411 research outputs found
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Stability in the Numerical Treatment of Volterra Integral and Integro-Differential Equations with emphasis on Finite Recurrence Relations.
In the last two decades the theory of Volterra integral equations and of integro-differential equations has developed extensively. New classes of methods for the numerical solution of such equations have been developed and at the same time there have been advances in the qualitative theory of these equations. More frequent use is being made of Volterra equations to model various physical and biological phenomenon as new insight has occurred into the asymptotic behaviour of solutions. In consequence, there has emerged a need for reliable and efficient methods for the numerical treatment of such equations.
This thesis is concerned with an aspect of numerical solution of Volterra integral and integro-differential equations. In Chapters 1 and 2 we are concerned with background material. We provide results on the classical theory of Volterra equations in Chapter 1 and on numerical methods in Chapter 2. The original material is contained in Chapters 3, 4 and 5. Here, stability results which involve the construction and analysis of finite-term recurrence relations are presented. The techniques relate to the treatment of Volterra integral and integro-differential equations. They permit the analysis of classical and 7-modified numerical methods.
The results presented should be viewed as a contribution towards an understanding of numerical stability for the methods considered. The area is one in which further work (subsequent to the present investigation and involving advanced techniques) has been performed and where open questions still remain.
The techniques which are employed in this thesis are applicable in other areas of numerical analysis and therefore have intrinsic interest
Numerical analysis of some integral equations with singularities
In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.Supported by a college bursary from the University of Chester
Numerical procedures for Volterra integral equations
This thesis investigates new finite difference methods for
the numerical solution of Volterra integral equations
Calibrated Adaptive Probabilistic ODE Solvers
Probabilistic solvers for ordinary differential equations assign a posterior
measure to the solution of an initial value problem. The joint covariance of
this distribution provides an estimate of the (global) approximation error. The
contraction rate of this error estimate as a function of the solver's step size
identifies it as a well-calibrated worst-case error, but its explicit numerical
value for a certain step size is not automatically a good estimate of the
explicit error. Addressing this issue, we introduce, discuss, and assess
several probabilistically motivated ways to calibrate the uncertainty estimate.
Numerical experiments demonstrate that these calibration methods interact
efficiently with adaptive step-size selection, resulting in descriptive, and
efficiently computable posteriors. We demonstrate the efficiency of the
methodology by benchmarking against the classic, widely used Dormand-Prince 4/5
Runge-Kutta method.Comment: 17 pages, 10 figures
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
Numerical analysis of some integral equations with singularities
In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.EThOS - Electronic Theses Online ServiceUniversity of ChesterGBUnited Kingdo
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
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