Polynomial approximation of differential equations

Lecture Notes in Physics: Monographs 8 ......Page 1
Polynomial Approximation of Differential Equations ......Page 3
Preface ......Page 5
CONTENTS ......Page 7
1.1 Sturm-Liouville problems ......Page 11
1.2 The Gamma function ......Page 12
1.3 Jacobi polynomials ......Page 14
1.4 Legendre polynomials ......Page 17
1.5 Chebyshev polynomials ......Page 19
1.6 Laguerre polynomials ......Page 22
1.7 Hermite polynomials ......Page 26
2.1 Inner products and norms ......Page 31
2.2 Orthogonal functions ......Page 33
2.3 Fourier coefficients ......Page 37
2.4 The projection operator ......Page 40
2.5 The maximum norm ......Page 41
2.6 Basis transformations ......Page 42
3.1 Zeroes of orthogonal polynomials ......Page 45
3.2 Lagrange polynomials ......Page 52
3.3 The interpolation operators ......Page 56
3.4 Gauss integration formulas ......Page 57
3.5 Gauss-Lobatto integration formulas ......Page 61
3.6 Gauss-Radau integration formulas ......Page 64
3.7 Clenshaw-Curtis integration formulas ......Page 65
3.8 Discrete norms ......Page 66
3.9 Discrete maximum norms ......Page 70
3.10 Scaled weights ......Page 72
4.1 Fourier transforms ......Page 75
4.2 Aliasing ......Page 78
4.3 Fast Fourier transform ......Page 80
4.4 Other fast methods ......Page 86
5.1 The Lebesgue integral ......Page 87
5.2 Spaces of measurable functions ......Page 91
5.3 Completeness ......Page 92
5.4 Weak derivatives ......Page 94
5.5 Transformation of measurable functions in R ......Page 95
5.6 Sobolev spaces in R ......Page 97
5.7 Sobolev spaces in intervals ......Page 100
6.1 The problem of best approximation ......Page 103
6.2 Estimates for the projection operator ......Page 106
6.3 Inverse inequalities ......Page 114
6.4 Other projection operators ......Page 117
6.5 Convergence of the Gaussian formulas ......Page 121
6.6 Estimates for the interpolation operator ......Page 125
6.7 Laguerre and Hermite functions ......Page 129
6.8 Refinements ......Page 133
7.1 Derivative matrices in the frequency space ......Page 135
7.2 Derivative matrices iji the physical space ......Page 139
7.3 Boundary conditions in the frequency space ......Page 145
7.4 Boundary conditions in the physical space ......Page 148
7.5 Derivatives of scaled functions ......Page 156
7.6 Numerical solution ......Page 158
8.1 Eigenvalues of first derivative operators ......Page 161
8.2 Eigenvalues of higher-order operators ......Page 167
8.3 Condition number ......Page 175
8.4 Preconditioners for second-order operators ......Page 178
8.5 Preconditioners for first-order operators ......Page 182
8.6 Convergence of eigenvalues ......Page 184
8.7 Multigrid method ......Page 189
9.1 General considerations ......Page 191
9.2 Approximation of linear equations ......Page 193
9.3 The weak formulation ......Page 199
9.4 Approximation of problems in the weak form ......Page 207
9.5 Approximation in unbounded domains ......Page 217
9.6 Other techniques ......Page 220
9.7 Boundary layers ......Page 221
9.8 Nonlinear equations ......Page 224
9.9 Systems of equations ......Page 227
9.10 Integral equations ......Page 228
10.1 The Gronwall inequality ......Page 231
10.2 Approximation of the heat equation ......Page 232
10.3 Approximation of linear first-order problems ......Page 239
10.4 Nonlinear time-dependent problems ......Page 245
10.5 Approximation of the wave equation ......Page 247
10.6 Time discretization ......Page 253
11.1 Introductory remarks ......Page 259
11.2 Non-overlapping multidomain methods ......Page 260
11.3 Solution techniques ......Page 270
11.4 Overlapping multidomain methods ......Page 273
12.1 A free boundary problem ......Page 275
12.2 An example in an unbounded domain ......Page 281
12.3 The nonlinear Schrodinger equation ......Page 284
12.4 Zeroes of Bessel functions ......Page 288
13.1 Poisson's equation ......Page 291
13.2 Approximation by the collocation method ......Page 292
13.3 Hints for the implementation ......Page 297
13.4 The incompressible Navier-Stokes equations ......Page 300
References ......Page 303
Index ......Page 315