Applied Numerical Mathematics最新文献

{"title":"Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of convection-diffusion type","authors":"Yanhua Liu,&nbsp;Xuesong Wang,&nbsp;Yao Cheng","doi":"10.1016/j.apnum.2024.06.023","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.023","url":null,"abstract":"<div><p>We develop a local discontinuous Galerkin (LDG) method for a fourth-order singularly perturbed problem of convection-diffusion type. The existence and uniqueness of the computed solution are verified. Using the Shishkin mesh we derive an optimal-order energy-norm error estimate which is uniformly valid in the perturbation parameter. Numerical experiments are also given to support our theoretical findings.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"205 ","pages":"Pages 16-37"},"PeriodicalIF":2.2,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low rank approximation in the computation of first kind integral equations with TauToolbox","authors":"Paulo B. Vasconcelos ,&nbsp;Laurence Grammont ,&nbsp;Nilson J. Lima","doi":"10.1016/j.apnum.2024.06.022","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.022","url":null,"abstract":"<div><p><span>Tau Toolbox</span> is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called <span>polynomial</span>, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.</p><p>In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.</p><p>Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.</p><p>The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"205 ","pages":"Pages 1-15"},"PeriodicalIF":2.2,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001703/pdfft?md5=49b2fb7a69e48f47a313e9bf4b9ddaa9&pid=1-s2.0-S0168927424001703-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Energetic spectral-element time marching methods for phase-field nonlinear gradient systems","authors":"Shiqin Liu ,&nbsp;Haijun Yu","doi":"10.1016/j.apnum.2024.06.021","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.021","url":null,"abstract":"<div><p>We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"205 ","pages":"Pages 38-59"},"PeriodicalIF":2.2,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An efficient hybrid conjugate gradient method with an adaptive strategy and applications in image restoration problems","authors":"Zibo Chen ,&nbsp;Hu Shao ,&nbsp;Pengjie Liu ,&nbsp;Guoxin Li ,&nbsp;Xianglin Rong","doi":"10.1016/j.apnum.2024.06.020","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.020","url":null,"abstract":"<div><p>In this study, we introduce a novel hybrid conjugate gradient method with an adaptive strategy called asHCG method. The asHCG method exhibits the following characteristics. (i) Its search direction guarantees sufficient descent property without dependence on any line search. (ii) It possesses strong convergence for the uniformly convex function using a weak Wolfe line search, and under the same line search, it achieves global convergence for the general function. (iii) Employing the Armijo line search, it provides an approximate guarantee for worst-case complexity for the uniformly convex function. The numerical results demonstrate promising and encouraging performances in both unconstrained optimization problems and image restoration problems.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 362-379"},"PeriodicalIF":2.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141542140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An optimized algorithm for numerical solution of coupled Burgers equations","authors":"Anurag Kaur ,&nbsp;V. Kanwar ,&nbsp;Higinio Ramos","doi":"10.1016/j.apnum.2024.06.019","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.019","url":null,"abstract":"<div><p>Investigation of the solutions of the coupled viscous Burgers system is crucial for realizing and understanding some physical phenomena in applied sciences. Particularly, Burgers equations are used in the modeling of fluid mechanics and nonlinear acoustics. In the present study, a modified meshless quadrature method based on radial basis functions is used to discretize the partial derivatives in the spatial variable. A technique to find the best value of the shape parameter is introduced. A high-resolution optimized hybrid block method is then used to solve the problem in the temporal variable. To validate the proposed method, several test problems are considered and the simulated results are compared with exact solutions and previous works. Moreover, a sensitivity analysis for parameter <em>c</em> is conducted, and the unconditional stability of the proposed algorithm has been validated.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 352-361"},"PeriodicalIF":2.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141542142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the stability of θ-methods for DDEs and PDDEs","authors":"Alejandro Rodríguez-Fernández ,&nbsp;Jesús Martín-Vaquero","doi":"10.1016/j.apnum.2024.06.018","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.018","url":null,"abstract":"<div><p>In this paper, the stability of <em>θ</em>-methods for delay differential equations is studied based on the test equation <span><math><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>−</mo><mi>A</mi><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>B</mi><mi>y</mi><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></math></span>, where <em>τ</em> is a constant delay and <em>A</em> is a positive definite matrix. It is mainly considered the case where the matrices <em>A</em> and <em>B</em> are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices <em>A</em> and <em>B</em> are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 312-328"},"PeriodicalIF":2.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141542141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The sine and cosine diffusive representations for the Caputo fractional derivative","authors":"Hassan Khosravian-Arab ,&nbsp;Mehdi Dehghan","doi":"10.1016/j.apnum.2024.06.017","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.017","url":null,"abstract":"<div><p>In recent years, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order <span><math><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mn>1</mn></math></span>. An error analysis of the newly presented methods together with some numerical examples is provided at the end.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 265-290"},"PeriodicalIF":2.2,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical threshold stability of a nonlinear age-structured reaction diffusion heroin transmission model","authors":"X. Liu ,&nbsp;M. Zhang ,&nbsp;Z.W. Yang","doi":"10.1016/j.apnum.2024.06.016","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.016","url":null,"abstract":"<div><p>This paper deals with the numerical threshold stability of a nonlinear age-space structured heroin transmission model. A semi-discrete system is established by spatially domain discretization of the original nonlinear age-space structured model. A threshold value is proposed in stability analysis of the semi-discrete system and named as a numerical basic reproduction number. Besides the role it plays in numerical threshold stability analysis, the numerical basic reproduction number can preserve qualitative properties of the exact basic reproduction number and converge to the latter while stepsizes vanish. A fully discrete system is established via a time discretization of the semi-discrete system, in which an implicit-explicit technique is implemented to ensure the preservation of the biological meanings (such as positivity) without CFL restriction. Some numerical experiments are exhibited in the end to confirm the conclusions and explore the final state.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 291-311"},"PeriodicalIF":2.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations","authors":"Saurabh Kumar ,&nbsp;Vikas Gupta ,&nbsp;Dia Zeidan","doi":"10.1016/j.apnum.2024.06.014","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.014","url":null,"abstract":"<div><p>In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 249-264"},"PeriodicalIF":2.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141438115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new class of quadrature rules for estimating the error in Gauss quadrature","authors":"Aleksandar V. Pejčev ,&nbsp;Lothar Reichel ,&nbsp;Miodrag M. Spalević ,&nbsp;Stefan M. Spalević","doi":"10.1016/j.apnum.2024.06.011","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.011","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The need to evaluate Gauss quadrature rules arises in many applications in science and engineering. It often is important to be able to estimate the quadrature error when applying an &lt;em&gt;ℓ&lt;/em&gt;-point Gauss rule, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;em&gt;f&lt;/em&gt; is an integrand of interest. Such an estimate often is furnished by applying another quadrature rule, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, with &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; nodes, and using the difference &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or its magnitude as an estimate for the quadrature error in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or its magnitude. The classical approach to estimate the error in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is to let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, with &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, be the Gauss-Kronrod quadrature rule associated with &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. However, it is well known that the Gauss-Kronrod rule associated with a Gauss rule &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; might not exist for certain measures that determine the Gauss rule and for certain numbers of nodes. This prompted M. M. Spalević &lt;span&gt;[1]&lt;/span&gt; to develop generalized averaged Gauss rules, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, with &lt;span&gt;&lt;math&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; nodes for estimating the error in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Similarly as for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-node Gauss-Kronrod rules, &lt;em&gt;ℓ&lt;/em&gt; nodes of the rule &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; agree with the nodes of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. However, generalized averaged Gauss rules are not internal for some measures. They therefore may not be applicable when the integrand only is define","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 206-221"},"PeriodicalIF":2.2,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}