numerically approximate the solution to a first order initial-value problem with the Adams-Moulton Method - Maple Help

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Student[NumericalAnalysis][AdamsMoulton] - numerically approximate the solution to a first order initial-value problem with the Adams-Moulton Method

Parameters

 ODE - equation; first order ordinary differential equation of the form $\frac{ⅆ}{ⅆt}y\left(t\right)=f\left(t,y\right)$ IC - equation; initial condition of the form y(a)=c, where a is the left endpoint of the initial-value problem t - name; the independent variable b - algebraic; the point for which to solve; the right endpoint of this initial-value problem opts - (optional) equations of the form keyword=value, where keyword is one of numsteps, output, comparewith, digits, plotoptions, or submethod; options for numerically solving the initial-value problem

Description

 • Given an initial-value problem consisting of an ordinary differential equation ODE, a range a <= t <= b, and an initial condition y(a) = c, the AdamsMoulton command computes an approximate value of y(b) using one of the Adams-Moulton Methods (a family of implicit multi-step methods).
 • If the second calling sequence is used, the independent variable t will be inferred from ODE.
 • The endpoints a and b must be expressions that can be evaluated to floating-point numbers. The initial condition IC must be of the form y(a)=c, where c can be evaluated to a floating-point number.
 • The AdamsMoulton command is a shortcut for calling the InitialValueProblem command with the method = AdamsMoulton option.

Notes

 • The Adams-Moulton implicit difference equation is solved at each step using fsolve, Maple's numeric root-finding routine. For example, if the third-order difference equation

${w}_{i+1}={w}_{i}+\frac{1}{12}h\left(5f\left({t}_{i+1},{w}_{i+1}\right)+8f\left({t}_{i},{w}_{i}\right)-f\left({t}_{i-1},{w}_{i-1}\right)\right)$

 is used, then at each iteration $i$, this equation will be numerically solved for ${w}_{i+1}$. The final ${w}_{N}$ computed is the approximate value of y(b).
 • The above equation is also (confusingly) called the "Two-Step" Adams-Moulton difference equation.
 • To approximate the solution to an initial-value problem using a method other than the Adams-Moulton Method, see InitialValueProblem.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
 > $\mathrm{AdamsMoulton}\left(\frac{ⅆ}{ⅆt}y\left(t\right)=\mathrm{cos}\left(t\right),y\left(0\right)=0.5,t=3,\mathrm{submethod}=\mathrm{step2}\right)$
 ${0.6578}$ (1)
 > $\mathrm{AdamsMoulton}\left(\frac{ⅆ}{ⅆt}y\left(t\right)=\mathrm{cos}\left(t\right),y\left(0\right)=0.5,t=3,\mathrm{submethod}=\mathrm{step2},\mathrm{output}=\mathrm{plot}\right)$