Résumé: In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more difficult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using both the obtained solution of the conformable time-derivative heat equation and the inverse Cole-Hopf transformation. This approach shows its efficiency to deal with this class of fractional nonlinear problems. Some numerical experiments are displayed to consolidate our approach.

Résumé: This paper focuses on investigating the maximum number of limit cycles bifurcating from the periodic orbits adapted to the cubic system given by x ̇= y− y x+ a 2, y ̇=− x+ x x+ a 2, where a is a positive number with a≠ 1. The study specifically examines the perturbation of this system within the class of all septic polynomial differential systems. Our main result demonstrates that the first-order averaging theory associated with the perturbed system yields a maximum of twenty-two limit cycles.

Résumé: This work concerns the study of the Burgers nonlinear equation for a perfect and a weakly viscous fluid. To solve the equation, we adopt the finite difference method combined with explicit and implicit schemes. We add to the original equation a numerical dispersion due to truncation errors (discretization errors). Then, we study the stability and convergence of the solution and make a comparison with some existing results. For illustration, the numerical simulations are given to support the theory.

Résumé: In this paper, we deal with a study of modified time-fractional Burgers equations. The idea is based on the use of a Cole-Hopf transformation which transforms the time-fractional modified Burgers equations into linear parabolic time fractional equations. To solve the latter, we use the Fourier transformation. Therefore, the solution of the modified time-fractional Burgers equations can be found by using the solution of parabolic equation and the inverse Cole-Hopf transformation.

Résumé: In this paper we consider a mixed formulation for study of the bilaplacian problem with obstacle constraints in conjunction with the penalization method. The idea is based on the decomposition of the bilaplacian operator into two coupled laplacians and of course by choosing suitable spaces. The numerical advantage is to allow the solution of coupled systems with nice matrices having the M-matrix property. Moreover, the computed solution of the problem requires less execution time with respect to the discrete system of the bilaplacian problem. For simulation, we test the e ciency of a variety of iterative relaxation methods and discuss their numerical performances.

Résumé: We study an iterative process to accelerate the successive approximations method in a monotonous convergence framework. It consists in interrupting the sequence of the successive approximations method produced at the kth iteration and substituting it by a combination of the element of the sequence produced at the iterate k+ 1 and an extrapolation vector. The latter uses a parameter which can be calculated mathematically. We illustrate numerically this process by studying a freeboundary problems class.

Résumé: The present study deals with the solution of a problem, defined in a three-dimensional domain, arising in fluid mechanics. Such problem is modelled with unilateral constraints on the boundary. Then, the problem to solve consists in minimizing a functional in a closed convex set. The characterization of the solution leads to solve a time-dependent variational inequality. An implicit scheme is used for the discretization of the time-dependent part of the operator and so we have to solve a sequence of stationary elliptic problems. For the solution of each stationary problem, an equivalent form of a minimization problem is formulated as the solution of a multivalued equation, obtained by the perturbation of the previous stationary elliptic operator by a diagonal monotone maximal multivalued operator. The spatial discretization of such problem by appropriate scheme leads to the solution of large scale algebraic …

Résumé: This paper focuses on the mathematical study of the existence of solitary gravity waves (solitons) and their characteristics (amplitude, velocity, ) generated by a piston wave maker lying upstream of a horizontal channel. The mathematical model requires both incompressibility condition, irrotational flow of no viscous fluid and Lagrange coordinates. By using both the inverse scattering method and a given initial potential we can transform the KdV equation into Sturm–Liouville spectral problem. The latter problem amounts to find negative discrete eigenvalues and associated eigenfunctions , where each calculated eigenvalue gives a soliton and the profile of the free surface. For solving this problem, we can use the Runge–Kutta method. For illustration, two examples of the wave maker movement are proposed. The numerical simulations show that the perturbation of wave maker with hyperbolic …