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This book illustrates how MAPLE can be used to supplement a
standard, elementary text in ordinary and partial differential
equation. MAPLE is used with several purposes in mind. The authors
are firm believers in the teaching of mathematics as an
experimental science where the student does numerous calculations
and then synthesizes these experiments into a general theory.
Projects based on the concept of writing generic programs test a
student's understanding of the theoretical material of the course.
A student who can solve a general problem certainly can solve a
specialized problem. The authors show MAPLE has a built-in program
for doing these problems. While it is important for the student to
learn MAPLES in built programs, using these alone removes the
student from the conceptual nature of differential equations. The
goal of the book is to teach the students enough about the computer
algebra system MAPLE so that it can be used in an investigative
way. The investigative materials which are present in the book are
done in desk calculator mode DCM, that is the calculations are in
the order command line followed by output line. Frequently, this
approach eventually leads to a program or procedure in MAPLE
designated by proc and completed by end proc. This book was
developed through ten years of instruction in the differential
equations course. Table of Contents 1. Introduction to the Maple
DEtools 2. First-order Differential Equations 3. Numerical Methods
for First Order Equations 4. The Theory of Second Order
Differential Equations with Con- 5. Applications of Second Order
Linear Equations 6. Two-Point Boundary Value Problems, Catalytic
Reactors and 7. Eigenvalue Problems 8. Power Series Methods for
Solving Differential Equations 9. Nonlinear Autonomous Systems 10.
Integral Transforms Biographies Robert P. Gilbert holds a Ph.D. in
mathematics from Carnegie Mellon University. He and Jerry Hile
originated the method of generalized hyperanalytic function theory.
Dr. Gilbert was professor at Indiana University, Bloomington and
later became the Unidel Foundation Chair of Mathematics at the
University of Delaware. He has published over 300 articles in
professional journals and conference proceedings. He is the
Founding Editor of two mathematics journals Complex Variables and
Applicable Analysis. He is a three-time Awardee of the
Humboldt-Preis, and. received a British Research Council award to
do research at Oxford University. He is also the recipient of a
Doctor Honoris Causa from the I. Vekua Institute of Applied
Mathematics at Tbilisi State University. George C. Hsiao holds a
doctorate degree in Mathematics from Carnegie Mellon University.
Dr. Hsiao is the Carl J. Rees Professor of Mathematics Emeritus at
the University of Delaware from which he retired after 43 years on
the faculty of the Department of Mathematical Sciences. Dr. Hsiao
was also the recipient of the Francis Alison Faculty Award, the
University of Delaware's most prestigious faculty honor, which was
bestowed on him in recognition of his scholarship, professional
achievement and dedication. His primary research interests are
integral equations and partial differential equations with their
applications in mathematical physics and continuum mechanics. He is
the author or co-author of more than 200 publications in books and
journals. Dr. Hsiao is world-renowned for his expertise in Boundary
Element Method and has given invited lectures all over the world.
Robert J. Ronkese holds a PhD in applied mathematics from the
University of Delaware. He is a professor of mathematics at the US
Merchant Marine Academy on Long Island. As an undergraduate, he was
an exchange student at the Swiss Federal Institute of Technology
(ETH) in Zurich. He has held visiting positions at the US Military
Academy at West Point and at the University of Central Florida in
Orlando.
This book illustrates how MAPLE can be used to supplement a
standard, elementary text in ordinary and partial differential
equation. MAPLE is used with several purposes in mind. The authors
are firm believers in the teaching of mathematics as an
experimental science where the student does numerous calculations
and then synthesizes these experiments into a general theory.
Projects based on the concept of writing generic programs test a
student's understanding of the theoretical material of the course.
A student who can solve a general problem certainly can solve a
specialized problem. The authors show MAPLE has a built-in program
for doing these problems. While it is important for the student to
learn MAPLES in built programs, using these alone removes the
student from the conceptual nature of differential equations. The
goal of the book is to teach the students enough about the computer
algebra system MAPLE so that it can be used in an investigative
way. The investigative materials which are present in the book are
done in desk calculator mode DCM, that is the calculations are in
the order command line followed by output line. Frequently, this
approach eventually leads to a program or procedure in MAPLE
designated by proc and completed by end proc. This book was
developed through ten years of instruction in the differential
equations course. Table of Contents 1. Introduction to the Maple
DEtools 2. First-order Differential Equations 3. Numerical Methods
for First Order Equations 4. The Theory of Second Order
Differential Equations with Con- 5. Applications of Second Order
Linear Equations 6. Two-Point Boundary Value Problems, Catalytic
Reactors and 7. Eigenvalue Problems 8. Power Series Methods for
Solving Differential Equations 9. Nonlinear Autonomous Systems 10.
Integral Transforms Biographies Robert P. Gilbert holds a Ph.D. in
mathematics from Carnegie Mellon University. He and Jerry Hile
originated the method of generalized hyperanalytic function theory.
Dr. Gilbert was professor at Indiana University, Bloomington and
later became the Unidel Foundation Chair of Mathematics at the
University of Delaware. He has published over 300 articles in
professional journals and conference proceedings. He is the
Founding Editor of two mathematics journals Complex Variables and
Applicable Analysis. He is a three-time Awardee of the
Humboldt-Preis, and. received a British Research Council award to
do research at Oxford University. He is also the recipient of a
Doctor Honoris Causa from the I. Vekua Institute of Applied
Mathematics at Tbilisi State University. George C. Hsiao holds a
doctorate degree in Mathematics from Carnegie Mellon University.
Dr. Hsiao is the Carl J. Rees Professor of Mathematics Emeritus at
the University of Delaware from which he retired after 43 years on
the faculty of the Department of Mathematical Sciences. Dr. Hsiao
was also the recipient of the Francis Alison Faculty Award, the
University of Delaware's most prestigious faculty honor, which was
bestowed on him in recognition of his scholarship, professional
achievement and dedication. His primary research interests are
integral equations and partial differential equations with their
applications in mathematical physics and continuum mechanics. He is
the author or co-author of more than 200 publications in books and
journals. Dr. Hsiao is world-renowned for his expertise in Boundary
Element Method and has given invited lectures all over the world.
Robert J. Ronkese holds a PhD in applied mathematics from the
University of Delaware. He is a professor of mathematics at the US
Merchant Marine Academy on Long Island. As an undergraduate, he was
an exchange student at the Swiss Federal Institute of Technology
(ETH) in Zurich. He has held visiting positions at the US Military
Academy at West Point and at the University of Central Florida in
Orlando.
This is the second edition of the book which has two additional new
chapters on Maxwell's equations as well as a section on properties
of solution spaces of Maxwell's equations and their trace spaces.
These two new chapters, which summarize the most up-to-date results
in the literature for the Maxwell's equations, are sufficient
enough to serve as a self-contained introductory book on the modern
mathematical theory of boundary integral equations in
electromagnetics. The book now contains 12 chapters and is divided
into two parts. The first six chapters present modern mathematical
theory of boundary integral equations that arise in fundamental
problems in continuum mechanics and electromagnetics based on the
approach of variational formulations of the equations. The second
six chapters present an introduction to basic classical theory of
the pseudo-differential operators. The aforementioned corresponding
boundary integral operators can now be recast as
pseudo-differential operators. These serve as concrete examples
that illustrate the basic ideas of how one may apply the theory of
pseudo-differential operators and their calculus to obtain
additional properties for the corresponding boundary integral
operators. These two different approaches are complementary to each
other. Both serve as the mathematical foundation of the boundary
element methods, which have become extremely popular and efficient
computational tools for boundary problems in applications. This
book contains a wide spectrum of boundary integral equations
arising in fundamental problems in continuum mechanics and
electromagnetics. The book is a major scholarly contribution to the
modern approaches of boundary integral equations, and should be
accessible and useful to a large community of advanced graduate
students and researchers in mathematics, physics, and engineering.
This is the second edition of the book which has two additional new
chapters on Maxwell's equations as well as a section on properties
of solution spaces of Maxwell's equations and their trace spaces.
These two new chapters, which summarize the most up-to-date results
in the literature for the Maxwell's equations, are sufficient
enough to serve as a self-contained introductory book on the modern
mathematical theory of boundary integral equations in
electromagnetics. The book now contains 12 chapters and is divided
into two parts. The first six chapters present modern mathematical
theory of boundary integral equations that arise in fundamental
problems in continuum mechanics and electromagnetics based on the
approach of variational formulations of the equations. The second
six chapters present an introduction to basic classical theory of
the pseudo-differential operators. The aforementioned corresponding
boundary integral operators can now be recast as
pseudo-differential operators. These serve as concrete examples
that illustrate the basic ideas of how one may apply the theory of
pseudo-differential operators and their calculus to obtain
additional properties for the corresponding boundary integral
operators. These two different approaches are complementary to each
other. Both serve as the mathematical foundation of the boundary
element methods, which have become extremely popular and efficient
computational tools for boundary problems in applications. This
book contains a wide spectrum of boundary integral equations
arising in fundamental problems in continuum mechanics and
electromagnetics. The book is a major scholarly contribution to the
modern approaches of boundary integral equations, and should be
accessible and useful to a large community of advanced graduate
students and researchers in mathematics, physics, and engineering.
This book is devoted to the mathematical foundation of boundary
integral equations. The combination of ?nite element analysis on
the boundary with these equations has led to very e?cient
computational tools, the boundary element methods (see e.g., the
authors [139] and Schanz and Steinbach (eds.) [267]). Although we
do not deal with the boundary element discretizations in this book,
the material presented here gives the mathematical foundation of
these methods. In order to avoid over generalization we have
con?ned ourselves to the treatment of elliptic boundary value
problems. The central idea of eliminating the ?eld equations in the
domain and - ducing boundary value problems to equivalent equations
only on the bou- ary requires the knowledge of corresponding
fundamental solutions, and this idea has a long history dating back
to the work of Green [107] and Gauss [95, 96]. Today the resulting
boundary integral equations still serve as a major tool for the
analysis and construction of solutions to boundary value problems.
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