INDEX

Adams method, 27, 157

Aitken’s δ² method, 35, 152

algebraic eigenvalue problem, 174

alternative theorem, 12, 39, 44, 46, 126

autonomous systems, 150, 152

Banach space, 175

biophysics, 162–172

block-tridiagonal form, 105

boundary-value problems for general systems, 13–21, 142–146

Brouwer fixed-point theorem, 21

cancellation of leading digits, 42–43, 46

collocation methods, 175–176

computing exercises, 160–162, 170–172

consistency, 22, 76

constructive existence proof, 111

continuation theorem, 147

continuity methods, 61, 143, 146–152, 161, 168

continuity parameters, 147

contracting mapping theorem, 30

contracting maps, 16, 38, 55, 85, 97, 110, 145, 176

convergence, 23

deferred approach to the limit, 78, 100

degree of precision, 113, 117

difference approximation, 131, 135

difference corrections, 78–83, 89, 100–102

differential inequality, 3, 10, 54

diffusion–kinetics equation, 162

eigenvalue–eigenvector problem, 132

eigenvalue problems, 13, 122–152

end corrections, 159

enzyme-catalyzed reaction, 162

error magnification, 42

Euclidean norm, 2, 5

Euler’s method, 24

existence and uniqueness theory, 7–13, 108–111, 142–147

expansion procedures, 173

explicit iteration, 86

extrapolation to zero mesh width, 78, 89, 100

false position, 176

finite difference methods, 72–105

for eigenvalue problems, 131–138, 149

example, 162–172

forced flow, 160–162

Fredholm theory, 151

functional, 176

functional iteration, 30, 50, 85, 118, 127

function space approximations, 173–177

fundamental solution, 15

Galerkin’s method, 105, 173, 177

Gaussian elimination, 74

Gaussian quadrature, 116, 141

generalized eigenvalue problems, 142–152

Green’s functions, 106–112, 138

h → 0 extrapolation, 78–81

example, 168–170

for systems, 100–104

Hilbert space, 173

ill-conditioned matrix, 46

problem, 61

imbedding, 150

implicit-function theorem, 146

implicit schemes, 86

initial-value methods, for eigenvalue problems, 125–131

for generalized eigenvalue problems, 142– 149

for linear second-order equations, 39–46

for nonlinear second-order equations, 47–54

for nonlinear systems, 54–69

initial-value problems, 1–7

numerical solution of, 21–30

inner iteration, 176

inner product, 174

integral equations, 106, 138–142, 151

numerical solution of, 112–121

integral operator, 110

interpolation, 175

inverse iteration, 132–133, 136

iterative solution of nonlinear systems, 30–37

Jacobian matrix, 4, 17, 33, 57, 88, 164

determinant, 146

kernel, 139

positive-definite, 141

Lipschitz condition, 2

local truncation error, 22, 76, 133, 136

loss of significance, 43

lubrication theory, 153–162

maximal pivots, 133

maximum norm, 1, 14

mesh width, 78, 89

method of bisection, 132

Michaelis constant, 162

midpoint rule, 29

molecular-diffusion coefficient, 162

multistep methods, 27

net function, 22, 132

Neumann iterates, 110

Newton’s method, 18, 33, 36, 53, 57, 60, 64, 87, 98, 103, 119, 129, 131, 144, 149, 156, 164, 176

nonlinear boundary conditions, 142, 149

nonlinear diffusivity, 170–172

nonlinear eigenvalue problems, 142–149, 154, 173

nonlinear second-order equations, 7–11, 47–54, 83–91

nonlinear systems, of algebraic equations, 30–35, 164

of differential equations, 1–5, 13–18, 54–70, 91–102, 142–149

of integral equations, 110–112

norms, matrix, 14 f, 134

vector, 1, 2, 176

operator, 138

order of accuracy, 23

orthogonality condition, 126

orthogonal transformations, 140

orthonormal basis, 174

outer iterations, 176

parallel computers, 64

parallel shooting, 61–71, 149

periodic boundary conditions, 91

periodic solutions, 37, 146, 152

Picard iteration, 2, 6

Poincaré continuation, 61, 146–152, 149, 150

positive definite, 132

positive solutions, 150

power series, 68–71

practical examples, 153–172

predictor–corrector, 27

pressure distribution, 154

quadrature, error, 113, 115

formula, 113, 159

quasilinearization, 104

reverse shooting, 54

Riemann sums, 5

Ritz method, 176–177

roundoff, 38

Runge–Kutta method, 26, 157

Schwarz inequality, 5

second–order equations, 39–47, 72–83

self–adjoint operator, 112, 126, 138

separated end conditions, 105

shooting, 143, 146, 153–162

Simpson’s rule, 116, 159

singularity of the integrand, 159

singular perturbation, 154, 158, 162

spectral norm, 134

stability, 23, 76, 84

starting values, 147, 157, 165

stretching the interval, 150

Sturm-Liouville problems, 122–124

Sturm sequence, 132, 136

symmetric integral equation, 138, 141

symmetric tridiagonal matrix, 132

thrust, 154

trapezoidal rule, 115, 116, 120, 134

tridiagonal matrix, 74, 88, 132, 136, 164

variational methods, 104, 151, 176

variational principles, 124

variational problems, 4, 9, 51, 58, 144, 156

equations, 4

systems, 17, 148

vector norms, 1, 2, 176

viscous compressible fluid, 153

weighted quadrature formula, 116

Gaussian, 113