Numerical solution of boundary value problems for ordinary differential equations / Uri M. Ascher, Robert M. M. Mattheij, Robert D. Russell.
Series Prentice Hall series in computational mathematicsEditor: Englewood Cliffs, N.J. : Prentice Hall, c1988Descripción: xxi, 595 p. : il. ; 25 cmISBN: 0136272665Otra clasificación: 65-01 (65L05 65L10 65L12 65L50 65L70 34-01)1 INTRODUCTION [1] 1.1 Boundary Value Problems for Ordinary Differential Equations [1] 1.1.1 Model problems [1] 1.1.2 General forms for the differential equations [3] 1.1.3 General forms for the boundary conditions [4] 1.2 Boundary Value Problems in Applications [7] 2 REVIEW OF NUMERICAL ANALYSIS AND MATHEMATICAL BACKGROUND [28] 2.1 Errors in Computation [28] 2.2 Numerical Linear Algebra [32] Eigenvalues, transformations, factorizations, projections [32] Norms, angles, and condition numbers 36 Gaussian elimination, LU-decomposition [39] Householder transformations: QU-decomposition [43] Least squares solution of overdetermined systems [45] The QR algorithm for eigenvalues [45] Error analysis [46] 2.3 Nonlinear Equations [48] Newton’s method [48] Fixed-point iteration and two basic theorems [49] Quasi-Newton methods [51] Quasilinearization [52] Polynomial Interpolation [55] 2.4.1 Forms for interpolation polynomials [55] 2.4.2 Osculatory interpolation [58] 2.5 Piecewise Polynomials, or Splines [59] 2.5.1 Local representations [59] 2.5.2 B-splines [61] 2.5.3 Spline interpolation [62] 2.6 Numerical Quadrature [63] 2.6.1 Basic rules [63] 2.6.2 Composite formulas [66] 2.7 Initial Value Ordinary Differential Equations [67] Numerical methods [68] Consistency, stability and convergence [71] Stiff problems [73] Error control and step selection [74] 2.8 Differential Operators and Their Discretizations [77] 2.8.1 Norms of functions, spaces, and operators [77] 2.8.2 Roundoff errors and truncation errors [80] 3 THEORY OF ORDINARY DIFFERENTIAL EQUATIONS [84] 3.1 Existence and Uniqueness Results [85] 3.1.1 Results for boundary value problems [85] 3.1.2 Results for initial value problems [87] 3.2 Green’s Functions [94] 3.2.1 A first-order system [94] 3.2.2 A higher-order ODE [96] 3.3 Stability of Initial Value Problems [99] 3.3.1 Stability and fundamental solutions [100] 3.3.2 The constant coefficient case [102] 3.3.3 The general linear case [104] 3.3.4 The nonlinear case [109] 3.4 Conditioning of Boundary Value Problems [110] 3.4.1 Linear problems and conditioning constants [111] 3.4.2 Dichotomy [115] 3.4.3 Well-conditioning and dichotomy [121] Exercises [127] 4 INITIAL VALUE METHODS [132] 4.1 Introduction: Shooting [132] 4.1.1 Shooting for a linear second-order problem [133] 4.1.2 Shooting for a nonlinear second-order problem [134] 4.1.3 Shooting—general application, limitations, extensions [134] 4.2 Superposition and Reduced Superposition [135] 4.2.1 Superposition [735] 4.2.2 Numerical accuracy [137] 4.2.3 Numerical stability [140] 4.2.4 Reduced superposition [143] 4.3 Multiple Shooting for Linear Problems [145] 4.3.1 The method [146] 4.3.2 Stability [149] 4.3.3 Practical considerations [757] 4.3.4 Compactification [753] 4.4 Marching Techniques for Multiple Shooting [155] 4.4.1 Reorthogonalization [756] 4.4.2 Decoupling [757] 4.4.3 Stabilized march [.767] 4.5 The Riccati Method [164] 4.5.1 Invariant imbedding [164] 4.5.2 Riccati transformations for general linear BVPs [766] 4.5.3 Method properties [168] 4.6 Nonlinear Problems [170] 4.6.1 Shooting for nonlinear problems [170] 4.6.2 Difficulties with single shooting [174] 4.6.3 Multiple shooting for nonlinear problems [175] Exercises [180] 5 FINITE DIFFERENCE METHODS [185] 5.1 Introduction [185] 5.1.1 A simple scheme for a second-order problem [187] 5.1.2 Simple one-step schemes for linear systems [190] 5.1.3 Simple schemes for nonlinear problems [194] 5.2 Consistency, Stability, and Convergence [198] 5.2.1 Linear problems [198] 5.2.2 Nonlinear problems [203] 5.3 Higher-Order One-Step Schemes [208] 5.3.1 Implicit Runge-Kutta schemes [210] 5.3.2 A subclass of Runge-Kutta schemes [213] 5.3.3 On the implementation of Runge-Kutta schemes [217] 5.4 Collocation Theory [218] 5.4.1 Linear problems [279] 5.4.2 Nonlinear problems [222] 5.5 Acceleration Techniques [226] 5.5.1 Error expansion [228] 5.5.2 Extrapolation [230] 5.5.3 Deferred corrections [234] 5.5.4 More deferred corrections [238] 5.6 Higher-Order ODEs [244] 5.6.1 More on a simple second-order scheme [245] 5.6.2 Collocation [247] 5.6.3 Collocation implementation using spline bases [259] 5.6.4 Conditioning of collocation matrices [262] 5.7 Finite Element Methods [266] 5.7.1 The Ritz method [267] 5.7.2 Other finite element methods [271] Exercises [272] 6 DECOUPLING [275] 6.1 Decomposition of Vectors [277] 6.2 Decoupling of the ODE [279] 6.2.1 Consistent fundamental solution [260] 6.2.2 The basic continuous decoupling algorithm [284] 6.3 Decoupling of One-Step Recursions [288] 6.3.1 Consistent fundamental solutions [290] 6.3.2 The basic discrete decoupling algorithm and additional considerations [291] 6.4 Practical Aspects of Consistency [293] 6.4.1 Consistency for separated BC [293] 6.4.2 Consistency for partially separated BC [295] 6.4.3 Consistency for general BC [296] 6.5 Closure and its Implications Exercises 299 [297] 7 SOLVING LINEAR EQUATIONS [303] 7.1 General Staircase Matrices and Condensation [306] 7.2 Algorithms for the Separated BC Case [308] 7.2.1 Gaussian elimination with partial pivoting [308] 7.2.2 Alternate row and column elimination [310] 7.2.3 Block tridiagonal elimination [313] 7.2.4 Stable compactification [316] 7.3 Stability for Block Methods [318] 7.4 Decomposition in the Nonseparated BC Case [319] 7.4.1 The LU-decomposition [320] 7A.2 A general decoupling algorithm [321] 7.5 Solution in More General Cases [322] 7.5.1 BVPs with parameters [322] 7.5.2 Multipoint BC [323] Exercises [324] 8 SOLVING NONLINEAR EQUATIONS [327] 8.1 Improving the Local Convergence of Newton’s Method [329] 8.1.1 Damped Newton [329] 8.1.2 Altering the Newton direction [338] 8.2 Reducing the Cost of the Newton Iteration [341] 8.2.1 Modified Newton [341] 8.2.2 Rank-1 updates [343] 8.3 Finding a Good Initial Guess [343] 8.3.1 Continuation [344] 8.3.2 Imbedding in a time-dependent problem [350] 8.4 Further Remarks on Discrete Nonlinear BVPS [353] Exercises [355] 9 MESH SELECTION [358] 9.1 Introduction [359] 9.1.1 Error equidistribution and monitoring [362] 9.2 Direct Methods [364] 9.2.1 Equidistributing local truncation error [365] 9.3 A Mesh Strategy for Collocation [367] 9.3.1 A practical mesh selection algorithm [367] 9.3.2 Numerical examples [371] 9.4 Transformation Methods [373] 9.4.1 Explicit method [373] 9.4.2 Implicit method [375] 9.5 General Considerations [380] 9.5.1 Some practical considerations [380] 9.5.2 Coordinating mesh selection and nonlinear iteration [381] 9.5.3 Other approaches [383] Exercises [384] 10 SINGULAR PERTURBATIONS [386] 10.1 Analytical Approaches [389] 10.1.1 A linear second-order ODE [390] 10.1.2 A nonlinear second-order ODE [395] 10.1.3 Linear first-order systems [397] 10.1.4 Nonlinear first-order systems [411] 10.2 Numerical Approaches [414] 10.2.1 Theoretical multiple shooting [415] 10.2.2 Stability and global error analysis, I [419] 10.2.3 Stability and global error analysis, II [422] 10.2.4 The discretization mesh as a stretching transformation [428] 10.3 Difference Methods [432] 10.3.1 One-sided schemes [432] 10.3.2 Symmetric schemes [440] 10.3.3 Exponential fitting [454] 10.4 Initial Value Methods [457] 10.4.1 Sequential shooting [457] 10.4.2 The Riccati method [460] Exercises [467] 11 SPECIAL TOPICS [469] 11.1 Reformulation of Problems in “Standard” Form [469] 11.1.1 Higher-order equations, parameters, nonseparated BC, multipoint BC [470] 11.1.2 Conditions at special points [471] 11.1.3 Integral relations [473] 11.2 Generalized ODEs and Differential Algebraic Equations [474] 11.3 Eigenvalue Problems [478] 11.3.1 Sturm-Liouville problems [478] 11.3.2 Eigenvalues of first-order systems [482] 11.4 BVPs with Singularities; Infinite Intervals [483] 11.4.1 Singularities of the first kind [484] 11.4.2 Infinite interval problems [486] 11.5 Path Following, Singular Points and Bifurcation [490] 11.5.1 Branching and stability [493] 11.5.2 Numerical techniques [496] 11.6 Highly Oscillatory Solutions [500] 11.7 Functional Differential Equations [505] 11.8 Method of Lines for PDEs [508] 11.9 Multipoint Problems [512] 11.10 On Code Design and Comparison [515] APPENDIX A: A MULTIPLE SHOOTING CODE [517] APPENDIX B: A COLLOCATION CODE [526]
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Incluye referencias bibliográficas (p. 566-585) e índice.
1 INTRODUCTION [1] --
1.1 Boundary Value Problems for Ordinary Differential --
Equations [1] --
1.1.1 Model problems [1] --
1.1.2 General forms for the differential equations [3] --
1.1.3 General forms for the boundary conditions [4] --
1.2 Boundary Value Problems in Applications [7] --
2 REVIEW OF NUMERICAL ANALYSIS AND MATHEMATICAL BACKGROUND [28] --
2.1 Errors in Computation [28] --
2.2 Numerical Linear Algebra [32] --
Eigenvalues, transformations, factorizations, projections [32] --
Norms, angles, and condition numbers 36 Gaussian elimination, LU-decomposition [39] --
Householder transformations: QU-decomposition [43] --
Least squares solution of overdetermined systems [45] --
The QR algorithm for eigenvalues [45] --
Error analysis [46] --
2.3 Nonlinear Equations [48] --
Newton’s method [48] --
Fixed-point iteration and two basic theorems [49] --
Quasi-Newton methods [51] --
Quasilinearization [52] --
Polynomial Interpolation [55] --
2.4.1 Forms for interpolation polynomials [55] --
2.4.2 Osculatory interpolation [58] --
2.5 Piecewise Polynomials, or Splines [59] --
2.5.1 Local representations [59] --
2.5.2 B-splines [61] --
2.5.3 Spline interpolation [62] --
2.6 Numerical Quadrature [63] --
2.6.1 Basic rules [63] --
2.6.2 Composite formulas [66] --
2.7 Initial Value Ordinary Differential Equations [67] --
Numerical methods [68] --
Consistency, stability and convergence [71] --
Stiff problems [73] --
Error control and step selection [74] --
2.8 Differential Operators and Their Discretizations [77] --
2.8.1 Norms of functions, spaces, and operators [77] --
2.8.2 Roundoff errors and truncation errors [80] --
3 THEORY OF ORDINARY DIFFERENTIAL EQUATIONS [84] --
3.1 Existence and Uniqueness Results [85] --
3.1.1 Results for boundary value problems [85] --
3.1.2 Results for initial value problems [87] --
3.2 Green’s Functions [94] --
3.2.1 A first-order system [94] --
3.2.2 A higher-order ODE [96] --
3.3 Stability of Initial Value Problems [99] --
3.3.1 Stability and fundamental solutions [100] --
3.3.2 The constant coefficient case [102] --
3.3.3 The general linear case [104] --
3.3.4 The nonlinear case [109] --
3.4 Conditioning of Boundary Value Problems [110] --
3.4.1 Linear problems and conditioning constants [111] --
3.4.2 Dichotomy [115] --
3.4.3 Well-conditioning and dichotomy [121] --
Exercises [127] --
4 INITIAL VALUE METHODS [132] --
4.1 Introduction: Shooting [132] --
4.1.1 Shooting for a linear second-order problem [133] --
4.1.2 Shooting for a nonlinear second-order problem [134] --
4.1.3 Shooting—general application, limitations, extensions [134] --
4.2 Superposition and Reduced Superposition [135] --
4.2.1 Superposition [735] --
4.2.2 Numerical accuracy [137] --
4.2.3 Numerical stability [140] --
4.2.4 Reduced superposition [143] --
4.3 Multiple Shooting for Linear Problems [145] --
4.3.1 The method [146] --
4.3.2 Stability [149] --
4.3.3 Practical considerations [757] --
4.3.4 Compactification [753] --
4.4 Marching Techniques for Multiple Shooting [155] --
4.4.1 Reorthogonalization [756] --
4.4.2 Decoupling [757] --
4.4.3 Stabilized march [.767] --
4.5 The Riccati Method [164] --
4.5.1 Invariant imbedding [164] --
4.5.2 Riccati transformations for general linear BVPs [766] --
4.5.3 Method properties [168] --
4.6 Nonlinear Problems [170] --
4.6.1 Shooting for nonlinear problems [170] --
4.6.2 Difficulties with single shooting [174] --
4.6.3 Multiple shooting for nonlinear problems [175] --
Exercises [180] --
5 FINITE DIFFERENCE METHODS [185] --
5.1 Introduction [185] --
5.1.1 A simple scheme for a second-order problem [187] --
5.1.2 Simple one-step schemes for linear systems [190] --
5.1.3 Simple schemes for nonlinear problems [194] --
5.2 Consistency, Stability, and Convergence [198] --
5.2.1 Linear problems [198] --
5.2.2 Nonlinear problems [203] --
5.3 Higher-Order One-Step Schemes [208] --
5.3.1 Implicit Runge-Kutta schemes [210] --
5.3.2 A subclass of Runge-Kutta schemes [213] --
5.3.3 On the implementation of Runge-Kutta schemes [217] --
5.4 Collocation Theory [218] --
5.4.1 Linear problems [279] --
5.4.2 Nonlinear problems [222] --
5.5 Acceleration Techniques [226] --
5.5.1 Error expansion [228] --
5.5.2 Extrapolation [230] --
5.5.3 Deferred corrections [234] --
5.5.4 More deferred corrections [238] --
5.6 Higher-Order ODEs [244] --
5.6.1 More on a simple second-order scheme [245] --
5.6.2 Collocation [247] --
5.6.3 Collocation implementation using spline bases [259] --
5.6.4 Conditioning of collocation matrices [262] --
5.7 Finite Element Methods [266] --
5.7.1 The Ritz method [267] --
5.7.2 Other finite element methods [271] --
Exercises [272] --
6 DECOUPLING [275] --
6.1 Decomposition of Vectors [277] --
6.2 Decoupling of the ODE [279] --
6.2.1 Consistent fundamental solution [260] --
6.2.2 The basic continuous decoupling algorithm [284] --
6.3 Decoupling of One-Step Recursions [288] --
6.3.1 Consistent fundamental solutions [290] --
6.3.2 The basic discrete decoupling algorithm and additional considerations [291] --
6.4 Practical Aspects of Consistency [293] --
6.4.1 Consistency for separated BC [293] --
6.4.2 Consistency for partially separated BC [295] --
6.4.3 Consistency for general BC [296] --
6.5 Closure and its Implications Exercises 299 [297] --
7 SOLVING LINEAR EQUATIONS [303] --
7.1 General Staircase Matrices and Condensation [306] --
7.2 Algorithms for the Separated BC Case [308] --
7.2.1 Gaussian elimination with partial pivoting [308] --
7.2.2 Alternate row and column elimination [310] --
7.2.3 Block tridiagonal elimination [313] --
7.2.4 Stable compactification [316] --
7.3 Stability for Block Methods [318] --
7.4 Decomposition in the Nonseparated BC Case [319] --
7.4.1 The LU-decomposition [320] --
7A.2 A general decoupling algorithm [321] --
7.5 Solution in More General Cases [322] --
7.5.1 BVPs with parameters [322] --
7.5.2 Multipoint BC [323] --
Exercises [324] --
8 SOLVING NONLINEAR EQUATIONS [327] --
8.1 Improving the Local Convergence of Newton’s Method [329] --
8.1.1 Damped Newton [329] --
8.1.2 Altering the Newton direction [338] --
8.2 Reducing the Cost of the Newton Iteration [341] --
8.2.1 Modified Newton [341] --
8.2.2 Rank-1 updates [343] --
8.3 Finding a Good Initial Guess [343] --
8.3.1 Continuation [344] --
8.3.2 Imbedding in a time-dependent problem [350] --
8.4 Further Remarks on Discrete Nonlinear BVPS [353] --
Exercises [355] --
9 MESH SELECTION [358] --
9.1 Introduction [359] --
9.1.1 Error equidistribution and monitoring [362] --
9.2 Direct Methods [364] --
9.2.1 Equidistributing local truncation error [365] --
9.3 A Mesh Strategy for Collocation [367] --
9.3.1 A practical mesh selection algorithm [367] --
9.3.2 Numerical examples [371] --
9.4 Transformation Methods [373] --
9.4.1 Explicit method [373] --
9.4.2 Implicit method [375] --
9.5 General Considerations [380] --
9.5.1 Some practical considerations [380] --
9.5.2 Coordinating mesh selection and nonlinear iteration [381] --
9.5.3 Other approaches [383] --
Exercises [384] --
10 SINGULAR PERTURBATIONS [386] --
10.1 Analytical Approaches [389] --
10.1.1 A linear second-order ODE [390] --
10.1.2 A nonlinear second-order ODE [395] --
10.1.3 Linear first-order systems [397] --
10.1.4 Nonlinear first-order systems [411] --
10.2 Numerical Approaches [414] --
10.2.1 Theoretical multiple shooting [415] --
10.2.2 Stability and global error analysis, I [419] --
10.2.3 Stability and global error analysis, II [422] --
10.2.4 The discretization mesh as a stretching transformation [428] --
10.3 Difference Methods [432] --
10.3.1 One-sided schemes [432] --
10.3.2 Symmetric schemes [440] --
10.3.3 Exponential fitting [454] --
10.4 Initial Value Methods [457] --
10.4.1 Sequential shooting [457] --
10.4.2 The Riccati method [460] --
Exercises [467] --
11 SPECIAL TOPICS [469] --
11.1 Reformulation of Problems in “Standard” Form [469] --
11.1.1 Higher-order equations, parameters, nonseparated BC, multipoint BC [470] --
11.1.2 Conditions at special points [471] --
11.1.3 Integral relations [473] --
11.2 Generalized ODEs and Differential Algebraic Equations [474] --
11.3 Eigenvalue Problems [478] --
11.3.1 Sturm-Liouville problems [478] --
11.3.2 Eigenvalues of first-order systems [482] --
11.4 BVPs with Singularities; Infinite Intervals [483] --
11.4.1 Singularities of the first kind [484] --
11.4.2 Infinite interval problems [486] --
11.5 Path Following, Singular Points and Bifurcation [490] --
11.5.1 Branching and stability [493] --
11.5.2 Numerical techniques [496] --
11.6 Highly Oscillatory Solutions [500] --
11.7 Functional Differential Equations [505] --
11.8 Method of Lines for PDEs [508] --
11.9 Multipoint Problems [512] --
11.10 On Code Design and Comparison [515] --
APPENDIX A: A MULTIPLE SHOOTING CODE [517] --
APPENDIX B: A COLLOCATION CODE [526] --
MR, 90h:65120
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