Explicit and implicit methods: Difference between revisions

Revision as of 13:24, 23 November 2009

Revision as of 04:16, 17 December 2009

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{{Unreferenced|date=December 2009}}

'''Explicit and implicit methods''' are approaches used in [[numerical analysis]] for obtaining numerical solutions of time-dependent [[ordinary differential equation|ordinary]] and [[partial differential equation]]s, as is required in [[computer simulation]]s of [[Process (science)|physical processes]].

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==Illustration using the forward and backward Euler methods==

Consider the [[ordinary differential equation]]

: <math>\frac{dy}{dt} = -y^2, \ y\in [0, a]\quad \quad (2)</math>

with the initial condition <math>y(0)=1.</math> Consider a grid <math>t_k=a\frac{k}{n}</math> for 0&le;0≤''k''&le;≤''n'', that is, the time step is <math>\Delta t=a/n,</math> and denote <math>y_k=y(t_k)</math> for each <math>k</math>. [[Discretization|Discretize]] this equation using the simplest explicit and implicit methods, which are the ''forward Euler'' and ''backward Euler '' methods (see [[numerical ordinary differential equations]]) and compare the obtained schemes.

;Forward Euler method:

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{{DEFAULTSORT:Explicit And Implicit Methods}}

[[Category:Numerical differential equations]]

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[[eo:Vikipedio:Projekto matematiko/Eksplicita kaj implica manieroj]]