Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Simultaneous time and chance discretization for stochastic differential equations
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
A new numerical method for SDEs and its application in circuit simulation
Journal of Computational and Applied Mathematics - Proceedings of the 8th international congress on computational and applied mathematics
Numerical solutions of stochastic differential equations — implementation and stability issues
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
SIAM Journal on Numerical Analysis
Stochastic differential algebraic equations of index 1 and applications in circuit simulation
Journal of Computational and Applied Mathematics
On parameter and state estimation for linear differential-algebraic equations
Automatica (Journal of IFAC)
Moment Invariants for the Analysis of 2D Flow Fields
IEEE Transactions on Visualization and Computer Graphics
Proceedings of the 48th Design Automation Conference
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We discuss differential-algebraic equations driven by Gaussian white noise, which are assumed to have noise-free constraints and to be uniformly of DAE-index 1.We first provide a rigorous mathematical foundation of the existence and uniqueness of strong solutions. Our theory is based upon the theory of stochastic differential equations (SDEs) and the theory of differential-algebraic equations (DAEs), to each of which our problem reduces on making appropriate simplifications.We then consider discretization methods; implicit methods are necessary because of the differential-algebraic structure, and we consider adaptations of such methods used for SDEs. The consequences of an inexact solution of the implicit equations, roundoff and truncation errors, are analysed by means of the mean-square numerical stability of general drift-implicit discretization schemes for SDEs. We prove that the convergence properties of our drift-implicit Euler scheme, split-step backward Euler scheme, trapezoidal scheme and drift-implicit Milstein scheme carry over from the corresponding properties of these methods applied to SDEs.Finally, we show how the theory applies to the transient noise simulation of electronic circuits.