Browsing by Author "Chattouh Abdeldjalil"

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    Error Analysis of Legendre-Galerkin Spectral Method for a Parabolic Equation with Dirichlet-Type Non-Local Boundary Conditions
    (2021) Chattouh Abdeldjalil; Saoudi Khaled
    An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper. The spatial discretization is based on Galerkin formulation and the Legendre orthogonal polynomials, while the time derivative is discretized by using the symmetric Euler finite difference schema. The stability and convergence of the semi-discrete spectral approximation are rigorously set up by following a novel approach to overcome difficulties caused by the non-locality of the boundary condition. Several numerical tests are included to confirm the efficacy of the proposed method and to support the theoretical results.
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    Legendre-Chebyshev pseudo-spectral method for the diffusion equation with non-classical boundary conditions
    (2020) Chattouh Abdeldjalil; Saoudi Khaled
    The present paper is devoted to the numerical approximation for the diffusion equation subject to non-local boundary conditions. For the space discretization, we apply the Legendre-Chebyshev pseudospectral method, so that, the problem under consideration is reduced to a system of ODEs which can be solved by the second order Crank-Nicolson schema. Optimal error estimates for the semi-discrete scheme are derived in L2-norm. Numerical tests are included to demonstrate the effectiveness of the proposed method.
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    Numerical solution for a class of parabolic integrodifferential equations subject to integral boundary conditions
    (2022) Chattouh Abdeldjalil
    Many physical phenomena can be modelled through nonlocal boundary value problems whose boundary conditions involve integral terms. In this work we propose a numerical algorithm, by combining second-order Crank-Nicolson schema for the temporal discretization and Legendre-Chebyshev pseudo-spectral method (LC-PSM) for the space discretization, to solve a class of parabolic integrodifferential equations subject to nonlocal boundary conditions. The approach proposed in this paper is based on Galerkin formulation and Legendre polynomials. Results on stability and convergence are established. Numerical tests are presented to support theoretical results and to demonstrate the accuracy and effectiveness of the proposed method.
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    On the Numerical Solution of a Semilinear Sobolev Equation Subject to Nonlocal Dirichlet Boundary Condition
    (2020) Chattouh Abdeldjalil; Saoudi Khaled
    A semilinear Sobolev equation ∂t(u − ∆u) − ∆u = f(∇u) with a Dirichlet-type integral boundary condition is investigated in this contribution. Using the Rothe method which is based on a semi-discretization of the problem under consideration with respect to the time variable, we prove the existence and uniqueness of a weak solution. Moreover, a suitable approach for the numerical solution based on Legendre spectral-method is presented.
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    Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition
    (2022) Chattouh Abdeldjalil; Saoudi Khaled; Nouar Maroua
    A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.