{"title":"A stable and high-accuracy numerical method for determining the time-dependent coefficient in the bioheat equation","authors":"Yan Qiao, Lin Sang, Hua Wu","doi":"10.1016/j.cam.2025.116528","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we propose a space–time spectral method for time-dependent coefficient identification of the inverse problem with the Ionkin-type nonlocal boundary and integral over-determination conditions. The Legendre–Galerkin method is applied in the spatial direction and the Legendre-tau method is applied in the time direction. And the method is also implemented by the explicit–implicit iterative method. The nonlinear term is collocated at the Chebyshev–Gauss–Lobatto points and computed explicitly by the fast Legendre transform. Tikhonov regularization is applied to employ the blood perfusion coefficient computation with the noisy perturbations. The adopted stabilization scheme presents a good performance in terms of accuracy, effectiveness and robustness on the inverse problem, especially for noisy perturbations. Numerical results are given to show the accuracy and stability of the approach and agree well with theory analysis. Optimal order convergence is also obtained through the estimates in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"464 ","pages":"Article 116528"},"PeriodicalIF":2.1000,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042725000433","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we propose a space–time spectral method for time-dependent coefficient identification of the inverse problem with the Ionkin-type nonlocal boundary and integral over-determination conditions. The Legendre–Galerkin method is applied in the spatial direction and the Legendre-tau method is applied in the time direction. And the method is also implemented by the explicit–implicit iterative method. The nonlinear term is collocated at the Chebyshev–Gauss–Lobatto points and computed explicitly by the fast Legendre transform. Tikhonov regularization is applied to employ the blood perfusion coefficient computation with the noisy perturbations. The adopted stabilization scheme presents a good performance in terms of accuracy, effectiveness and robustness on the inverse problem, especially for noisy perturbations. Numerical results are given to show the accuracy and stability of the approach and agree well with theory analysis. Optimal order convergence is also obtained through the estimates in the -norm.
期刊介绍:
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.