Abstract
We provide the derivation of a new formula for the approximation of an integral Markov process arising in the approximation of stochastic differential equations. This formula extends an existing formula derived in [1]. We have shown numerically that the leading order approximation of the differential equation with noise by solving an associated averaged problem and estimating the difference between them and the result is illustrated through some examples.
Highlights
Nonlinear ordinary and partial differential equations arise in various fields of sciences, fluid mechanics, solid state physics, plasma physics, nonlinear optics, and mathematical biology
We provide the derivation of a new formula for the approximation of an integral Markov process arising in the approximation of stochastic differential equations
We have shown numerically that the leading order approximation of the differential equation with noise by solving an associated averaged problem and estimating the difference between them and the result is illustrated through some examples
Summary
Nonlinear ordinary and partial differential equations arise in various fields of sciences, fluid mechanics, solid state physics, plasma physics, nonlinear optics, and mathematical biology. It is important to mention that noisy systems can be modeled in several ways: for example, Langevin’s equation describes a linear physical system to which white noise is added, and the linear theory for it has been extended to nonlinear stochastic differential equations with additive white noise [8]. It is observed that computer simulations of this type of stochastic ordinary differential equation with standard methods have some issues needed to be explored so that the reader will be benefitted while solving these type problems numerically
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