Partial Differential Equations with Numerical Methods (Texts in Applied Mathematics (45))

The Book Is Suitable For Advanced Undergraduate And Beginning Graduate Students Of Applied Mathematics And Engineering. The Main Theme Is The Integration Of The Theory Of Linear Pdes And The Numerical Solution Of Such Equations. For Each Type Of Pde, Elliptic, Parabolic, And Hyperbolic, The Text Contains One Chapter On The Mathematical Theory Of The Differential Equation, Followed By One Chapter On Finite Difference Methods And One On Finite Element Methods. As Preparation, The Two-point Boundary Value Problem And The Initial-value Problem For Odes Are Discussed In Separate Chapters. There Is Also One Chapter On The Elliptic Eigenvalue Problem And Eigenfunction Expansion. The Presentation Does Not Presume A Deep Knowledge Of Mathematical And Functional Analysis. Some Background On Linear Functional Analysis And Sobolev Spaces, And Also On Numerical Linear Algebra, Is Reviewed In Two Appendices. A Two-point Boundary Value Problem -- Elliptic Equations -- Finite Difference Methods For Elliptic Equations -- Finite Element Methods For Elliptic Equations -- The Elliptic Eigenvalue Problem -- Initial-value Problems For Ordinary Differential Equations -- Parabolic Equations -- Finite Difference Methods For Parabolic Problems -- The Finite Element Method For A Parabolic Problem -- Hyperbolic Equations -- Finite Difference Methods For Hyperbolic Equations -- The Finite Element Method For Hyperbolic Equations -- Some Other Classes Of Numerical Methods. By Stig Larsson, Vidar Thomée. The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The Text Would Be Suitable For Advanced Undergraduate And Beginning Graduate Students Of Applied Mathematics And Engineering. The Presentation Does Not Presume A Deep Knowledge Of Mathematical And Functional Analysis. The Required Background On Linear Functional Analysis And Sobolev Spaces Is Reviewed In An Appendix.--book Jacket. Introduction -- A Two-point Boundary Value Problem -- Elliptic Equations -- Finite Difference Methods For Elliptic Equations -- Finite Element Methods For Elliptic Equations -- The Elliptic Eigenvalue Problem -- Initial-value Problems For Odes -- Parabolic Equations -- Finite Difference Methods For Parabolic Problems -- The Finite Element Method For A Parabolic Problem -- Hyperbolic Equations -- Finite Difference Methods For Hyperbolic Equations -- The Finite Element Method For Hyerbolic Equations -- Some Other Classes Of Numerical Methods -- Some Tools From Mathematical Analysis -- Orientation On Numerical Linear Algebra. Stig Larsson And Vidar Thomée. Includes Bibliographical References And Index. Front Matter....Pages I-XI Introduction....Pages 1-14 A Two-Point Boundary Value Problem....Pages 15-24 Elliptic Equations....Pages 25-41 Finite Difference Methods for Elliptic Equations....Pages 43-49 Finite Element Methods for Elliptic Equations....Pages 51-76 The Elliptic Eigenvalue Problem....Pages 77-94 Initial-Value Problems for Ordinary Differential Equations....Pages 95-108 Parabolic Equations....Pages 109-127 Finite Difference Methods for Parabolic Problems....Pages 129-148 The Finite Element Method for a Parabolic Problem....Pages 149-161 Hyperbolic Equations....Pages 163-183 Finite Difference Methods for Hyperbolic Equations....Pages 185-199 The Finite Element Method for Hyperbolic Equations....Pages 201-216 Some Other Classes of Numerical Methods....Pages 217-224 Back Matter....Pages 225-261 Annotation On the occasion of W. Zimmermann's 70th birthday some eminent scientists gave review talks in honor of one of the great masters of quantum field theory. It was decided to write them up and publish them in this book, together with reprints of some seminal papers of the laureate. Thus, this volume deepens our understanding of anomalies, algebraic renormalization theory, axiomatic field theory and of much more while illuminating the past and present state of affairs and pointing to interesting problems for future research Covers the integration of the theory of linear Partial Differential Equations and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, this text contains chapters on the mathematical theory of the differential equation, finite difference methods, and finite element methods In this text we study boundary value and initial-boundary value problems for partial differential equations, that are significant in applications, from both a theoretical and a numerical point of view.