African Journal of Science, Technology, Innovation and Development

This paper deals with a class of Kirchhoff type problems on a double phase setting with a small perturbation.It is well known that (AR)condition is an important technique to apply the Mountain Pass theorem.A legitimate question arises, can we ensure the existence of weak solutions in the case where (AR)-condition is not satisfied?Our goal is to give a positive answer to this question.We present a new sufficient assumption weaker than (AR)-condition for which the considered problem admits at least two weak solutions.The proof rely on variational arguments based on the Mountain Pass Theorem with Cerami condition.Our results extend the results already proven by other authors.

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the typewhere f ∈ L 1 (Ω) and F ∈ N i=1 L p i (•) (Ω), and a i (x, u, ξ) are Carathéodory functions from Ω × IR × IR N into IR , which satisfy assumptions of growth, coercivity and strict monotonicity.We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces.Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods.The problems of the type (1) are very interesting from the purely mathematical point of view.On the other hand, such equations (1) appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.

In this paper, we develop a Haar wavelet-based reconstruction method to recover missing data on an inaccessible part of the boundary from measured data on another accessible part.The technique is developed to solve inverse Cauchy problems governed by the Poisson equation which is severely ill-posed.The new method is mathematically simple to implement and can be easily applied to Cauchy problems governed by other partial differential equations appearing in various fields of natural science, engineering, and economics.To take into account the ill-conditioning of the obtained linear system due to the ill-posedness of the Cauchy problem, a preconditioning strategy combined with a regularization has been developed.Comparing the numerical results produced by a meshless method based on the polynomial expansion with those produced by the proposed technique illustrates the superiority of the latter.Other numerical results show its effectiveness.

In this paper, we aim to study an inverse problem for determining two time-independent coefficients in a fractional diffusion system from the final measurements.First, we prove the well-posedness of the state problem, and then we show some regularity results for the solution of the direct system using the Mittag-Leffler function.Then, we reformulate our inverse problem into an optimal control one.Afterwards, we establish the existence of the minimizer and prove the stability estimate for two coefficients with respect to the final data.The descent method is proposed as a numerical one based on the gradient calculus via the adjoint state and we compare it with the conjugate gradient method.Finally, we will present some numerical tests that shows the efficiency of the proposed methods.

We consider the problem of identifying an unknown portion $ \Gamma $ of the boundary of a $ d $-dimensional ($ d = 2, 3 $) body $ \Omega $ by a pair of Cauchy data $ (f, g) $ on the accessible part $ \Sigma $ of the boundary of a harmonic function $ u $. On the unknown boundary, a Robin homogeneous condition is assumed. For a fixed constant impedance $ \alpha $, it was shown by Cakoni and Kress in [1] through concrete examples that a single measurement of $ (f, g) $ on $ \Sigma $ can give rise to infinitely many different domains $ \Omega $. Nonetheless, shape optimization techniques can provide fair detections of the unknown boundary given a single pair of Cauchy data as we will showcase here. On this purpose, the inverse problem is recast into three different shape optimization formulations and the shape derivative of the cost function associated with each formulations are obtained. The shape gradient informations are then utilized in a Sobolev gradient-based scheme via finite element method to solve the optimization problems. Numerical results are provided to illustrate the feasibility of the proposed numerical methods in two and three spatial dimensions.

Electrodeposition is a low-cost and malleable technique for manufacturing a wide diversity of equipment and materials including coatings and films. It was initially operated to prepare metallic mirrors and corrosion resistant surfaces among other things. We consider a 1D (1 dimensional) stationary problem, that is a mathematical model derived from nickel-iron alloy electrodeposition's chemical reactions.The aim of this paper is to provide a mathematical study of the stationary one dimensional case problem using the topological degree.

This work develops a novel class of numerical approximation algorithms for highly nonlinear stochastic differential equations. It is inspired by a stochastic approximation/optimization algorithm. The idea is the generation of random-varying truncation bounds. The algorithms are suited in case the coefficients have faster than linear growth resulting in the finite explosion time in implementing the usual Euler-Maruyama scheme, and are easier to be implemented in contrast to the existing approaches. In this paper, weak convergence and weak convergence rates of the algorithms are established using the martingale problem formulation. Some numerical examples are presented for demonstration. Finally, remarks on algorithms with additional random switching and algorithms with decreasing step sizes are given.

In this paper, we establish a blow-up criterion for the local smooth solutions to the three-dimensional incompressible Boussinesq equations in the framework of Besov spaces $ \dot{B}_{\infty ,\infty }^{0} $ via partial horizontal derivatives of two velocity components $ \partial _{1}u_{1} $ and $ \partial _{2}u_{2} $. Our result can be regarded as the borderline case of the work by F. Wu (Bull. Braz. Math. Soc., New Series (52) 2021, 267-279).