Volume 6. Ordinary Differential Equations and Integral Equations

1st Edition


  • J.D. Pryce
  • C.T.H. Baker
  • G. Monegato
  • G. vanden Berghe
  • Ordinary Differential Equations and Integral Equations

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    This volume contains contributions in the area of differential equations and integral equations. Many numerical methods have arisen in response to the need to solve "real-life" problems in applied mathematics, in particular problems that do not have a closed-form solution. Contributions on both initial-value problems and boundary-value problems in ordinary differential equations appear in this volume. Numerical methods for initial-value problems in ordinary differential equations fall naturally into two classes: those which use one starting value at each step (one-step methods) and those which are based on several values of the solution (multistep methods).
    John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century.
    Rob Corless and Lawrence Shampine talk about established technology, namely software for initial-value problems using Runge-Kutta and Rosenbrock methods, with interpolants to fill in the solution between mesh-points, but the 'slant' is new - based on the question, "How should such software integrate into the current generation of Problem Solving Environments?"
    Natalia Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the nth power of square matrices.
    The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods. Related is the study of chaotic behaviour.
    Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems.
    Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta metho

    Table of Contents

    Preface. Numerical methods for ordinary differential equations in the 20th century (J.C. Butcher). Initial value problems for ODE's in problem solving environments (L.F. Shampine, R.M. Corless). Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems (N. Borovykh, M.N. Spijker). Numerical bifurcation analysis for ODEs (W. Govaerts). Preserving algebraic invariants with Runge-Kutta methods (A. Iserles, A. Zanna). Performance of two methods for solving separable Hamiltonian systems (V. Antohe, I. Gladwell). Order stars and stiff integrators (E. Hairer, G. Wanner). Exponentially fitted Runge-Kutta methods (G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke). Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs (J.R. Cash). Software and algorithms for sensitivity analysis of large-scale differential algebraic systems (S. Li, L. Petzold). Compensating for order variation in mesh refinement for direct transcription methods (J.T. Betts, N. Biehn, S.L. Campbell, W.P. Huffman). Continuous numerical methods for ODEs with defect control (W.H. Enright). Numerical solutions of stochastic differential equations - implementations and stability issues (K. Burrage, P. Burrage, T. Mitsui). Numerical modelling in biosciences using delay differential equations (G.A. Bocharov, F.A. Rihan). Dynamics of constrained differential delay equations (J. Norbury, R.E. Wilson). A perspective on the numerical treatment of Volterra equations (C.T.H. Baker) Numerical stability of nonlinear delay differential equations of neutral type (A. Bellen, N. Guglielmi, M. Zennaro). Numerical bijurcation analysis of delay differential equations (K. Engelborghs, T. Luzyanina, D. Roose). How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation? (N.J. Ford, V. Wulf). Designing efficient software for solving delay differential eq


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    © 2001
    North Holland
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