Sayed A. Dahy, H. M. El-Hawary, Alaa Fahim, Tarek Aboelenen
{"title":"基于调质分数Sturm-Liouville特征问题的高阶谱配置方法","authors":"Sayed A. Dahy, H. M. El-Hawary, Alaa Fahim, Tarek Aboelenen","doi":"10.1007/s40314-023-02475-8","DOIUrl":null,"url":null,"abstract":"Abstract This paper presents an accurate exponential tempered fractional spectral collocation method (TFSCM) to solve one-dimensional and time-dependent tempered fractional partial differential equations (TFPDEs). We use a family of tempered fractional Sturm–Liouville eigenproblems (TFSLP) as a basis and the fractional Lagrange interpolants (FLIs) that generally satisfy the Kronecker delta (KD) function at the employed collocation points. Firstly, we drive the corresponding tempered fractional differentiation matrices (TFDMs). Then, we treat with various linear and nonlinear TFPDEs, among them, the space-tempered fractional advection and diffusion problem, the time-space tempered fractional advection–diffusion problem (TFADP), the multi-term time-space tempered fractional problems, and the time-space tempered fractional Burgers’ equation (TFBE) to investigate the numerical capability of the fractional collocation method. The study includes a numerical examination of the produced condition number $$\\kappa (A)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the $$L^\\infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -norm error and exponential rate of spectral convergence.","PeriodicalId":50649,"journal":{"name":"Computational & Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.5000,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"High-order spectral collocation method using tempered fractional Sturm–Liouville eigenproblems\",\"authors\":\"Sayed A. Dahy, H. M. El-Hawary, Alaa Fahim, Tarek Aboelenen\",\"doi\":\"10.1007/s40314-023-02475-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract This paper presents an accurate exponential tempered fractional spectral collocation method (TFSCM) to solve one-dimensional and time-dependent tempered fractional partial differential equations (TFPDEs). We use a family of tempered fractional Sturm–Liouville eigenproblems (TFSLP) as a basis and the fractional Lagrange interpolants (FLIs) that generally satisfy the Kronecker delta (KD) function at the employed collocation points. Firstly, we drive the corresponding tempered fractional differentiation matrices (TFDMs). Then, we treat with various linear and nonlinear TFPDEs, among them, the space-tempered fractional advection and diffusion problem, the time-space tempered fractional advection–diffusion problem (TFADP), the multi-term time-space tempered fractional problems, and the time-space tempered fractional Burgers’ equation (TFBE) to investigate the numerical capability of the fractional collocation method. The study includes a numerical examination of the produced condition number $$\\\\kappa (A)$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the $$L^\\\\infty $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -norm error and exponential rate of spectral convergence.\",\"PeriodicalId\":50649,\"journal\":{\"name\":\"Computational & Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2023-10-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational & Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s40314-023-02475-8\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational & Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40314-023-02475-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
High-order spectral collocation method using tempered fractional Sturm–Liouville eigenproblems
Abstract This paper presents an accurate exponential tempered fractional spectral collocation method (TFSCM) to solve one-dimensional and time-dependent tempered fractional partial differential equations (TFPDEs). We use a family of tempered fractional Sturm–Liouville eigenproblems (TFSLP) as a basis and the fractional Lagrange interpolants (FLIs) that generally satisfy the Kronecker delta (KD) function at the employed collocation points. Firstly, we drive the corresponding tempered fractional differentiation matrices (TFDMs). Then, we treat with various linear and nonlinear TFPDEs, among them, the space-tempered fractional advection and diffusion problem, the time-space tempered fractional advection–diffusion problem (TFADP), the multi-term time-space tempered fractional problems, and the time-space tempered fractional Burgers’ equation (TFBE) to investigate the numerical capability of the fractional collocation method. The study includes a numerical examination of the produced condition number $$\kappa (A)$$ κ(A) of the linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the $$L^\infty $$ L∞ -norm error and exponential rate of spectral convergence.
期刊介绍:
Computational & Applied Mathematics began to be published in 1981. This journal was conceived as the main scientific publication of SBMAC (Brazilian Society of Computational and Applied Mathematics).
The objective of the journal is the publication of original research in Applied and Computational Mathematics, with interfaces in Physics, Engineering, Chemistry, Biology, Operations Research, Statistics, Social Sciences and Economy. The journal has the usual quality standards of scientific international journals and we aim high level of contributions in terms of originality, depth and relevance.