Results in Applied Mathematics最新文献

{"title":"A fitted mesh robust numerical method and analysis for the singularly perturbed parabolic PDEs with a degenerate coefficient","authors":"Hassan J. Al Salman ,&nbsp;Fasika Wondimu Gelu ,&nbsp;Ahmed A. Al Ghafli","doi":"10.1016/j.rinam.2024.100519","DOIUrl":"10.1016/j.rinam.2024.100519","url":null,"abstract":"<div><div>In this study, we present a nearly second-order central finite difference approach for solving a singularly perturbed parabolic problem with a degenerate coefficient. The approach uses a Crank–Nicolson method to discretize the time direction on the uniform mesh and a second-order central finite difference method on the Shishkin mesh in the space direction. The solution to the problem shows a parabolic boundary layer around <span><math><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>. Our error estimates indicate that the suggested approach is nearly second-order <span><math><mi>ɛ</mi></math></span>-uniformly convergent both in space and time directions. Some numerical results have been generated to validate the theoretical findings. Extensive comparisons have been carried out, demonstrating that the current approach is more accurate than previous methods in the literature.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100519"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing the coarseness measure of a bicolored point set over guillotine partitions","authors":"José Fernández Goycoolea ,&nbsp;Luis H. Herrera ,&nbsp;Pablo Pérez-Lantero ,&nbsp;Carlos Seara","doi":"10.1016/j.rinam.2024.100503","DOIUrl":"10.1016/j.rinam.2024.100503","url":null,"abstract":"<div><div>The coarseness of a set of points in the plane colored red and blue is a measure of how well the points are mixed together. It has appealing theoretical properties, including a connection to the set of points tendency to accept a good clustering partition. Yet, it is computationally expensive to compute exactly. In this paper, the notion of computing the coarseness using a guillotine partition approach is introduced, and efficient algorithms for computing this guillotine coarseness are presented: a top-down approach and a dynamic programming approach, both of them achieving polynomial time and space complexities. Finally, an even faster <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>n</mi><mo>)</mo></mrow></mrow></math></span> polynomial-time algorithm to compute a reduced version of the measurement named two-level guillotine coarseness is presented using geometric data structures for faster computations. These restrictions establish lower bounds for the general guillotine coarseness that allow the development of more efficient algorithms for computing it.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100503"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A numerical technique for a class of nonlinear fractional 2D Volterra integro-differential equations","authors":"F. Afiatdoust ,&nbsp;M.H. Heydari ,&nbsp;M.M. Hosseini ,&nbsp;M. Mohseni Moghadam","doi":"10.1016/j.rinam.2024.100510","DOIUrl":"10.1016/j.rinam.2024.100510","url":null,"abstract":"<div><div>The present study focuses on designing a multi-step technique, known as the block-by-block technique, to provide the numerical solution for a category of nonlinear fractional two-dimensional Volterra integro-differential equations. The proposed technique is a block-by-block method based on Romberg’s numerical integration formula, which simultaneously obtains highly accurate solutions at certain nodes without requiring initial starting values. The convergence analysis of the established method for the aforementioned equations is investigated using Gronwall’s inequality. Several numerical tests are presented to demonstrate the accuracy, speed, and good performance of the procedure.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100510"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The numerical solution of a Fredholm integral equations of the second kind by the weighted optimal quadrature formula","authors":"Abdullo Hayotov ,&nbsp;Samandar Babaev","doi":"10.1016/j.rinam.2024.100508","DOIUrl":"10.1016/j.rinam.2024.100508","url":null,"abstract":"<div><div>This work considers the optimal quadrature formula in a Hilbert space for the numerical approximation of the integral equations. It discusses the sequence of solving integral equations with quadrature formulas. An optimal quadrature formula with weight is constructed in the Hilbert space. The algorithms for solving the integral equation are given using the constructed optimal quadrature formula and trapezoidal rule. Several integral equations are solved based on these algorithms.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100508"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Induction of patterns through crowding in a cross-diffusion model","authors":"Mohammed Aldandani ,&nbsp;John Ward ,&nbsp;Fordyce A. Davidson","doi":"10.1016/j.rinam.2024.100506","DOIUrl":"10.1016/j.rinam.2024.100506","url":null,"abstract":"<div><div>In this paper we focus on pattern formation in systems of interacting populations. We show that if one considers these populations to be “crowded” in a way that is defined below, then cross-diffusion terms appear naturally. Moreover, we show that these additional cross-diffusion terms can generate stable spatial patterns that are not manifest in the corresponding standard “dilute” formulation. This result demonstrates the need for care when choosing standard Fickian diffusion as the default in applications to population dynamics.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100506"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the improved convergence of lifted distributional Gauss curvature from Regge elements","authors":"Jay Gopalakrishnan ,&nbsp;Michael Neunteufel ,&nbsp;Joachim Schöberl ,&nbsp;Max Wardetzky","doi":"10.1016/j.rinam.2024.100511","DOIUrl":"10.1016/j.rinam.2024.100511","url":null,"abstract":"<div><div>Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100511"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact solutions of time-delay integer- and fractional-order advection equations","authors":"C.N. Angstmann,&nbsp;S.-J.M. Burney,&nbsp;D.S. Han,&nbsp;B.I. Henry,&nbsp;Z. Xu","doi":"10.1016/j.rinam.2024.100514","DOIUrl":"10.1016/j.rinam.2024.100514","url":null,"abstract":"<div><div>Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours that are fundamentally different to standard propagating wave solutions.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100514"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weakly coupled system of semilinear structural σ-evolution models with δ- visco-elastic damping","authors":"Abdelhamid Mohammed Djaouti ,&nbsp;Mourad Kainane mezadek ,&nbsp;Mohamed Kainane mezadek ,&nbsp;Ali M.A. Bany Awad","doi":"10.1016/j.rinam.2024.100490","DOIUrl":"10.1016/j.rinam.2024.100490","url":null,"abstract":"<div><div>This paper focuses on the study of global existence (in time) of solutions to a weakly coupled system of Cauchy problem for semilinear <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-evolution models with <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-visco-elastic damping. The system consists of two equations, one involving the function <span><math><mi>u</mi></math></span> and the other involving the function <span><math><mi>v</mi></math></span>. The equations are characterized by a classical power nonlinearity and a derivative-type nonlinearity. The main objective is to investigate the relationship between the regularity assumptions on the initial data and the range of permissible exponents <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in the power nonlinearity. The paper considers the system in a spatial domain <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and a time domain <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>, with specific conditions on the parameters <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, and <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, under the symmetry property as well as the exponents <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. The initial data <span><math><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></math></span> are required to satisfy certain conditions in terms of their integrability and Sobolev regularity.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100490"},"PeriodicalIF":1.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Legendre–Galerkin spectral method for option pricing under regime switching models","authors":"Abdelmajid Ezzine,&nbsp;Abdellah Alla,&nbsp;Nadia Raissi","doi":"10.1016/j.rinam.2024.100505","DOIUrl":"10.1016/j.rinam.2024.100505","url":null,"abstract":"<div><div>The aim of this paper is to investigate an efficient spectral method for pricing European call options under regime-switching models. The main characteristic of this model is to incorporate the change in behavior of the underlying assets depending on different market states. The option pricing problem is modeled as a system of coupled Black–Scholes PDEs. The spatial discretization of the problem is performed using the Legendre–Galerkin spectral method based on Fourier-like basis functions, while the temporal discretization is based on a Crank–Nicolson scheme. Furthermore, the stability and convergence analysis are carried out for both the semi-and fully discretization of the resulted coupled PDE system. Finally, numerical experiments are illustrated to demonstrate the practical application potential of the discussed approach and its efficiency in real world cases.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100505"},"PeriodicalIF":1.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the exponential stability of uniformly damped wave equations and their structure-preserving discretization","authors":"H. Egger ,&nbsp;S. Kurz ,&nbsp;R. Löscher","doi":"10.1016/j.rinam.2024.100502","DOIUrl":"10.1016/j.rinam.2024.100502","url":null,"abstract":"<div><div>We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for mild solutions using Lyapunov-type arguments. For the formulation of our results, we use the language of Hilbert complexes which provides all the tools required for our analysis and is also general enough to cover a number of interesting examples. Some of these are briefly discussed in the course of the manuscript. The functional analytic setting and the main arguments in our proofs are chosen such that they transfer almost verbatim to the discrete setting. We thus obtain corresponding decay results for numerical approximations of a variety of problems obtained by compatible discretization strategies which can be seen as our main contribution.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100502"},"PeriodicalIF":1.4,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}