Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Ascher, Uri M.; Mattheij, R.M.M.; Russell, Robert D.

doi:10.1137/1.9781611971231

Cited by 1,044 publications

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“…The problem is thus a two-point boundary value problem. Using a collocation code COLNEW [1,4] (obtained via NETLIB), the optimality system was solved. Figures 2-4 represent the graphs of the solution to the optimality system (7-10) coupled with (12)(13)(14)(15)(16) at three different early treatment initiations.…”

Section: Numerical Resultsmentioning

confidence: 99%

Optimal control of the chemotherapy of HIV

Kirschner¹,

Lenhart²,

Serbin³

1997

Journal of Mathematical Biology

427

324

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Abstract. Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an objective function based on a combination of maximizing benefit based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of four ordinary differential equations and four adjoint ordinary differential equations.

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Section: Numerical Resultsmentioning

confidence: 99%

Optimal control of the chemotherapy of HIV

Kirschner¹,

Lenhart²,

Serbin³

1997

Journal of Mathematical Biology

427

324

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“…Hence its computation with explicit methods might generate large global errors and high computational costs (see e.g. Chapter 2.7 of Ascher et al 1995). To avoid this, instead of (2) we propose to solve…”

Section: Ode Solvermentioning

confidence: 99%

Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer–Resource, Cannibalistic and Trophic Models

Sanz

Getto

2016

Bull Math Biol

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With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur. The methods combine curve continuation, ODE solvers and test functions. Finally we apply the method to the above models using existing data for Daphnia magna consuming Algae, and for Perca fluviatilis feeding on Daphnia magna. In particular we validate the methods by deriving expressions for equilibria and bifurcations with respect to which we compute errors, and by comparing the obtained curves with curves that were computed earlier with other methods. We also present new curves to show how the methods can easily be applied to derive new biological insight. Schemes of algorithms are included.

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“…The search for a method to solve problems (1.1) is strongly motivated by numerous applications from physics (see [3] Our aim is to investigate the convergence of shooting procedures (see [1] or [18]) for the approximate solution of (1.1), based on an efficient numerical solution of the associated singular initial value problems.…”

Section: Preliminariesmentioning

confidence: 99%

“…Our aim is to investigate the convergence of shooting procedures (see [1] or [18]) for the approximate solution of (1.1), based on an efficient numerical solution of the associated singular initial value problems.…”

Section: Y (T) = M (T) T Y(t) + F (T Y(t)) T ∈ (0 1] (11a)mentioning

confidence: 99%

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The convergence of shooting methods for singular boundary value problems

Koch

Weinmüller²

2001

Math. Comp.

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Abstract. We investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems. Particular attention is paid to the well-posedness of the process. It is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available. Moreover, convergence results for a perturbed Newton iteration are discussed. PreliminariesWe discuss the numerical solution of the following nonlinear singular boundary value problems of the first order:where y and f are vector-valued functions of dimension n, M is an n × n matrix and g : R n × R n → R r , r ≤ n, is a smooth function.The search for a method to solve problems (1.1) is strongly motivated by numerous applications from physics (see [3] Our aim is to investigate the convergence of shooting procedures (see [1] or [18]) for the approximate solution of (1.1), based on an efficient numerical solution of the associated singular initial value problems.The unsatisfactory convergence properties of direct higher order methods are the main motivation to use shooting methods for the solution of singular boundary value problems. For collocation schemes only the stage order can be observed; the superconvergence does not hold in general, see [16]. In the presence of a singularity, other direct higher order methods (finite differences) show order reductions and become inefficient. Moreover, the standard acceleration techniques based on low-order methods do not work efficiently either, because in general, a proper asymptotic error expansion for the basic scheme does not exist, cf. [12]. Moreover, multiple shooting seems to be a particularly attractive alternative, because within its framework one can use different controlling mechanisms close to and away from

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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Cited by 1,044 publications

References 0 publications

Optimal control of the chemotherapy of HIV

Optimal control of the chemotherapy of HIV

Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer–Resource, Cannibalistic and Trophic Models

The convergence of shooting methods for singular boundary value problems

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