偏微分方程导论Introduction to partial differential equations

2004-11  

Springer Verlag  

Tveito, Aslak/ Winther, Ragnar  

392  

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This book teaches the basic methods of partial differential equations and introduces related important ideas associated with the analysis of numerical methods for those partial differential equations。 Standard topics such as separation of variables， Fourier analysis， maximum principles， and energy estimates are included。 Numerical methods are introduced in parallel to the classical theory。 The numerical experiments are used to illustrate properties of differential equations and theory for finite difference approximations is developed。 Numerical methods are included in the book to show the significance of computations in partial differential equations and to illustrate the strong interaction between mathematical theory and the development of numerical methods。 Great care has been taken throughout the book to seek a sound balance between the analytical and numerical techniques。 The authors present the material at an easy pace with well-organized exercises ranging from the straightforward to the challenging。 In addition， special projects are included， containing step by step hints and instructions， to help guide students in the correct way of approaching partial differential equations。 The text would be suitable for advanced undergraduate and graduate courses in mathematics and engineering。 Necessary prerequisites for this text are basic calculus and linear algebra。 Some elementary knowledge of ordinary differential equations is also preferable。

1　Setting the Scene 1.1 What Is a Differential Equation？ 1.1.1 Concepts 1.2 The Solution and Its Properties 1.2.1 An Ordinary Differential Equation 1.3 A Numerical Method 1.4 Cauchy Problems 1.4.1 First Order Homogeneous Equations 1.4.2 First Order Nonhomogeneous Equations 1.4.3 The Wave Equation 1.4.4 The Heat Equation 1.5 Exercises 1.6 Projects2 TWO-Point Boundary Value Problems 2.1 Poisson’S Equation in OBe Dimension 2.1.1 Green’S Function 2.1.2 Smoothness of the Solution 2.1.3 A Maximum Principle 2.2 A Finite Difference Approximation 2.2.1 Taylor Series 2.2.2 A System of Algebraic Equations. 2.2.3 Gaussian Elimination for Tridiagonal Linear Systems 2.2.4 Diagonal Dominant Matrices　　2.2.5 Positive Definite Matrices　2.3 Continuous and Discrete Solutions 2.3.1 Difference and Differential Equations. 2.3.2 Symmetry 2.3.3 Uniqueness 2.3.4 A Maximum Principle for the Discrete Problem. 2.3.5 Convergence of the Discrete Solutions　2.4 Eigenvalue Problems 2.4.1 The COntinuous Eigenvalue Problem 2.4.2 The Discrete Eigenvalue Problem　2.5 Exercises 　2.6 Projects3 The Heat Equation 3.1 A Brief Overview 3.2 Separation of Variables 3.3 The Principle of Superposition 3.4 F0urier Coefficients 3.5 Other Boundary Conditions 3.6 The Neumann Problem 3.6.1 The Eigenvalue Problem 3.6.2 Particular Solutions 3.6.3 A F0rmal Solution 3.7 Energy Arguments 3.8 Differentiation of Integrals 3.9 Exercises 3.10 Projects4 Finite Difference Schemes for the Heat Equation 4.1 An Explicit Scheme 4.2 F0urier Analysis of the Numerical Solution 4.2.1 Particular Solutions 4.2.2 Comparison of the Analytical and Discrete Solution 4.2.3 Stability Considerations 4.2.4 The Accuracy oftim Approximation 4.2.5 Summary of the Comparison 4.3 Von Neumann’S Stability Analysis 4.3.1 Particular Solutions：Continuous and Discrete 4.3.2 Examples 4.3.3 A Nonlinear Problem 4.4 An Implicit Scheme 4.4.1 Stability Anysis 4.5 Numerical Stability by Energy Arguments 4.6 Exercises5 The Wave Equation6 Maximum Principles7 Poisson’s Equation in Two Space Dimensions8 Orthogonality and General Fourier Series9 Convergence of Fourier Series10 The Heat Equation Revisited11 Reaction-Diffusion Equations12 Application of the Fourier TransformReferencesIndex

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