Bernatz R. Fourier Series and Numerical Methods for Partial Differential Equations

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CONTENTS

2.1.3 Inner Products

2.2 The Integral as an Inner Product

2.2.1 Piecewise Continuous Functions

2.2.2 Inner Product on C

(a, 6)

2.3 Principle of Superposition

2.3.1 Finite Case

2.3.2 Infinite Case

2.3.3 Hubert Spaces

2.4 General Fourier Series

2.5 Fourier Sine Series on (0, c)

2.5.1 Odd, Periodic Extensions

2.6 Fourier Cosine Series on (0, c)

2.6.1 Even, Periodic Extensions

2.7 Fourier Series on

(—c,c)

2.7.1 2c-Periodic Extensions

2.8 Best Approximation

2.9 Bessel's Inequality

2.10 Piecewise Smooth Functions

2.11 Fourier Series Convergence

2.11.1 Alternate Form

2.11.2 Riemann-Lebesgue Lemma

2.11.3 A Dirichlet Kernel Lemma

2.11.4 A Fourier Theorem

2.12 2c-Periodic Functions

2.13 Concluding Remarks

Exercises

Sturm-Liouville Problems

3.1 Basic Examples

3.2 Regular Sturm-Liouville Problems

3.3 Properties

3.3.1 Eigenfunction Orthogonality

3.3.2 Real Eigenvalues

3.3.3 Eigenfunction Uniqueness

3.3.4 Non-negative Eigenvalues

3.4 Examples

3.4.1 Neumann Boundary Conditions

3.4.2 Robin and Neumann BCs

CONTENTS VII

3.4.3 Periodic Boundary Conditions 83

3.5 Bessel's Equation 85

3.6 Legendre's Equation 90

Exercises 93

Heat Equation 97

4.1 Heat Equation in ID 97

4.2 Boundary Conditions 100

4.3 Heat Equation in 2D 101

4.4 Heat Equation in 3D 103

4.5 Polar-Cylindrical Coordinates 105

4.6 Spherical Coordinates 108

Exercises 108

Heat Transfer ¡n ID 113

5.1 Homogeneous IBVP 113

5.1.1 Example: Insulated Ends 114

5.2 Semihomogeneous PDE 116

5.2.1 Variation of Parameters 117

5.2.2 Example: Semihomogeneous IBVP 119

5.3 Nonhomogeneous Boundary Conditions 120

5.3.1 Example: Nonhomogeneous Boundary Condition 122

5.3.2 Example: Time-Dependent Boundary Condition 125

5.3.3 Laplace Transforms 127

5.3.4 Duhamel's Theorem 128

5.4 Spherical Coordinate Example 131

Exercises 133

Heat Transfer in 2D and 3D 139

6.1 Homogeneous 2D IBVP 139

6.1.1 Example: Homogeneous IBVP 142

6.2 Semihomogeneous 2D IBVP 143

6.2.1 Example: Internal Source or Sink 146

6.3 Nonhomogeneous 2D IBVP 147

6.4 2D BVP: Laplace and Poisson Equations 150

6.4.1 Dirichlet Problems 150

6.4.2 Dirichlet Example 154

6.4.3 Neumann Problems 156

VIH CONTENTS

6.4.4 Neumann Example 159

6.4.5 Dirichlet, Neumann BC Example 163

6.4.6 Poisson Problems 166

6.5 Nonhomogeneous 2D Example 169

6.6 Time-Dependent BCs 170

6.7 Homogeneous 3D IBVP 173

Exercises 176

Wave Equation 181

7.1 Wave Equation in ID 181

7.1.1 d'Alembert's Solution 184

7.1.2 Homogeneous IBVP: Series Solution 187

7.1.3 Semihomogeneous IBVP 190

7.1.4 Nonhomogeneous IBVP 193

7.1.5 Homogeneous IBVP in Polar Coordinates 195

7.2 Wave Equation in 2D 199

7.2.1 2D Homogeneous Solution 199

Exercises 202

Numerical Methods: an Overview 207

8.1 Grid Generation 208

8.1.1 Adaptive Grids 210

8.1.2 Multilevel Methods 212

8.2 Numerical Methods 214

8.2.1 Finite Difference Method 214

8.2.2 Finite Element Method 216

8.2.3 Finite Analytic Method 217

8.3 Consistency and Convergence 218

The Finite Difference Method 219

9.1 Discretization 219

9.2 Finite Difference Formulas 222

9.2.1 First Partíais 222

9.2.2 Second Partíais 222

9.3 ID Heat Equation 223

9.3.1 Explicit Formulation 223

9.3.2 Implicit Formulation 224

9.4 Crank-Nicolson Method 226

CONTENTS IX

9.5

9.6

9.7

9.8

9.9

9.10

Error and Stability

9.5.1 Error Types

9.5.2 Stability

Convergence in Practice

ID Wave Equation

9.7.1 Implicit Formulation

9.7.2 Initial Conditions

2D Heat Equation in Cartesian Coordinates

Two-Dimensional Wave Equation

2D Heat Equation in Polar Coordinates

Exercises

226

227

231

233

234

239

244

Finite Element Method 249

10.1 General Framework 250

10.2 ID Elliptical Example 252

10.2.1 Reformulations 252

10.2.2 Equivalence in Forms 253

10.2.3 Finite Element Solution 255

10.3 2D Elliptical Example 257

10.3.1 Weak Formulation 257

10.3.2 Finite Element Approximation 258

10.4 Error Analysis 261

10.5 ID Parabolic Example 264

10.5.1 Weak Formulation 264

10.5.2 Method of Lines 265

10.5.3 Backward Euler's Method 266

Exercises 268

Finite Analytic Method 271

11.1 ID Transport Equation 272

11.1.1 Finite Analytic Solution 273

11.1.2 FA and FD Coefficient Comparison 275

11.1.3 Hybrid Finite Analytic Solution 279

11.2 2D Transport Equation 280

11.2.1 FA Solution on Uniform Grids 282

11.2.2 The Poisson Equation 287

11.3 Convergence and Accuracy 290

Exercises 291

X CONTENTS

Appendix A: FA ID Case 295

Appendix B: FA 2D Case 303

B.l The Case φ = \ 308

B.2 The Case φ = -Bx + Ay 309

References 311

Index

315

PREFACE

The importance of partial differential equations in modeling phenomena in engi-

neering and the physical, natural, and social sciences is known by many students and

practitioners in these fields. I came to know this in my studies of atmospheric science.

A dependent variable, such as air temperature, is generally a function of two or more

independent variables (time and three spatial coordinates). Our desire to understand

and predict the state of natural systems, such as our atmosphere, frequently begins

with equations involving rates of change (partial derivatives) of these quantities with

respect to the independent variables. Once these equations have evolved from the

conceptual models of such systems, the next challenge is "solving" these partial

differential equations in qualitative and quantitative ways.

This book on solution techniques for partial differential equations has evolved over

the last six years and several offerings of

introductory course on partial differential

equations at Luther College. Students enter the course with a background typical of

most junior- or senior-level mathematics or physical science majors including two

to three calculus courses, an introductory linear algebra course, and a one-semester

course in ordinary differential equations. With this common foundation, the book

intends to strengthen and extend the reader's knowledge and appreciation of the power

of linear spaces and linear transformations for purposes of understanding and solving

a wide range of equations including many important partial differential equations.

The notions of infinite dimensional vector spaces, scalar product, and norm lead to,

XÜ PREFACE

perhaps, the reader's initial introduction to Hubert spaces through the theoretical

development of Fourier series and properties of convergence. These somewhat

abstract foundations are important aspects of developing an undergraduate's more

general problem solving skill.

Most "real-world" partial differential equations, because of their nonlinear nature,

do not lend themselves to solution techniques of separation of variables, orthogonal

eigenfunction bases, and Fourier series solutions. Consequently, three different nu-

merical solution techniques are introduced in the final third of

the

book. The versatile

finite difference method is introduced first because of its relative understandable and

easy implementation. The finite element method is a popular method used by many

sanctioners in a variety of fields. Yet, it has a formidable theoretical foundation

including concepts of infinite-dimensional function spaces and finite-dimensional

subspaces. The third method for numerical solutions is the finite analytic method

wherein separation of variables Fourier series methods are applied to locally lin-

earized versions of the original partial differential equation.

Admittedly, I do not cover all of this material in a one-semester course. Usually,

Chapters

are covered, and then topics are chosen from Chapters 6 and

Chapter

9 on finite differences is covered, and then either an introduction to finite elements

or the finite analytic method completes the semester.

of Maple work sheets has been developed over the years. They are useful for solving

many of the exercises ranging from one-dimensional problems using Fourier se-

ries to multidimensional problems using the various numerical techniques. The

work sheets are available for users of the book through the textbook web site:

http://faculty.luther.edu/ bernatzr/PDE Text/index.html

RICHARD BERNATZ

Decorah, Iowa

January 29, 2010

ACKNOWLEDGMENTS

I am grateful for all the bright and enthusiastic students at Luther College who have

plowed their way through the partial differential equations course and the primitive

versions of this manuscript. I thank them for their patience while working with an

evolving text, as well as their suggestions for improving the work. In particular, the

careful reading of the text and the additional exercises developed by Aaron Peterson

were especially helpful. I want to thank Christian Anderson for giving me the

thought that it may be worth making this manuscript available to a wider audience.

In addition, I appreciate the guidance provided by the staff at John Wiley. I thank

Susanne Steitz-Filler, in particular, for the assistance she provided to make the goal

of publishing this manuscript a reality.

R.A.B.

XIII

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CHAPTER 1

INTRODUCTION

This chapter introduces general topics concerning partial differential equations (PDEs).

It begins with basic terminology associated with PDEs and then describes how PDEs

are classified. The PDEs common to scientific and engineering fields are introduced.

Next, the notion of initial and boundary value problems is introduced. The chapter

ends with a brief discussion of various solution techniques, with an introductory

example of the method of separation of variables. This example also serves as

motivation for developing the material on Fourier series in Chapter 2.

1.1 TERMINOLOGY AND NOTATION

Suppose u is a function of

the

spatial variable x and time t so that u = u(x, t). Recall

that the partial derivative of u with respect to x is defined as

du u(x + h,t)— u{x,t)

ίΛ 1Χ

— = lim 1 (1.1)

OX h-^0 h

Fourier Series and Numerical Methods for Partial Differential Equations, 1

First Edition. By Richard Bernatz