贝尔拉普拉斯变换的计算

IF 0.6 Q3 MATHEMATICS
D. Caratelli, C. Cesarano, P. Ricci
{"title":"贝尔拉普拉斯变换的计算","authors":"D. Caratelli, C. Cesarano, P. Ricci","doi":"10.14658/PUPJ-DRNA-2021-1-7","DOIUrl":null,"url":null,"abstract":"since it converts a function of a real variable t (often representing the time) to a function of a complex variable s (complex frequency). This transform is used for solving differential equations, since it transforms differential into algebraic equations and convolution into multiplication. It can be applied to local integrable functions on [0,+∞) and it converges in each half plane Re(s)> a, the constant a, also known as the convergece abscissa, depends on the growth behavior of f (t). A large number of LT can be found in the literature and, together with the respective antitransforms, are usually used in the solution of the most diverse differential problems. The numerical computation of the link between transforms and antitransforms was considered, for example, by F.G. Tricomi [22, 23], which highlighted the link with the series expansions in Laguerre polynomials. These results have been extended to more general expansions [17], however the results of Tricomi have been proven numerically more convenient by the point of view of numerical complexity. Recently, extensions of the LT have been considered in [16] and in [6] the numerical computation was carried out by approximating the respective kernels by means of expansions in a general Dirichlet series. Extensions of LT (called Laguerre-Laplace transforms) have been obtained in the first place [6] by replacing the exponential with the Laguerre-type exponentials, introduced in [8] and previously studied in [10], [11], [13]. Subsequently the kernel was replaced by an expansion whose coefficients are combination of Bell polynomials, exploiting a transformation, introduced in [16], which uses the Blissard formula, a typical tool of the umbral calculus [19, 20]. In order to validate the computational methods, some examples were taken into consideration, by the first author, with the aid of the computer algebra program Mathematica©. The rapid decay of the considered kernels allows to extend the integration interval to a right neighborhood of the origin. However, the higher computational complexity using the theoretical approach based on the generating function of the generalized Lucas polynomials suggested to approximate the original kernel by a truncation of a general Dirichlet’s series, and to use the matrix pencil method for evaluating the best coefficients. The results obtained confirm the correctness of the procedure introduced.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Computation of the Bell-Laplace transforms\",\"authors\":\"D. Caratelli, C. Cesarano, P. Ricci\",\"doi\":\"10.14658/PUPJ-DRNA-2021-1-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"since it converts a function of a real variable t (often representing the time) to a function of a complex variable s (complex frequency). This transform is used for solving differential equations, since it transforms differential into algebraic equations and convolution into multiplication. It can be applied to local integrable functions on [0,+∞) and it converges in each half plane Re(s)> a, the constant a, also known as the convergece abscissa, depends on the growth behavior of f (t). A large number of LT can be found in the literature and, together with the respective antitransforms, are usually used in the solution of the most diverse differential problems. The numerical computation of the link between transforms and antitransforms was considered, for example, by F.G. Tricomi [22, 23], which highlighted the link with the series expansions in Laguerre polynomials. These results have been extended to more general expansions [17], however the results of Tricomi have been proven numerically more convenient by the point of view of numerical complexity. Recently, extensions of the LT have been considered in [16] and in [6] the numerical computation was carried out by approximating the respective kernels by means of expansions in a general Dirichlet series. Extensions of LT (called Laguerre-Laplace transforms) have been obtained in the first place [6] by replacing the exponential with the Laguerre-type exponentials, introduced in [8] and previously studied in [10], [11], [13]. Subsequently the kernel was replaced by an expansion whose coefficients are combination of Bell polynomials, exploiting a transformation, introduced in [16], which uses the Blissard formula, a typical tool of the umbral calculus [19, 20]. In order to validate the computational methods, some examples were taken into consideration, by the first author, with the aid of the computer algebra program Mathematica©. The rapid decay of the considered kernels allows to extend the integration interval to a right neighborhood of the origin. However, the higher computational complexity using the theoretical approach based on the generating function of the generalized Lucas polynomials suggested to approximate the original kernel by a truncation of a general Dirichlet’s series, and to use the matrix pencil method for evaluating the best coefficients. The results obtained confirm the correctness of the procedure introduced.\",\"PeriodicalId\":51943,\"journal\":{\"name\":\"Dolomites Research Notes on Approximation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2021-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Dolomites Research Notes on Approximation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.14658/PUPJ-DRNA-2021-1-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dolomites Research Notes on Approximation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.14658/PUPJ-DRNA-2021-1-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2

摘要

因为它将实变量t(通常表示时间)的函数转换为复变量s(复频率)的函数。这个变换用于求解微分方程,因为它将微分方程转化为代数方程,将卷积转化为乘法。它可以应用于[0,+∞)上的局部可积函数,它收敛于每个半平面Re(s)> a,常数a,也称为收敛横坐标,取决于f (t)的生长行为。在文献中可以找到大量的LT,并且与相应的反变换一起,通常用于解决最多样化的微分问题。例如,F.G. Tricomi[22,23]考虑了变换与反变换之间联系的数值计算,突出了与拉盖尔多项式级数展开的联系。这些结果已推广到更一般的展开[17],但从数值复杂性的角度来看,Tricomi的结果在数值上更方便。最近,在[16]和[6]中考虑了LT的扩展,在一般的Dirichlet级数中,通过展开式逼近各自的核进行了数值计算。LT的扩展(称为Laguerre-Laplace变换)首先通过用laguerre型指数代替指数得到了[6],该指数在[8]中引入,并在[10],[11],[13]中进行了研究。随后,核被一个展开式取代,其系数是贝尔多项式的组合,利用[16]中引入的一个变换,该变换使用了Blissard公式,这是一个典型的本影微积分工具[19,20]。为了验证计算方法,第一作者在计算机代数程序Mathematica©的帮助下考虑了一些例子。所考虑的核的快速衰减允许将积分区间扩展到原点的右邻域。然而,基于广义卢卡斯多项式生成函数的理论方法计算复杂度较高,建议通过对一般狄利克雷级数的截断来近似原始核,并使用矩阵铅笔法来评估最佳系数。所得结果证实了所介绍方法的正确性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Computation of the Bell-Laplace transforms
since it converts a function of a real variable t (often representing the time) to a function of a complex variable s (complex frequency). This transform is used for solving differential equations, since it transforms differential into algebraic equations and convolution into multiplication. It can be applied to local integrable functions on [0,+∞) and it converges in each half plane Re(s)> a, the constant a, also known as the convergece abscissa, depends on the growth behavior of f (t). A large number of LT can be found in the literature and, together with the respective antitransforms, are usually used in the solution of the most diverse differential problems. The numerical computation of the link between transforms and antitransforms was considered, for example, by F.G. Tricomi [22, 23], which highlighted the link with the series expansions in Laguerre polynomials. These results have been extended to more general expansions [17], however the results of Tricomi have been proven numerically more convenient by the point of view of numerical complexity. Recently, extensions of the LT have been considered in [16] and in [6] the numerical computation was carried out by approximating the respective kernels by means of expansions in a general Dirichlet series. Extensions of LT (called Laguerre-Laplace transforms) have been obtained in the first place [6] by replacing the exponential with the Laguerre-type exponentials, introduced in [8] and previously studied in [10], [11], [13]. Subsequently the kernel was replaced by an expansion whose coefficients are combination of Bell polynomials, exploiting a transformation, introduced in [16], which uses the Blissard formula, a typical tool of the umbral calculus [19, 20]. In order to validate the computational methods, some examples were taken into consideration, by the first author, with the aid of the computer algebra program Mathematica©. The rapid decay of the considered kernels allows to extend the integration interval to a right neighborhood of the origin. However, the higher computational complexity using the theoretical approach based on the generating function of the generalized Lucas polynomials suggested to approximate the original kernel by a truncation of a general Dirichlet’s series, and to use the matrix pencil method for evaluating the best coefficients. The results obtained confirm the correctness of the procedure introduced.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.70
自引率
7.70%
发文量
0
审稿时长
8 weeks
期刊介绍: Dolomites Research Notes on Approximation is an open access journal that publishes peer-reviewed papers. It also publishes lecture notes and slides of the tutorials presented at the annual Dolomites Research Weeks and Workshops, which have been organized regularly since 2006 by the Padova-Verona Research Group on Constructive Approximation and Applications (CAA) in Alba di Canazei (Trento, Italy). The journal publishes, on invitation, survey papers and summaries of Ph.D. theses on approximation theory, algorithms, and applications.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信