Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Circuit noise evaluation by Padé approximation based model-reduction techniques
ICCAD '97 Proceedings of the 1997 IEEE/ACM international conference on Computer-aided design
A fast spectral algorithm for nonlinear wave equations with linear dispersion
Journal of Computational Physics
Spectral methods in MatLab
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
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Stochastic ordinary and partial differential equations (SOPDEs) in various forms arise and are successfully utilized in the modeling of a variety of physical and engineered systems such as telecommunication systems, electronic circuits, cosmological systems, financial systems, meteorological and climate systems. While the theory of stochastic partial and especially ordinary differential equations is more or less well understood, there has been much less work on practical formulations and computational approaches to solving these equations. In this paper, we concentrate on the stochastic non-linear Schrödinger equation (SNLSE) that arises in the analysis of wave propagation phenomena, mainly motivated by its predominant role as a modeling tool in the design of optically amplified long distance fiber telecommunication systems. We present novel formulations and computational methods for the stochastic characterization of the solution of the SNLSE. Our formulations and techniques are not aimed at computing individual realizations, i.e., sample paths, for the solution of the SNLSE á la Monte Carlo. Instead, starting with the SNLSE, we derive new systems of differential equations and develop associated computational techniques. The numerical solutions of these new equations directly produce the ensemble-averaged stochastic characterization desired for the solution of the SNLSE, in a non-Monte Carlo manner without having to compute many realizations needed for ensemble-averaging.