solutions with spacial periodic boundary conditions and lower regularity assumptions on the initial data. More precisely, for initial data $u_0 \in H_P^2(Ω)$ with $w_0:=(I-Δ) u_0 \in L^2(Ω)$, we prove the existence of a weak solution that is global in time. And for initial data $u_0 \in H_P^3(Ω)$ with $w_0:=(I-Δ) u_0 \in H_P^1(Ω) \cap L^\infty(Ω)$, we prove the existence of a unique strong solution that is local in time. Our proofs are based on the existence of fixed-point ordered pairs $\{u_N,w_N\}$ that solve Petrov-Galerkin HM systems, constructed using spacial Fourier basis. Through appropriate a-priori estimates combined with compactness arguments, we reach when $N\to\infty$ limit point solutions $(u,w)$ to the (HM) system.

math.AP↗

arXiv2021

A Finite-Element Model for the Hasegawa-Mima Wave Equation

In a recent work, two of the authors have formulated the non-linear space-time Hasegawa-Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair $(u,w)$, with $w=(I-Δ)u$. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa-Mima model, obtaining global existence of solutions in $H^2$ on any time interval $[0,T]$ for all T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system with a nonrestrictive condition on the time step. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.

math.NA↗