Numerical Solution of Nonlinear Boundary Value Problems with Applications

by Kubicek, Milan; Hlavacek, Vladimir

• ISBN13: 9780486463001
• ISBN10: 0486463001
• Format: Paperback
• Copyright: 2008-02-29
• Publisher: Dover Publications

Summary

A survey of the development, analysis, and application of numerical techniques in solving nonlinear boundary value problems, this text presents numerical analysis as a working tool for physicists and engineers. Topics include initial and boundary value problems for ordinary differential equations and the numerical realization of parametric studies. 1983 edition.

Author Biography

Milan Kubicek, Department of Chemical Engineering, Prague Institute of Chemical Technology Vladimir Hlavacek, Department of Chemical Engineering, State University of New York at Buffalo

Table of Contents

Preface	p. vii
Occurrence and Solution of Nonlinear Boundary Value Problems in Engineering and Physics	p. 1
Occurrence of nonlinear problems for ordinary differential equations	p. 1
Existence of a solution	p. 2
Problems of a multiplicity of solutions	p. 3
Nonlinear phenomena	p. 3
Types of nonlinear boundary value problems in chemical engineering	p. 6
Types of nonlinear boundary value problems in physics	p. 8
References	p. 9
Initial and Boundary Value Problems for Ordinary Differential Equations, Solution of Algebraic Equations	p. 11
Numerical solution of initial value problems	p. 11
A short summary of commonly used methods	p. 12
One-step methods: Runge-Kutta methods	p. 13
Linear multistep methods	p. 17
Numerical solution of stiff initial value problems	p. 19
Dependence of solution of initial value problems on the initial condition and on a parameter	p. 22
Numerical solution of linear and nonlinear algebraic equations	p. 25
Gaussian elimination and its modifications	p. 25
Band and almost-band matrices	p. 26
Solution of a single transcendental equation	p. 26
Solution of systems of nonlinear equations	p. 27
Dependence of the solution of a set of nonlinear equations on a parameter	p. 33
Boundary value problems	p. 34
Existence and uniqueness of solution	p. 34
Two-point boundary value problem of the pth-order	p. 37
Bibliography	p. 38
Linear Boundary Value Problems	p. 41
Finite-difference methods	p. 41
Superposition of solution: Method of complementary functions	p. 49
Method of adjoints	p. 55
Method of splitting the differential operator: the factorization method	p. 57
Invariant imbedding approach	p. 63
Discussion	p. 65
Problems	p. 68
Bibliography	p. 70
Numerical Methods for Nonlinear Boundary Value Problems	p. 72
Finite-difference methods	p. 72
Construction of the finite-difference analogy	p. 73
Finite-difference methods using approximations of higher order	p. 75
Aspects of the effective use of finite-difference methods	p. 86
Numerical solution of finite-difference equations	p. 89
Remarks on the relative merits of the finite-difference technique	p. 94
Problems	p. 95
Bibliography	p. 97
Method of Green's functions	p. 99
Construction of a Green's function	p. 99
Method of successive approximations	p. 102
Newton-Kantorovich linearization	p. 103
Discussion	p. 108
Problems	p. 109
Bibliography	p. 110
Newton-Kantorovich method and the quasilinearization technique	p. 110
Newton-Kantorovich method	p. 111
Third-order Newton-Kantorovich method	p. 119
Discussion	p. 123
Problems	p. 126
Bibliography	p. 128
Method of false transient for the solution of nonlinear boundary value problems	p. 129
Introduction	p. 129
False transient equations	p. 130
Numerical solution of false transient equations	p. 131
Examples	p. 135
Discussion	p. 151
Problems	p. 152
Bibliography	p. 153
One-parameter imbedding technique	p. 154
Introduction	p. 154
Details of one-parameter imbedding technique	p. 154
Continuous version of the Newton-Kantorovich method for NBVP	p. 161
Multiloop one-parameter imbedding for the NBVP	p. 166
One-loop one-parameter imbedding procedure for the NBVP	p. 175
Numerical aspects of methods	p. 179
Discussion	p. 184
Problems	p. 195
Bibliography	p. 198
Shooting methods	p. 199
Problem of order 1	p. 200
Problems of higher order	p. 205
Discussion	p. 217
Problems	p. 227
Bibliography	p. 232
Numerical Realization of Parametric Studies in Nonlinear Boundary Value Problems	p. 234
Conversion of boundary value problems to initial value problems and parameter mapping techniques	p. 235
Sequential use of standard methods	p. 253
Method of parametric differentiation	p. 259
Description of the method	p. 259
Numerical realization of the method	p. 260
Examples	p. 270
Method of differentiation with respect to boundary conditions	p. 275
General parameter mapping technique	p. 280
GPM algorithm for one second-order equation	p. 280
GPM algorithm for two second-order equations	p. 284
Predictor-corrector GPM algorithm for two second-order equations	p. 289
GPM algorithm for a system of first-order differential equations	p. 296
Evaluation of branching points in nonlinear boundary value problems	p. 302
Branching points for one second-order equation	p. 303
Branching points for two second-order equations	p. 306
Branching points for a system of nonlinear algebraic equations	p. 309
Branching points for a system of first-order differential equations	p. 311
Problems	p. 314
Bibliography	p. 318
Index	p. 321
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