Ahmed Bchatnia

Ahmed Bchatnia

Qassim University
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Ahmed Bchatnia has a total of 63 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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    Optimal control of mixing efficiency in a viscous fluid: A Navier–Stokes and advection–diffusion approach

    by Ghazi Bel Mufti, Ahmed Bchatnia, Fabrice Golfier

    We study an optimal control problem motivated by the physical process of mixing a soluble substance into a viscous fluid, such as dissolving sugar in coffee by stirring with a spoon. The fluid motion is modeled by the incompressible Navier–Stokes equations, while the transport of the dissolved substance is described by an advection–diffusion equation. The control variable represents the imposed stirring action, modeled here as a distributed body force acting throughout the fluid domain. The objective is to achieve fast and efficient homogenization while minimizing the energetic cost of stirring. We first establish the well-posedness of the coupled system under suitable assumptions on the data and the control. Using the direct method of the calculus of variations, we prove the existence of at least one optimal control. Subsequently, we derive first-order necessary optimality conditions through an extended Lagrangian formalism, which yields a coupled system involving the primal state (velocity, pressure, and concentration) and the adjoint states. The gradient of the cost function is explicitly computed, providing the foundation for gradient-based optimization algorithms. Finally, we implement a gradient-based numerical method and validate it on a two-dimensional test case: for a localized concentration blob stirred by a physically bounded distributed force, the optimal control reduces the terminal spatial variance of the concentration by approximately 99% relative to the uncontrolled (purely diffusive) case, and by approximately 96% relative to a constant forcing of comparable energy. The influence of the regularization parameter on the temporal structure of the optimal forcing is also discussed. The proposed framework offers a rigorous and quantitatively validated approach for optimizing mixing processes in various industrial and natural applications.

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    Global Solution, Decay, and Blow-Up of a Viscoelastic Kirchhoff System with Logarithmic Nonlinearity and Acoustic Fractional Delay Boundary Conditions

    by Zahid Ullah, Jianghao Hao, Ahmed Bchatnia, Abdul Aziz

    We examine a nonlinear viscoelastic Kirchhoff model endowed with a logarithmic source. Assuming suitable hypotheses and under the inclusion of nonlinear distributed delay feedback terms, in addition to the acoustic and fractional boundary feedback terms, we obtain global existence, general decay estimates, and finite-time blow-up for negative initial energy, all for a wide range of relaxation kernels. The Kirchhoff term K(∥∇W∥22) significantly changes the structure of the Nehari manifold and requires a new energy functional based on the primitive K^. Our analysis identifies the precise blow-up mechanism and establishes sharp dimension-dependent restrictions on the exponent p. Our outcomes broaden and enhance recent contributions on classical wave equations to the physically meaningful Kirchhoff framework.

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    Cross-Tempered Fractional Damping in Coupled Viscoelastic Wave Equations: Global Existence and Long-Time Behavior

    by Iqra Kanwal, Jianghao Hao, Ahmed Bchatnia, Muhammad Fahim Aslam, Muhammad Afnan

    This paper studies a coupled system of viscoelastic wave equations with frictional damping, cross-tempered fractional damping, viscoelastic memory, and logarithmic source nonlinearities. The fractional damping acts across the two components, so that the fractional feedback in each equation is generated by the velocity of the other component. To handle the memory and fractional terms, we introduce suitable history and diffusive variables and reformulate the problem as an evolution system in an extended energy space. Under appropriate assumptions on the relaxation kernels, fractional parameters, and nonlinear exponent, we establish the local well-posedness of mild and strong solutions using semigroup theory. We then use a potential-well argument to prove global existence for initial data in the stable set. Under an additional decay condition on the relaxation kernels, an appropriate Lyapunov functional is constructed to establish the exponential decay of the energy. Finally, numerical simulations based on a finite-difference scheme and a physics-informed neural network (PINN) are used to illustrate the predicted decay behavior.

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    Exponential Decay of the Local Energy for Solutions of the Damped Critical Wave Equation Outside the Union of Two Strictly Convex Obstacles

    by Naima Mehenaoui, Saleh Fahad Aljurbua, Ahmed Bchatnia

    This paper is concerned with the defocusing energy-critical wave equation with a localized semilinearity and damping, set in the exterior of two disjoint, strictly convex obstacles in R3. The trapping nature of this geometry, due to a periodic ray bouncing between the obstacles, is the main obstruction to decay. Under an exterior geometric control condition, the local energy is shown to decay exponentially, uniformly over data supported in a fixed ball with energy below any prescribed level. The rate is exactly that of the damped linear flow, and hence independent of the data and of the energy level; only the constant depends on the latter. The approach relies on Lafontaine’s lossless Strichartz estimates, a Lax–Phillips and microlocal defect measure analysis of the damped linear flow, and a Duhamel–Gronwall scheme.

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    Optimal Control of Wave Energy Dissipation via a Mobile Damping Actuator

    by Ahmed Bchatnia, Saleh Fahad Aljurbua

    This work studies an optimal control problem for a wave equation with a moving localized damping, modeled by the characteristic function of a ball whose center evolves according to a controlled second-order dynamical system. The objective is to minimize the H1-norm of the time derivative of the wave at a final time, while penalizing the control effort through a quadratic cost on the acceleration. We first establish the local well-posedness of the coupled system, consisting of a damped wave equation and an ordinary differential equation for the damping center. Under suitable compactness and regularity assumptions on the admissible controls, we prove the existence of at least one optimal control via the direct method of the calculus of variations. We then derive first-order necessary optimality conditions through an extended Lagrangian formalism, yielding a coupled system involving the primal state, two adjoint states, and a pointwise relation linking the optimal control to the adjoint variable. The gradient of the cost functional is computed explicitly, showing that the sensitivity is concentrated on the boundary of the moving damping zone. Finally, a numerical implementation based on a gradient descent algorithm is proposed, using a Newmark scheme for the wave equation and a Verlet scheme for the trajectory, with regularization of the nonsmooth indicator function. The resulting algorithm provides a systematic approach for computing optimal damping trajectories in applications such as vibration suppression and noise control.

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    Well-Posedness and Stability of a Coupled Wave System with Fractional Delay Damping: Application to Two Submarine Cables on a Heterogeneous Seabed

    by Ahmed Bchatnia, Abdelkader Benaissa, Abderrahmane Beniani, Papadopoulos Pericles

    This article investigates a coupled system of wave equations in Rn featuring a delay term within internal fractional feedback. Using semigroup theory, we establish the existence and uniqueness of solutions under a suitable condition relating the weight of the delay term in the fractional feedback to the weight of the term without delay. Second, we demonstrate that the system is strongly stable. Finally, we establish a polynomial decay rate using multiplier techniques combined with the frequency domain method.

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    Optimal Polynomial Stability of the Porous Elastic System with One Dynamic Boundary Control

    by Abderrahmane Kasmi, Saleh Fahad Aljurbua, Ahmed Bchatnia

    This paper investigates the asymptotic stability of a one-dimensional porous elastic system subject to a single boundary control of the fractional derivative type. The system consists of two coupled hyperbolic equations describing the displacement of an elastic solid and the volume fraction, with boundary conditions that include a generalized Caputo fractional derivative of order α∈(0,1) at x=L. This configuration has not been previously addressed in the literature. Using semigroup theory, we first reformulate the system as an abstract Cauchy problem and prove that the associated operator generates a C0-semigroup of contractions on a suitable energy space, establishing global well-posedness. Under explicit and generic conditions on the physical parameters and the length L, we prove strong stability via the Arendt–Batty criterion, showing that all solutions tend to zero in the energy norm as t→∞. The main result is a polynomial decay rate: there exists c>0 such that ∥SA(t)U0∥H≤ct−1/6∥U0∥D(A) for all initial data in the domain of the generator. The proof relies on the Borichev–Tomilov theorem and a detailed contradiction argument based on asymptotic expansions of the resolvent operator. Notably, the decay rate is independent of any relation between the wave propagation speeds, which distinguishes our result from many previous studies on porous elastic or Timoshenko systems.

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    Blow-up in a fractionally delayed and strongly damped logarithmic plate equation: theoretical and numerical analysis

    by Iqra Kanwal, Jianghao Hao, Luqman Bashir, M. Fahim Aslam, Ahmed Bchatnia

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    Pseudo-Almost Automorphic 𝓒0-Solutions: Well-Posedness and Asymptotic Behaviour in Evolution Equations with Nonlocal Constraints

    by Bassem Meknani, Reem Alrebdi, Ahmed Bchatnia

    This research establishes an innovative analytical framework for examining C0-solutions of pseudo-almost automorphic type in evolution systems governed by nonlocal initial conditions within Banach spaces. Our methodological approach is founded upon the compactness properties of the resolvent operator (I−A)−1 and the semigroup generated by A. By integrating semigroup theory with fixed-point methodologies, we develop a unified strategy that successfully overcomes challenges arising from nonlocal initial specifications. The practical applicability of our theoretical framework is verified through its implementation on a transport equation with nonlocal initial history, thereby demonstrating both the existence of solutions and their distinctive character.

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    Asymptotic behavior of the wave equation under time-dependent fractional damping

    by Younes Bidi, Saleh Fahad Aljurbua, Ahmed Bchatnia, Abderrahmane Beniani

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