{"title":"分数阶演化方程中恢复源的准可逆性方法","authors":"Liangliang Sun, Zhaoqi Zhang, Yunxin Wang","doi":"10.1007/s13540-025-00370-z","DOIUrl":null,"url":null,"abstract":"<p>In this paper, a quasi-reversibility method is used to solve an inverse spatial source problem of multi-term time-space fractional parabolic equation by observation at the terminal measurement data. We are mainly concerned with the case where the time source can be changed sign, which is practically important but has not been well explored in literature. Under certain conditions on the time source, we establish the uniqueness of the inverse problem, and also a Hölder-type conditional stability of the inverse problem is firstly given. Meanwhile, we prove a stability estimate of optimal order for the inverse problem. Then some convergence estimates for the regularized solution are proved under an a-priori and an a-posteriori regularization parameter choice rule. Finally, several numerical experiments illustrate the effectiveness of the proposed method in one-dimensional case.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"44 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The quasi-reversibility method for recovering a source in a fractional evolution equation\",\"authors\":\"Liangliang Sun, Zhaoqi Zhang, Yunxin Wang\",\"doi\":\"10.1007/s13540-025-00370-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, a quasi-reversibility method is used to solve an inverse spatial source problem of multi-term time-space fractional parabolic equation by observation at the terminal measurement data. We are mainly concerned with the case where the time source can be changed sign, which is practically important but has not been well explored in literature. Under certain conditions on the time source, we establish the uniqueness of the inverse problem, and also a Hölder-type conditional stability of the inverse problem is firstly given. Meanwhile, we prove a stability estimate of optimal order for the inverse problem. Then some convergence estimates for the regularized solution are proved under an a-priori and an a-posteriori regularization parameter choice rule. Finally, several numerical experiments illustrate the effectiveness of the proposed method in one-dimensional case.</p>\",\"PeriodicalId\":48928,\"journal\":{\"name\":\"Fractional Calculus and Applied Analysis\",\"volume\":\"44 1\",\"pages\":\"\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2025-01-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fractional Calculus and Applied Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s13540-025-00370-z\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fractional Calculus and Applied Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13540-025-00370-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The quasi-reversibility method for recovering a source in a fractional evolution equation
In this paper, a quasi-reversibility method is used to solve an inverse spatial source problem of multi-term time-space fractional parabolic equation by observation at the terminal measurement data. We are mainly concerned with the case where the time source can be changed sign, which is practically important but has not been well explored in literature. Under certain conditions on the time source, we establish the uniqueness of the inverse problem, and also a Hölder-type conditional stability of the inverse problem is firstly given. Meanwhile, we prove a stability estimate of optimal order for the inverse problem. Then some convergence estimates for the regularized solution are proved under an a-priori and an a-posteriori regularization parameter choice rule. Finally, several numerical experiments illustrate the effectiveness of the proposed method in one-dimensional case.
期刊介绍:
Fractional Calculus and Applied Analysis (FCAA, abbreviated in the World databases as Fract. Calc. Appl. Anal. or FRACT CALC APPL ANAL) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order. The high standards of its contents are guaranteed by the prominent members of Editorial Board and the expertise of invited external reviewers, and proven by the recently achieved high values of impact factor (JIF) and impact rang (SJR), launching the journal to top places of the ranking lists of Thomson Reuters and Scopus.