Kouakou Cyrille N'Dri, Ardjouma Ganon, G. Yoro, K. A. Touré
In this paper, we study numerical approximations of a semilinear parabolic problem in one-dimension, of which the nonlinearity appears both in source term and in Neumann boundary condition. By a semidiscretization using finite difference method, we obtain a system of ordinary differential equations which is an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches in a finite time and estimate its semidiscrete quenching time. Convergence of the numerical quenching time to the theoretical one is established. Next, we show that the quenching rate of the numerical scheme is different from the continuous one. Finally, we give some numerical results to illustrate our analysis.
{"title":"Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux","authors":"Kouakou Cyrille N'Dri, Ardjouma Ganon, G. Yoro, K. A. Touré","doi":"10.33993/jnaat522-1325","DOIUrl":"https://doi.org/10.33993/jnaat522-1325","url":null,"abstract":"In this paper, we study numerical approximations of a semilinear parabolic problem in one-dimension, of which the nonlinearity appears both in source term and in Neumann boundary condition. By a semidiscretization using finite difference method, we obtain a system of ordinary differential equations which is an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches in a finite time and estimate its semidiscrete quenching time. Convergence of the numerical quenching time to the theoretical one is established. Next, we show that the quenching rate of the numerical scheme is different from the continuous one. Finally, we give some numerical results to illustrate our analysis.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"142 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139149354","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}
The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform is the main part of the singular integral equations on the real line. Therefore, approximations of the Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of the singular integrals in the case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is by far poorer than the one on bounded intervals. The case of the Hilbert Transform has been considered very little. This article is devoted to the approximation of the Hilbert transform in Lebesgue spaces by operators which introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. In this paper, we prove that the approximating operators are bounded maps in Lebesgue spaces and strongly converges to the Hilbert transform in these spaces.
{"title":"Approximation of the Hilbert transform in the Lebesgue spaces","authors":"Rashid Aliev, Lale Alizade","doi":"10.33993/jnaat522-1312","DOIUrl":"https://doi.org/10.33993/jnaat522-1312","url":null,"abstract":"The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform is the main part of the singular integral equations on the real line. Therefore, approximations of the Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of the singular integrals in the case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is by far poorer than the one on bounded intervals. The case of the Hilbert Transform has been considered very little. This article is devoted to the approximation of the Hilbert transform in Lebesgue spaces by operators which introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. In this paper, we prove that the approximating operators are bounded maps in Lebesgue spaces and strongly converges to the Hilbert transform in these spaces.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139149695","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}
Hammed Anuoluwapo Abbas, K. Aremu, O. Oyewole, A. Mebawondu, O. Narain
In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.
{"title":"Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces","authors":"Hammed Anuoluwapo Abbas, K. Aremu, O. Oyewole, A. Mebawondu, O. Narain","doi":"10.33993/jnaat522-1351","DOIUrl":"https://doi.org/10.33993/jnaat522-1351","url":null,"abstract":"In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"27 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139151636","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}
M. Benbourhim, A. Bouhamidi, Pedro González-Casanova
This paper establishes convergence rates and error estimates for the pseudo-polyharmonic div-curl and elastic interpolation. This type of interpolation is based on a combination of the divergence and the curl of a multivariate vector field and minimizing an appropriate functional energy related to the divergence and curl. Convergence rates and error estimates are established when the interpolated vector field is assumed to be in the classical fractional vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. The error estimates introduced in this work are sharp and the rate of convergence depends algebraically on the fill distance of the scattered data nodes. More precisely, the order of convergence depends, essentially, on the smoothness of the target vector field, on the dimension of the Euclidean space and on the null space of corresponding Sobolev semi-norm.
{"title":"Convergence and error estimates for pseudo-polyharmonic div-curl and elastic interpolation on a bounded domain","authors":"M. Benbourhim, A. Bouhamidi, Pedro González-Casanova","doi":"10.33993/jnaat521-1306","DOIUrl":"https://doi.org/10.33993/jnaat521-1306","url":null,"abstract":"This paper establishes convergence rates and error estimates for the pseudo-polyharmonic div-curl and elastic interpolation. This type of interpolation is based on a combination of the divergence and the curl of a multivariate vector field and minimizing an appropriate functional energy related to the divergence and curl. Convergence rates and error estimates are established when the interpolated vector field is assumed to be in the classical fractional vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. The error estimates introduced in this work are sharp and the rate of convergence depends algebraically on the fill distance of the scattered data nodes. More precisely, the order of convergence depends, essentially, on the smoothness of the target vector field, on the dimension of the Euclidean space and on the null space of corresponding Sobolev semi-norm.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121170436","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}
A survey of some classical results from the theory of trigonomtrical series is presented, especially the case of Fourier series. Some new proofs are presented, and Riemann's theory of trigonometrical series is is given special attention.
{"title":"Notes regarding classical Fourier series","authors":"P. Bracken","doi":"10.33993/jnaat521-1307","DOIUrl":"https://doi.org/10.33993/jnaat521-1307","url":null,"abstract":"A survey of some classical results from the theory of trigonomtrical series is presented, especially the case of Fourier series. Some new proofs are presented, and Riemann's theory of trigonometrical series is is given special attention.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131045556","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}
In this paper, we propose an algorithm based on branch and bound method to underestimate the objective function and reductive transformation which is transformed the all multivariable functions on univariable functions. We also demonstrate several quadratic lower bound functions are proposed which they are better/preferable than the others well-known in literature. We obtain that our experimental results are more effective when we face different nonconvex functions.
{"title":"New technique for solving multivariate global optimization","authors":"Djamel Aaid, Ö. Özer","doi":"10.33993/jnaat521-1287","DOIUrl":"https://doi.org/10.33993/jnaat521-1287","url":null,"abstract":"In this paper, we propose an algorithm based on branch and bound method to underestimate the objective function and reductive transformation which is transformed the all multivariable functions on univariable functions. We also demonstrate several quadratic lower bound functions are proposed which they are better/preferable than the others well-known in literature. We obtain that our experimental results are more effective when we face different nonconvex functions.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123621875","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}
We consider the unique continuation (data assimilation) problem for the Helmholtz equation and study its numerical approximation based on physics-informed neural networks (PINNs). Exploiting the conditional stability of the problem, we first give a bound on the generalization error of PINNs. We then present numerical experiments in 2d for different frequencies and for geometric configurations with different stability bounds for the continuation problem. The results show that vanilla PINNs provide good approximations even for noisy data in configurations with robust stability (both low and moderate frequencies), but may struggle otherwise. This indicates that more sophisticated techniques are needed to obtain PINNs that are frequency-robust for inverse problems subject to the Helmholtz equation.
{"title":"Solving ill-posed Helmholtz problems with physics-informed neural networks","authors":"Mihai Nechita","doi":"10.33993/jnaat521-1305","DOIUrl":"https://doi.org/10.33993/jnaat521-1305","url":null,"abstract":"We consider the unique continuation (data assimilation) problem for the Helmholtz equation and study its numerical approximation based on physics-informed neural networks (PINNs). Exploiting the conditional stability of the problem, we first give a bound on the generalization error of PINNs. We then present numerical experiments in 2d for different frequencies and for geometric configurations with different stability bounds for the continuation problem. The results show that vanilla PINNs provide good approximations even for noisy data in configurations with robust stability (both low and moderate frequencies), but may struggle otherwise. This indicates that more sophisticated techniques are needed to obtain PINNs that are frequency-robust for inverse problems subject to the Helmholtz equation.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132632389","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}
In some old results, we find estimates the best approximation (E_{n,p}(f)) of a periodic function satisfying (f^{(r)}inmathbb{L}^p_{2pi}) in terms of the norm of (f^{(r)}) (Favard inequality). In this work, we look for a similar result under the weaker assumption (f^{(r)}in mathbb{L}^q_{2pi}), with (1
{"title":"New estimates related with the best polynomial approximation","authors":"J. Bustamante","doi":"10.33993/jnaat521-1313","DOIUrl":"https://doi.org/10.33993/jnaat521-1313","url":null,"abstract":"In some old results, we find estimates the best approximation (E_{n,p}(f)) of a periodic function satisfying (f^{(r)}inmathbb{L}^p_{2pi}) in terms of the norm of (f^{(r)}) (Favard inequality). In this work, we look for a similar result under the weaker assumption (f^{(r)}in mathbb{L}^q_{2pi}), with (1<q<p<infty). We will present inequalities of the form (E_{n,p}(f)leq C(n)Vert D^{(r)}fVert_q), where (D^{(r)}) is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129742056","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}
This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.
{"title":"Non-homogeneous impulsive time fractional heat conduction equation","authors":"A. Aghili","doi":"10.33993/jnaat521-1316","DOIUrl":"https://doi.org/10.33993/jnaat521-1316","url":null,"abstract":"This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133283276","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}
In this article, we present a general form of the fixed point method for processing the large and sparse linear complementarity problem, as well as a general condition for the method's convergence when the system matrix is a (P)-matrix and some sufficient conditions for the proposed method when the system matrix is a (H_+)-matrix or symmetric positive definite matrix.
{"title":"A note on fixed point method and linear complementarity problem","authors":"B. Kumar, Deepmala, Arup K Das","doi":"10.33993/jnaat521-1290","DOIUrl":"https://doi.org/10.33993/jnaat521-1290","url":null,"abstract":"In this article, we present a general form of the fixed point method for processing the large and sparse linear complementarity problem, as well as a general condition for the method's convergence when the system matrix is a (P)-matrix and some sufficient conditions for the proposed method when the system matrix is a (H_+)-matrix or symmetric positive definite matrix.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123287414","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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