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# Transform Methods for Solving Partial Differential Equations (Symbolic Numeric Computation) ebook

## by Dean G. Duffy

Transform methods provide a bridge between the commonly used method of separation variables and numerical techniques for solving linear partial differential equations

Transform methods provide a bridge between the commonly used method of separation variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables and numerical transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion. Because the problem retains some of its analytic aspects, one can gain greater physical insight than typically obtained from a purely numerical approach.

Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations

Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion, and because the problem retains some of its analytic aspect, one can gain greater physical insight than typically obtained from a purely numerical approach.

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Start by marking Transform Methods for Solving Partial Differential Equations as Want to Read: Want to Read savin. ant to Read. Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. A revised format that makes the book easier to use as a reference: problems are classified according to type of region, type of coordinate system, and type of partial differential equation.

Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations.

A textbook or reference for applied physicists or mathematicians; geophysicists; or civil, mechanical, or electrical engineers. It assumes the usual undergraduate sequence of mathematics in engineering or the sciences, the traditional calculus, differential equations, and Fourier and Laplace transforms. It explains how to use those and the Hankel transforms to solve linear partial differential equations that are encountered in engineering and sciences.

This essential text/reference draws from the latest literature on transform methods to provide in-depth discussions on the joint transform problem, the Cagniard-de Hoop method, and the Wiener-Hopf technique. Some 1,500 references are included as well. Format Hardback 512 pages. Dimensions 156 x 235 x 3. 5mm 907g. Publication date 16 Feb 1994. Publisher Taylor & Francis Inc.

Many differential equations cannot be solved using symbolic computation ("analysis"). In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.

Transform methods for solving partial differential equations. In the present paper, a new method of solving Bessel's differential equation is given using the L2-transform.

Symbolic & Numeric Computation. Transform Methods for Solving Partial Differential Equations. This new book deals with the construction of finite-difference (FD) algorithms for three main types of equations: elliptic equations, heat equations, and gas dynamic equations in Lagrangian form. These methods can be applied to domains of arbitrary shapes. The construction of FD algorithms for all.

Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion, and because the problem retains some of its analytic aspect, one can gain greater physical insight than typically obtained from a purely numerical approach.

Transform Methods for Solving Partial Differential Equations, Second Edition illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. The author has expanded the second edition to provide a broader perspective on the applicability and use of transform methods and incorporated a number of significant refinements:

New in the Second Edition:

· Expanded scope that includes numerical methods and asymptotic techniques for inverting particularly complicated transforms

· Discussions throughout the book that compare and contrast transform methods with separation of variables, asymptotic methods, and numerical techniques

· Many added examples and exercises taken from a wide variety of scientific and engineering sources

· Nearly 300 illustrations--many added to the problem sections to help readers visualize the physical problems

· A revised format that makes the book easier to use as a reference: problems are classified according to type of region, type of coordinate system, and type of partial differential equation

· Updated references, now arranged by subject instead of listed all together

As reflected by the book's organization, content, and many examples, the author's focus remains firmly on applications. While the subject matter is classical, this book gives it a fresh, modern treatment that is exceptionally practical, eminently readable, and especially valuable to anyone solving problems in engineering and the applied sciences.

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Transform Methods for Solving Partial Differential Equations, Second Edition by Dean G. Duffy (Chapman & Hall/CRC) illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. This second edition is expanded to provide a broader perspective on the applicability and use of transform methods. It classifies the problems presented in every chapter by type of region, coordinate system and partial differential equation. Many of the problems included in the book are illustrated to show the reader what they will look like physically. Unlike many mathematics texts, this book provides a step-by-step analysis of problems taken from the actual scientific and engineering literature.

Transform methods provide a bridge between the commonly used method of separation variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables and numerical transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion. Because the problem retains some of its analytic aspects, one can gain greater physical insight than typically obtained from a purely numerical approach.

Transform Methods for Solving Partial Differential Equations, Second Edition illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. The author has expanded the second edition to provide a broader perspective on the applicability and use of transform methods and incorporated a number of significant refinements:

New in the Second Edition:

* Expanded scope that includes numerical methods and asymptotic techniques for inverting particularly complicated transforms

* Discussions throughout the book that compare and contrast transform methods with separation of variables, asymptotic methods, and numerical techniques Many added examples and exercises taken from a wide variety of scientific and engineering sources

* Nearly 300 illustrations - many added to the problem sections to help readers visualize the physical problems

* A revised format that makes the book easier to use as a reference: problems are classified according to type of region, type of coordinate system, and type of partial differential equation

* Updated references, now arranged by subject

While the subject matter is classical, this fresh, modern treatment that is exceptionally practical, eminently and especially valuable to anyone solvying problems in engineering

Purpose. This book illustrates the use of Laplace, Fourier and Hankel transforms for solving linear partial differential equations that are encountered in engineering and sciences. To this end, this new edition features updated references as well as many new examples and exercises taken from a wide variety of sources. Of particular importance is the inclusion of numerical methods and asymptotic techniques for inverting particularly complicated transforms.

Transform methods provide an alternative and bridge between the commonly employed methods of separation of variables and numerical methods in solving linear partial differential equations. The relationship between the techniques grouped as: Numerical

Techniques, Separation of Variables, Transform Methods and Asymptotic Analysis.

Transform methods are similar to separation of variables because they often yield closed form solutions via the powerful method of contour integration. Indeed, all of the results from separation of variables could also be derived using transform methods. Moreover, transform methods can handle a wider class of problems, such as those involving time-dependent boundary conditions, where separation of variables would fail.

Even in those cases when the inverse of the transform cannot be found analytically, transform methods can still be used profitably. A wide variety of numerical and asymptotic methods now exist for their inversion. Because some analytic aspects of the problem are retained, it is easier to obtain greater physical insight than from a purely numerical approach.

Prerequisites. The book assumes the usual undergraduate sequence of mathematics in engineering or the sciences: the traditional calculus and differential equations. A course in complex variables and Fourier and Laplace transforms is also essential. Finally some knowledge of Bessel functions is desirable to completely understand the book.

Audience. This book may be used as either a textbook or a reference book for applied physicists, geophysicists, civil, mechanical or electrical engineers and applied mathematicians.

Chapter Overview. The purpose of Chapter 1 is two-fold. The first four sections (and Section 1.7) serve as a refresher on the background material: linear ordinary differential equations, transform methods and complex variables., The amount of time spent with this material depends upon the background of the class. At least one class period should be spent on each section. Section 1.5 and Section 1.6 cover multivalued complex functions. These sections can be omitted if you only plan to teach Chapter 2 and Chapter 3. Otherwise, several class periods will be necessary to master this material because most students have never seen it. Due to the complexity of the problems, it is suggested that take-home problems are given to test the student's knowledge.

Chapter 2 through Chapter 5 are the meat-and-potatoes of the book.

The material is subdivided according to whether we invert a single-valued or multivalued transform. Each chapter is then subdivided into two parts. The first part deals with simply the mechanics of how to invert the transform while the second part actually applies the transform methods to solving partial differential equations. Undergraduates with a strong mathematics background should be able to handle Chapter 2 and Chapter 3 while Chapter 4 and Chapter 5 are really graduate-level material. The constant theme is the repeated application of the residue theorem to invert Fourier and Laplace transforms.

In Chapter 6 we solve partial differential equations by repeated applications of transform methods. We are now in advanced topics and this material is really only suitable for graduate students. The first two sections are straightforward, brute-force applications of Laplace and Fourier transforms in solving partial differential equations where we hope that we can invert both transforms to find the solution. Section 6.3 and Section 6.4 are devoted to the very clever inversion techniques of Cagniard and De Hoop. For too long this interesting work has been restricted to the seismic, acoustic and electrodynamic communities.

In Chapter 7 we treat the classic Wiener-Hopf problem. This is very difficult material because of the very complicated analysis that is usually involved. Duffy breaks the chapter into two parts: Section 7.1 deals with finite domains while Section 7.2 applies to infinite and semi-infinite domains.

Features. This is an unabashedly applied book because the intended audience is problem solvers in engineering and the applied sciences. However, references are given to other books that do cover any unproven point, should the reader be interested. Also Duffy tries to give some human touch to this field by including references to the original works and photographs of some of the leading figures.

It is always difficult to write a book that satisfies both the student and the researcher. The student usually wants all of the gory details while the researcher wants the answer now. Duffy tries to accommodate both by centering most of the text around examples of increasing difficulty. There are plenty of details for the student, but the researcher may quickly leaf through the examples to find the material that interests him.

As anyone who has taken a course knows, the only way that you know a subject is by working the problems. For that reason Duffy has included several hundred well-crafted problems, most of which were taken from the scientific and engineering literature. When possible, these problems are grouped according to some common property - such as a cylindrical domain. Because many of these problems are difficult, The author has included detailed solutions to most of them. The student is asked however to refrain from looking at the solution before he has really tried to solve it on his own. No pain; no gain. The remaining problems have intermediate results so that the student has confidence that he is on the right track. The researcher also might look at these problems because his problem might already have been solved.

A new feature of this book is the inclusion of sections on the numerical in-version of Laplace, Hankel and Fourier transforms. I have included MATLAB code for the reader's use. A quick glance at the scripts reveals their "Fortran"-like structure. This was done for a reason. For those who know MATLAB well, it is easy to optimize the scripts using MATLAB syntax. For the Fortran and C crowd, the scripts are easily convertible into those languages.

Finally, an important aspect of this book is the numerous references that can serve as further grist for the student or point the researcher toward a solution of his problem. Of course, we must strike a balance between having a book of references and leaving out some interesting papers. The criteria for inclusion were three-fold. First, the paper had to have used the technique and not merely chanted the magic words; quoting results was unacceptable. Second, the papers had to compute both the forward and inverse transforms. The use of asymptotic or numerical methods to invert the transform excluded the reference.
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This book presents solutions of a broad range of problems arising in electric, mechanic and civil engineering and formalized in (generalized) linear, partial differential equations.

This book gives a short introduction into the mathematical foundation of the tools used. It lacks a definition of "transform methods" and their algebraic properties. Thereby it does not tell the reader, why transform methods are appropriate for the presented problems and superior to other approaches, what they are indeed.

Though the bibliography mentions authors from the early 18th century to now some important have not been included, like Doetsch.

Mr. Doetsch, famous for his "Laplace Transformation", published a similiar sampler in the late sixties of the last century. Most of the examples presented by Duffy to show the usage of Fourier and Laplace transformation seem to be taken from Doetsch.

This book seems not sufficient for teaching, as it does neither derive the ideas from a well formalized foundation, nor does it develop the solutions from that well formalized foundation. Too often deus ex macchina has been applied and the very question remains: Why?

This book does neither seem to be suited as a cook book, as the solved problems have not been indexed.

My personal impression of this book is completed by the first, obvious error on page xiii: delta(t-a) should be infinite for t=a, not t=0, and delta(t-a) should be zero for t<>a, not t<>0. Author:
Dean G. Duffy
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Mathematics
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Publisher:
Chapman and Hall/CRC; 2 edition (June 26, 2004)
Pages:
728 pages
Rating:
4.3
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