Wavelet Based Approximation Schemes for Singular Integral Equations
暫譯: 基於小波的奇異積分方程近似方案
Panja, Madan Mohan, Mandal, Birendra Nath
- 出版商: CRC
- 出版日期: 2020-09-25
- 售價: $6,720
- 貴賓價: 9.5 折 $6,384
- 語言: 英文
- 頁數: 300
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 0367199173
- ISBN-13: 9780367199173
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商品描述
Many mathematical problems in science and engineering are defined by ordinary or partial differential equations with appropriate initial-boundary conditions. Among the various methods, boundary integral equation method (BIEM) is probably the most effective. It's main advantage is that it changes a problem from its formulation in terms of unbounded differential operator to one for an integral/integro-differential operator, which makes the problem tractable from the analytical or numerical point of view. Basically, the review/study of the problem is shifted to a boundary (a relatively smaller domain), where it gives rise to integral equations defined over a suitable function space. Integral equations with singular kernels areamong the most important classes in the fields of elasticity, fluid mechanics, electromagnetics and other domains in applied science and engineering. With the advancesin computer technology, numerical simulations have become important tools in science and engineering. Several methods have been developed in numerical analysis for equations in mathematical models of applied sciences.
Widely used methods include: Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM) and Galerkin Method (GM). Unfortunately, none of these are versatile. Each has merits and limitations. For example, the widely used FDM and FEM suffers from difficulties in problem solving when rapid changes appear in singularities. Even with the modern computing machines, analysis of shock-wave or crack propagations in three dimensional solids by the existing classical numerical schemes is challenging (computational time/memory requirements). Therefore, with the availability of faster computing machines, research into the development of new efficient schemes for approximate solutions/numerical simulations is an ongoing parallel activity. Numerical methods based on wavelet basis (multiresolution analysis) may be regarded as a confluence of widely used numerical schemes based on Finite Difference Method, Finite Element Method, Galerkin Method, etc. The objective of this monograph is to deal with numerical techniques to obtain (multiscale) approximate solutions in wavelet basis of different types of integral equations with kernels involving varieties of singularities appearing in the field of elasticity, fluid mechanics, electromagnetics and many other domains in applied science and engineering.
商品描述(中文翻譯)
許多科學和工程中的數學問題是由具有適當初邊界條件的常微分方程或偏微分方程定義的。在各種方法中，邊界積分方程法（Boundary Integral Equation Method, BIEM）可能是最有效的。它的主要優勢在於將問題的表述從無界微分算子轉換為積分/積分微分算子，這使得從分析或數值的角度來看，問題變得可處理。基本上，問題的研究/分析被轉移到邊界（相對較小的區域），在那裡產生了定義在合適函數空間上的積分方程。具有奇異核的積分方程是彈性、流體力學、電磁學及其他應用科學和工程領域中最重要的類別之一。隨著計算機技術的進步，數值模擬已成為科學和工程中的重要工具。數值分析中已經開發出幾種方法來處理應用科學數學模型中的方程。
廣泛使用的方法包括：有限差分法（Finite Difference Method, FDM）、有限元素法（Finite Element Method, FEM）、有限體積法（Finite Volume Method, FVM）和Galerkin方法（Galerkin Method, GM）。不幸的是，這些方法都不是通用的。每種方法都有其優點和局限性。例如，廣泛使用的FDM和FEM在奇異點出現快速變化時，解決問題會遇到困難。即使在現代計算機上，利用現有的經典數值方案分析三維固體中的衝擊波或裂紋擴展也是一項挑戰（計算時間/內存需求）。因此，隨著更快計算機的出現，開發新型高效方案以獲得近似解/數值模擬的研究是一項持續的平行活動。基於小波基（Wavelet Basis， multiresolution analysis）的數值方法可以被視為基於有限差分法、有限元素法、Galerkin方法等的廣泛使用數值方案的匯流。這本專著的目的是處理數值技術，以獲得不同類型的積分方程的（多尺度）近似解，這些方程的核涉及在彈性、流體力學、電磁學及許多其他應用科學和工程領域中出現的各種奇異性。
作者簡介
M M Panja has a MSc in Applied Mathematics (1987) from Calcutta University, India, and a PhD (1993) from Visva-Bharati University, India. He investigated the origin of (hidden) geometric phase on quantum mechanical problems and initiated studies on Lie group theoretic approach of differential equations during his postdoctoral research. His investigations (2007) on approximation theory based on multiresolution analysis, has been published several international journals. His current research interests are (i) multiscale approximation based on wavelets, and (ii) similarity (exact) solution of mathematical models involving differential and integral operators.
B N Mandal has a MSc in Applied Mathematics (1966) and a PhD (1973) from Calcutta University, India. He was a postdoctoral Commonwealth Fellow at Manchester University, 1973-75. He was faculty at Calcutta University, 1970-89 and later at Indian Statistical Institute (ISI), Kolkata, 1989-2005. He was a NASI Senior Scientist, 2009-14 in ISI. His research work encompasses several areas of applied mathematics including water waves, integral transforms, integral equations, inventory problems, wavelets etc. He has published a number of works with reputable publishers. He has supervised PhD theses of more than 20 candidates and has more than 275 research publications.
作者簡介(中文翻譯)
M M Panja 擁有印度加爾各答大學的應用數學碩士學位（1987年）和印度維斯瓦-巴拉提大學的博士學位（1993年）。他研究了量子力學問題中（隱藏的）幾何相位的起源，並在其博士後研究期間開始了關於微分方程的李群理論方法的研究。他在2007年對基於多解析度分析的近似理論的研究已發表於多本國際期刊。他目前的研究興趣包括 (i) 基於小波的多尺度近似，以及 (ii) 涉及微分和積分運算子的數學模型的相似（精確）解。
B N Mandal 擁有印度加爾各答大學的應用數學碩士學位（1966年）和博士學位（1973年）。他曾於1973年至1975年擔任曼徹斯特大學的英聯邦博士後研究員。他於1970年至1989年在加爾各答大學任教，隨後於1989年至2005年在印度統計學研究所（ISI）任教。他於2009年至2014年在ISI擔任NASI高級科學家。他的研究工作涵蓋應用數學的多個領域，包括水波、積分變換、積分方程、庫存問題、小波等。他已與多家知名出版社發表了多篇作品，並指導了超過20位候選人的博士論文，擁有超過275篇研究出版物。
