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xiv
8.1.4
8.2
Boundary Value Problems and Orthogonal Functions .... 257
Other Boundary Conditions .............. 257
8.2.1
Sturm-Liouville Problems ............... 261
8.2.2
The Mean Square Distance .................. 264
8.3
General Fourier Series ..................... 267
8.4
A Poincar´e Inequality ..................... 273
8.5
Exercises
8.6
............................ 276
9 Convergence of Fourier Series
285
9.1 Contents Changing the Scale ................... 256
Different Notions of Convergence ............... 285
9.2 Pointwise Convergence ..................... 290
9.3 Uniform Convergence ..................... 296
9.4 Mean Square Convergence ................... 300
9.5 Smoothness and Decay of Fourier Coefficients ........ 302
9.6 Exercises ............................ 307
10 The Heat Equation Revisited 313
10.1 Compatibility Conditions ................... 314
10.2 Fourier’s Method: A Mathematical Justification ....... 319
10.2.1 The Smoothing Property ............... 319
10.2.2 The Differential Equation ............... 321
10.2.3 The Initial Condition ................. 323
10.2.4 Smooth and Compatible Initial Functions ...... 325
10.3 Convergence of Finite Difference Solutions .......... 327
10.4 Exercises ............................ 331
11 Reaction-Diffusion Equations 337
11.1 The Logistic Model of Population Growth .......... 337
11.1.1 A Numerical Method for the Logistic Model ..... 339
11.2 Fisher’s Equation ........................ 340
11.3 A Finite Difference Scheme for Fisher’s Equation ...... 342
11.4 An Invariant Region ...................... 343
11.5 The Asymptotic Solution ................... 346
11.6 Energy Arguments ....................... 349
11.6.1 An Invariant Region .................. 350
11.6.2 Convergence Towards Equilibrium .......... 351
11.6.3 Decay of Derivatives .................. 352
11.7 Blowup of Solutions ...................... 354
11.8 Exercises ............................ 357
11.9 Projects ............................. 360
12 Applications of the Fourier Transform 365
12.1 The Fourier Transform ..................... 366
12.2 Properties of the Fourier Transform ............. 368