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423
The Treatment of Derivative Discontinuities in Differential Equations
, 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 ..."
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Cited by 2 (1 self)
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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 CamassaHolm equation
"... Abstract. This paper develops a new approach in the analysis of the CamassaHolm 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 ..."
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Cited by 98 (7 self)
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Abstract. This paper develops a new approach in the analysis of the CamassaHolm 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 RungeKutta Methods
 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 twobody problems of electrodynamics. In this thesis, we present ..."
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Cited by 5 (0 self)
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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 RungeKutta formulas, defect error control, and an automatic handling technique
Automatic Fréchet differentiation for the numerical solution of boundaryvalue problems
 CODEN ACMSCU. ISSN 00983500 (print), 15577295 (electronic). Kim:2012:ASS
, 2012
"... A new solver for nonlinear boundaryvalue 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 ..."
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Cited by 6 (1 self)
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, the derivatives involved are Fréchet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operatorvalued 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
, 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 ..."
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Cited by 5 (3 self)
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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
 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 ..."
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Cited by 5 (0 self)
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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
 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 ..."
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Cited by 7 (2 self)
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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
, 1989
"... AbstractPreviously 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 twophase complex homotopy continuation method for obtaining th ..."
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AbstractPreviously 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 twophase complex homotopy continuation method for obtaining
A Multiquadric Solution for the Shallow Water Equations
 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 shallowwater equations. The numerical solutions are evaluated at scattered collocation points and the spatial partial derivatives are formed directly from part ..."
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Cited by 20 (9 self)
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A computational algorithm based on the multiquadric, which is a continuously differentiable radial basis function, is devised to solve the shallowwater 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.
"... 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 statespace form is used to predict the future dynamic behavior of the nonlinear 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 algorithm not only solves the differentiation equations of the network but also produces the sensitivity for the training
Results 1  10
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423