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Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Marian Smoluchowski in 1905, although Louis Bachelier was the first person credited with modeling Brownian motion in 1900, giving a very early example of a stochastic differential equation now known as Bachelier model.
Software engineering is a field within computer science focused on designing, developing, testing, and maintaining of software applications.It involves applying engineering principles and computer programming expertise to develop software systems that meet user needs.
Embedded software is computer software, written to control machines or devices that are not typically thought of as computers, commonly known as embedded systems.It is typically specialized for the particular hardware that it runs on and has time and memory constraints. [1]
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itô SDE. The zero-drift equation with constant diffusion can be considered as a model of classical Brownian motion : ∂ ∂ t p ( x , t ) = D 0 ∂ 2 ∂ x 2 [ p ( x , t ) ] . {\displaystyle {\frac {\partial }{\partial t}}p(x,t)=D_{0}{\frac {\partial ^{2 ...
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). [1]
Stochastic integrals can rarely be solved in analytic form, making stochastic numerical integration an important topic in all uses of stochastic integrals. Various numerical approximations converge to the Stratonovich integral, and variations of these are used to solve Stratonovich SDEs (Kloeden & Platen 1992).
One of the most studied SPDEs is the stochastic heat equation, [3] which may formally be written as = +, where is the Laplacian and denotes space-time white noise.Other examples also include stochastic versions of famous linear equations, such as the wave equation [4] and the Schrödinger equation.