Moulay Rchid Sidi Ammi

Diamond- Jensen's Inequality on Time Scales

The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-Open image in new window derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensens inequality on time scales via the diamond-Open image in new window integral and present some corollaries, including Holders and Minkowskis diamond-Open image in new window integral inequalities.The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond- derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensens inequality on time scales via the diamond- integral and present some corollaries, including Hölders and Minkowskis diamond- integral inequalities.

Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed.

Optimal control of nonlocal thermistor equations

We are concerned with the optimal control problem of the well known nonlocal thermistor problem, i.e. in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature. Existence of an optimal control is proved. The optimality system consisting of the state system coupled with adjoint equations is derived, together with a characterisation of the optimal control. Uniqueness of solution to the optimality system, and therefore, the uniqueness of the optimal control, is established. The last part is devoted to numerical simulations.

Semidiscretization for a nonlocal parabolic problem

A time discretization technique by Euler forward scheme is proposed to deal with a nonlocal parabolic problem. Existence and uniqueness of the approximate solution are proved.

Existence and uniqueness results for a fractional Riemann–Liouville nonlocal thermistor problem on arbitrary time scales

Abstract Using a fixed point theorem in a proper Banach space, we prove existence and uniqueness results of positive solutions for a fractional Riemann–Liouville nonlocal thermistor problem on arbitrary nonempty closed subsets of the real numbers.

Galerkin spectral method for the fractional nonlocal thermistor problem

We develop and analyse a numerical method for the time-fractional nonlocal thermistor problem. By rigorous proofs, some error estimates in different contexts are derived, showing that the combination of the backward differentiation in time and the Galerkin spectral method in space leads, for an enough smooth solution, to an approximation of exponential convergence in space.

Existence of Three Positive Solutions to Some p-Laplacian Boundary Value Problems

We obtain, by using the Leggett-Williams fixed point theorem, sufficient conditions that ensure the existence of at least three positive solutions to some p-Laplacian boundary value problems on time scales.

Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem

We obtain necessary optimality conditions for a semi-discretized optimal control problem for the classical system of nonlinear partial differential equations modelling the water-oil (isothermal dead-oil model).

Regularity of solutions to higher-order integrals of the calculus of variations

We obtain new regularity conditions for problems of calculus of variations with higher-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main regularity result asserts that autonomous integral functionals with a Lagrangian having coercive partial derivatives with respect to the higher-order derivatives admit only minimisers with essentially bounded derivatives.

Numerical analysis of some time-fractional partial differential equations for noise removal

In this paper, we consider the numerical resolution of two time-fractional partial differential equations for processing image denoising, which are obtained from some classical partial differential equations for processing image denoising, edge preservation and compression by replacing the first-order time derivative with a fractional derivative of order α, with 0 < α < 1. Discrete schemes of these models are introduced based on the finite differences method. Then, stability and error estimates are derived on the approximate solutions. Numerical experiments are presented to show the robustness of these new time-fractional models to obtain better results in image denoising and restoration.