Abstract The aim of this paper is to present the mathematical and numerical study of a nonlocal nonlinear model based on the variable exponent p(x)-Laplacian for removing Cauchy noise, which is a type of impulsive and non-Gaussian degradation. The proposed model benefits from the performance of the nonlocal approach to preserve small details and textures, and the efficiency of the variable exponent to reduce the execution time. To demonstrate the reliability of our proposed model, we provide some experimental denoising results and illustrate the comparison with some models from the literature.
{"title":"A Nonlocal Model for Reconstructing Images Corrupted by Cauchy Noise","authors":"F. Bendaida","doi":"10.2478/mjpaa-2023-0003","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0003","url":null,"abstract":"Abstract The aim of this paper is to present the mathematical and numerical study of a nonlocal nonlinear model based on the variable exponent p(x)-Laplacian for removing Cauchy noise, which is a type of impulsive and non-Gaussian degradation. The proposed model benefits from the performance of the nonlocal approach to preserve small details and textures, and the efficiency of the variable exponent to reduce the execution time. To demonstrate the reliability of our proposed model, we provide some experimental denoising results and illustrate the comparison with some models from the literature.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48424936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this work and in the context of PDE constrained optimization problems, we are interested in identification of a parameter in the diffusion equation proposed in [1]. We propose to identify this parameter automatically by a gradient descent algorithm to improve the restoration of a noisy image. Finally, we give numerical results to illustrate the performance of the automatic selection of this parameter and compare our numerical results with other image denoising approaches or algorithms.
{"title":"Parameters identification for a nonlinear partial differential equation in image denoising","authors":"A. E. Mourabit","doi":"10.2478/mjpaa-2023-0010","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0010","url":null,"abstract":"Abstract In this work and in the context of PDE constrained optimization problems, we are interested in identification of a parameter in the diffusion equation proposed in [1]. We propose to identify this parameter automatically by a gradient descent algorithm to improve the restoration of a noisy image. Finally, we give numerical results to illustrate the performance of the automatic selection of this parameter and compare our numerical results with other image denoising approaches or algorithms.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47007283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we propose to compare three a posteriori error estimators namely equilibrated, star-based and residual based for the Poisson problem and the Stokes problem with lowest-order Crouzeix-Raviart finite element discretization. The numerical results are presented to compare the performance of the three estimators in an adaptive refinement strategy.
{"title":"Numerical comparison of three a posteriori error estimators for nonconforming finite element method","authors":"B. Achchab, A. Agouzal, K. Bouihat, A. Majdoubi","doi":"10.2478/mjpaa-2023-0001","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0001","url":null,"abstract":"Abstract In this paper, we propose to compare three a posteriori error estimators namely equilibrated, star-based and residual based for the Poisson problem and the Stokes problem with lowest-order Crouzeix-Raviart finite element discretization. The numerical results are presented to compare the performance of the three estimators in an adaptive refinement strategy.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46939497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract To have a more realistic model, in this paper, This manuscript is devoted to investigating a fractional-order mathematical model of Kouidere et al. That describes the dynamics of spread of African swine fever virus (ASFV). The aim of this work is to protect susceptible pigs from the virus, In our model, by including three controls which represent: the iron fencing and spraying pesticides and get rid. The aims of this paper is to reduce the number of infected pigs and ticks by using optimal control strategy and fractinal order derivation. Pontryagin’s maximal principle is used to describe optimal controls with Caputo time-fractional derivative and the optimal system is resolved in an iterative manner. Numerical simulations are presented based on the presented method. We finished tis article with a conclusion.
{"title":"Fractional optimal control problem for a mathematical modeling of African swine fever virus transmission","authors":"A. Kouidere, O. Balatif, M. Rachik","doi":"10.2478/mjpaa-2023-0007","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0007","url":null,"abstract":"Abstract To have a more realistic model, in this paper, This manuscript is devoted to investigating a fractional-order mathematical model of Kouidere et al. That describes the dynamics of spread of African swine fever virus (ASFV). The aim of this work is to protect susceptible pigs from the virus, In our model, by including three controls which represent: the iron fencing and spraying pesticides and get rid. The aims of this paper is to reduce the number of infected pigs and ticks by using optimal control strategy and fractinal order derivation. Pontryagin’s maximal principle is used to describe optimal controls with Caputo time-fractional derivative and the optimal system is resolved in an iterative manner. Numerical simulations are presented based on the presented method. We finished tis article with a conclusion.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69233473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the history of option pricing, Black-Scholes model is one of the most significant models. In this paper, we present a new numerical strategy for valuing American option pricing problems governed by Black-Scholes model (BSM). Numerical computations are carried out to show the efficiency and robustness of the proposed method. We compare our numerical solution with the ones based on Finite Element Method (FEM) and the Enriched Finite Element Method (PUFEM). Our result shows the efficiency of the proposed strategy. In addition, that approach can be used to treat nonlinear evolutionary problems.
{"title":"About the valuation of American option under Black-Scholes model : a numerical study","authors":"R. Malek","doi":"10.2478/mjpaa-2023-0005","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0005","url":null,"abstract":"Abstract In the history of option pricing, Black-Scholes model is one of the most significant models. In this paper, we present a new numerical strategy for valuing American option pricing problems governed by Black-Scholes model (BSM). Numerical computations are carried out to show the efficiency and robustness of the proposed method. We compare our numerical solution with the ones based on Finite Element Method (FEM) and the Enriched Finite Element Method (PUFEM). Our result shows the efficiency of the proposed strategy. In addition, that approach can be used to treat nonlinear evolutionary problems.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42774876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem: { Δ2pu=λm(x)u in Ω,∂u∂v=∂(Δu)∂v=…=∂(Δ2p-1u)∂v=0 on ∂Ω. left{ {matrix{ {{Delta ^{2p}}u = lambda mleft( x right)u,,,in,,,Omega ,} cr {{{partial u} over {partial v}} = {{partial left( {Delta u} right)} over {partial v}} = ldots = {{partial left( {{Delta ^{2p - 1}}u} right)} over {partial v}} = 0,,,on,,,partial Omega .} cr } } right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, p ∈ ℕ*, m ∈ L∞ (Ω), and Δ2pu := Δ (Δ...( Δu)), 2p times the operator Δ.
{"title":"Existence of solutions for 4p-order PDES with Neumann boundary conditions","authors":"N. Moradi, F. Moradi, S. E. Habib, M. Addam","doi":"10.2478/mjpaa-2023-0004","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0004","url":null,"abstract":"Abstract In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem: { Δ2pu=λm(x)u in Ω,∂u∂v=∂(Δu)∂v=…=∂(Δ2p-1u)∂v=0 on ∂Ω. left{ {matrix{ {{Delta ^{2p}}u = lambda mleft( x right)u,,,in,,,Omega ,} cr {{{partial u} over {partial v}} = {{partial left( {Delta u} right)} over {partial v}} = ldots = {{partial left( {{Delta ^{2p - 1}}u} right)} over {partial v}} = 0,,,on,,,partial Omega .} cr } } right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, p ∈ ℕ*, m ∈ L∞ (Ω), and Δ2pu := Δ (Δ...( Δu)), 2p times the operator Δ.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44640896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ahmed Nafidi, Ilyasse Makroz, B. Achchab, R. Gutiérrez-Sánchez
Abstract We propose a novel diffusion process having a mean function equal to the Pareto probability density function up to a constant of proportionality. We examine the probabilistic properties of the proposed model. Then, referring to the problem of statistical inference, we describe the approach employed to tackle the issue of obtaining parameter estimates by maximizing the likelihood function based on discrete sampling. This estimation reduces to solving a set of complex equations, that is accomplished using the simulated annealing algorithm. A simulation study is also given to validate the methodology presented. Finally, using a real-world example of the Moroccan child mortality rate, we obtain the fits and forecasts by employing the suggested stochastic process and nonlinear regression model.
{"title":"Stochastic Pareto diffusion process : Statistical analysis and computational issues. Simulation and Application","authors":"Ahmed Nafidi, Ilyasse Makroz, B. Achchab, R. Gutiérrez-Sánchez","doi":"10.2478/mjpaa-2023-0009","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0009","url":null,"abstract":"Abstract We propose a novel diffusion process having a mean function equal to the Pareto probability density function up to a constant of proportionality. We examine the probabilistic properties of the proposed model. Then, referring to the problem of statistical inference, we describe the approach employed to tackle the issue of obtaining parameter estimates by maximizing the likelihood function based on discrete sampling. This estimation reduces to solving a set of complex equations, that is accomplished using the simulated annealing algorithm. A simulation study is also given to validate the methodology presented. Finally, using a real-world example of the Moroccan child mortality rate, we obtain the fits and forecasts by employing the suggested stochastic process and nonlinear regression model.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44975151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we study the cyclicity problem with respect to the forward shift operator Sb acting on the de Branges–Rovnyak space ℋ (b) associated to a function b in the closed unit ball of H∞ and satisfying log(1− |b| ∈ L1(𝕋). We present a characterisation of cyclic vectors for Sb when b is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of Sb in this case, but we provide here an elementary proof. We also study the situation where b has the form b = (1+ I)/2, where I is a non-constant inner function such that the associated model space KI = ℋ (I) has an orthonormal basis of reproducing kernels.
摘要本文研究了前移算子Sb作用于H∞闭单位球中与函数b相关的de Branges-Rovnyak空间H (b)上且满足log(1−|b|∈L1(交)的循环性问题。给出了当b是非有限Blaschke积的有理函数时Sb的循环向量的刻画。在这种情况下，这个特征可以由在[22]中给出的Sb不变子空间的描述推导出来，但是我们在这里提供一个初等证明。我们还研究了b的形式为b = (1+ I)/2的情况，其中I是一个非常数内函数，使得相关模型空间KI = h (I)具有再现核的标准正交基。
{"title":"Cyclicity in de Branges–Rovnyak spaces","authors":"E. Fricain, S. Grivaux","doi":"10.2478/mjpaa-2023-0016","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0016","url":null,"abstract":"Abstract In this paper, we study the cyclicity problem with respect to the forward shift operator Sb acting on the de Branges–Rovnyak space ℋ (b) associated to a function b in the closed unit ball of H∞ and satisfying log(1− |b| ∈ L1(𝕋). We present a characterisation of cyclic vectors for Sb when b is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of Sb in this case, but we provide here an elementary proof. We also study the situation where b has the form b = (1+ I)/2, where I is a non-constant inner function such that the associated model space KI = ℋ (I) has an orthonormal basis of reproducing kernels.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46379369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we investigate the limit points set of descent spectrum of upper triangular operator matrices MC=(AC0B) {M_C} = left( {matrix{A hfill & C hfill cr 0 hfill & B hfill cr } } right) . We prove that acc(σdes(MC)) ∪ Waccσdes = acc(σdes(A)) ∪ acc(σdes(B)) where Waccσdes is the union of certain holes in acc(σdes(MC)), which happen to be subsets of acc(σasc(B)) ∩ acc(σdes(A)). Furthermore, several sufficient conditions for acc(σdes(MC)) = acc(σdes(A)) ∪ acc(σdes(B)) holds for every C ∈ ℬ(Y, X) are given.
{"title":"Limit points for descent spectrum of operator matrices","authors":"H. Boua, M. Karmouni, A. Tajmouati","doi":"10.2478/mjpaa-2022-0024","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0024","url":null,"abstract":"Abstract In this paper, we investigate the limit points set of descent spectrum of upper triangular operator matrices MC=(AC0B) {M_C} = left( {matrix{A hfill & C hfill cr 0 hfill & B hfill cr } } right) . We prove that acc(σdes(MC)) ∪ Waccσdes = acc(σdes(A)) ∪ acc(σdes(B)) where Waccσdes is the union of certain holes in acc(σdes(MC)), which happen to be subsets of acc(σasc(B)) ∩ acc(σdes(A)). Furthermore, several sufficient conditions for acc(σdes(MC)) = acc(σdes(A)) ∪ acc(σdes(B)) holds for every C ∈ ℬ(Y, X) are given.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47523801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is { -div(b(x)| ∇u |p-2∇u(1+| u |)γ)+| ∇u |p| u |θ=f,in Ω,u=0,on ∂Ω, left{ {matrix{ { - divleft( {bleft( x right){{{{left| {nabla u} right|}^{p - 2}}nabla u} over {left( {1 + left| u right|} right)gamma }}} right) + {{{{left| {nabla u} right|}^p}} over {{{left| u right|}^theta }}} = f,} hfill & {in,Omega ,} hfill cr {u = 0,} hfill & {on,partial Omega ,} hfill cr } } right. swhere Ω is a bounded open subset in ℝN, 1 ≤ θ < 2, p > 2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f.
研究了一类具有退化矫顽力和奇异梯度低阶项的非线性椭圆方程的存在性和正则性结果。模型问题是{div (b (x) | |∇u p 2∇u (1 u + | |)γ)+ |∇u p | | |θ= f,在Ω,u = 0,在∂Ω,左 {{矩阵{{- div 离开({b 离开(x 右){{{{左|{微分算符u} 右|}^ {p - 2}} 微分算符u} /{离开({1 + 左| u |} ) 伽马}}}右)+{{{{左|{微分算符u} 右|}^ p}} /{{{左| u 右|}^ θ}}}= f,} hfill &{ω,} hfill cr {u = 0} hfill &{、ω部分} hfill cr}} 。其中Ω是一个有界开子集，1≤θ < 2, p > 2和γ > 0。我们将证明，即使低阶项是奇异项，在各种关于源f可和性的假设下，我们得到了正解的存在性和正则性。
{"title":"Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term","authors":"H. Khelifi","doi":"10.2478/mjpaa-2022-0022","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0022","url":null,"abstract":"Abstract We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is { -div(b(x)| ∇u |p-2∇u(1+| u |)γ)+| ∇u |p| u |θ=f,in Ω,u=0,on ∂Ω, left{ {matrix{ { - divleft( {bleft( x right){{{{left| {nabla u} right|}^{p - 2}}nabla u} over {left( {1 + left| u right|} right)gamma }}} right) + {{{{left| {nabla u} right|}^p}} over {{{left| u right|}^theta }}} = f,} hfill & {in,Omega ,} hfill cr {u = 0,} hfill & {on,partial Omega ,} hfill cr } } right. swhere Ω is a bounded open subset in ℝN, 1 ≤ θ < 2, p > 2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47857091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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