Results 1 - 10 of 423

The Treatment of Derivative Discontinuities in Differential Equations

by C.A.H. Paul, C. A. H. Paul , 1999
"... The assumption of sufficiently smooth derivatives underlies much of the analysis of numerical methods for both ordinary and delay differential equations. However, derivative discontinuities can arise in ordinary differential equations and usually do arise in delay differential equations. In this ..."
Abstract - Cited by 2 (1 self) - Add to MetaCart
discontinuities in delay differential algebraic equations. Key words. differential equations, discontinuities, numerical solution 1 Introduction When solving a differential equation numerically, the assumption of a continuous solution with sufficiently smooth derivatives underlies much of the convergence

Global conservative solutions of the Camassa-Holm equation

by Alberto Bressan, Adrian Constantin
"... Abstract. This paper develops a new approach in the analysis of the Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new va ..."
Abstract - Cited by 98 (7 self) - Add to MetaCart
Abstract. This paper develops a new approach in the analysis of the Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new

Numerical Solution of Retarded and Neutral Delay Differential Equations using Continuous Runge-Kutta Methods

by Hiroshi Hayashi, Hiroshi Hayashi - Department of Computer Science, University of Toronto , 1996
"... A delay differential equation (DDE) can provide us with a realistic model of many phenomena arising in applied mathematics. For example, a DDE can be used for the modeling of population dynamics, the spread of infectious diseases, and two-body problems of electrodynamics. In this thesis, we present ..."
Abstract - Cited by 5 (0 self) - Add to MetaCart
a numerical method for solving DDEs and analysis for the numerical solution. We have developed the method by adapting recently developed techniques for initial value ordinary differential equations (continuous Runge-Kutta formulas, defect error control, and an automatic handling technique

Automatic Fréchet differentiation for the numerical solution of boundary-value problems

by Asgeir Birkisson, Tobin A. Driscoll - CODEN ACMSCU. ISSN 0098-3500 (print), 1557-7295 (electronic). Kim:2012:ASS , 2012
"... A new solver for nonlinear boundary-value problems (BVPs) in MATLAB is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton’s method in function space, where instead of the usual Jacobian matrices, t ..."
Abstract - Cited by 6 (1 self) - Add to MetaCart
, the derivatives involved are Fréchet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Fréchet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable

Computational Complexity of Numerical Solutions of Initial Value Problems for Differential Algebraic Equations

by Silvana Ilie , 2005
"... We investigate the cost of solving initial value problems for differential algebraic equations depending on the number of digits of accuracy requested. A recent result showed that the cost of solving initial value problems (IVP) for ordinary differential equations (ODE) is polynomial in the number o ..."
Abstract - Cited by 5 (3 self) - Add to MetaCart
of differential algebraic equations. We consider DAE of constant index to which the method applies. The DAE is allowed to be of arbitrary index, fully implicit and have derivatives of order higher than one. Similarly, by considering a realistic model, we show that the cost of computing the solution of IVP

Analysis of Continuous Moving Mesh Equations

by Jeremy H. Smith, Andrew M. Stuart - SIAM J. Sci. Statist. Comput , 1996
"... We consider a continuous formulation of moving mesh equations based on a relaxation of an equidistribution principle. Under natural assumptions on the monitor function, we derive bounds on the departure from equidistribution of the evolving mesh. Furthermore, we derive stability bounds for solutions ..."
Abstract - Cited by 5 (0 self) - Add to MetaCart
We consider a continuous formulation of moving mesh equations based on a relaxation of an equidistribution principle. Under natural assumptions on the monitor function, we derive bounds on the departure from equidistribution of the evolving mesh. Furthermore, we derive stability bounds

Applications and numerical analysis of partial differential Volterra equations: a brief survey

by Simon Shaw, J. R. Whiteman - Comput. Methods Appl. Mech. Engrg , 1997
"... This article contains a concise survey of the numerical analysis of Volterra equations, and leads up to some recent results on a posteriori error estimation for finite element approximations. 1 Introduction Modern techniques for the numerical solution of problems involving partial differential equat ..."
Abstract - Cited by 7 (2 self) - Add to MetaCart
equations are concerned not only with the discretization and approximate solution of continuous problems, but also fundamentally with assessing the integrity of the discrete solutions. As a result, emphasis in error analysis has moved from solely deriving a priori theoretical error estimates, to encompass

NONLOCAL AUTOMATED SENSITIVITY ANALYSIS

by R. Kalaba L, L. Tesfatsion , 1989
"... Abstract--Previously a complete ordinary differential equation (ODE) system was developed for tracking the solution x(~) of a parameterized system of nonlinear equations 0 = ~'(x,~) over an ~-interval [~0, ~]. This paper develops a two-phase complex homotopy continuation method for obtaining th ..."
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Abstract--Previously a complete ordinary differential equation (ODE) system was developed for tracking the solution x(~) of a parameterized system of nonlinear equations 0 = ~'(x,~) over an ~-interval [~0, ~]. This paper develops a two-phase complex homotopy continuation method for obtaining

A Multiquadric Solution for the Shallow Water Equations

by Yiu-Chung Hon , Kwok Fai Cheung, Xian-Zhong Mao, Edward J. Kansa - ASCE J. HYDRAULIC ENGINEERING , 1999
"... A computational algorithm based on the multiquadric, which is a continuously differentiable radial basis function, is devised to solve the shallow-water equations. The numerical solutions are evaluated at scattered collocation points and the spatial partial derivatives are formed directly from part ..."
Abstract - Cited by 20 (9 self) - Add to MetaCart
A computational algorithm based on the multiquadric, which is a continuously differentiable radial basis function, is devised to solve the shallow-water equations. The numerical solutions are evaluated at scattered collocation points and the spatial partial derivatives are formed directly from

Nonlinear System Identification for Predictive Control using Continuous Time Recurrent Neural Networks and Automatic Differentiation.

by unknown authors
"... In this paper, a continuous time recurrent neural network (CTRNN) is developed to be used in nonlinear model predictive control (NMPC) context. The neural network represented in a general nonlinear state-space form is used to predict the future dynamic behavior of the non-linear process in real time ..."
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time. An efficient training algorithm for the proposed network is developed using automatic differentiation (AD) techniques. By automatically generating Taylor coefficients, the al-gorithm not only solves the differentiation equations of the network but also produces the sensitivity for the training
Results 1 - 10 of 423