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Student[NumericalAnalysis]

 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 $\frac{ⅆ}{ⅆt}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y\left(t\right)=f\left(t,y\right)$ 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

Options

 • comparewith = [list]
 A list of method-submethod pairs; the method specified in the method option will be compared graphically with these methods. This option may only be used if output is set to either plot or information.
 It must be of the form
 comparewith = [[method_1, submethod_1], [method_2, submethod_2]]
 If either method lacks applicable submethods, the corresponding submethod_n entry should be omitted.
 Lists of all supported methods and their submethods are found in the InitialValueProblem help page, under the descriptions for the method and submethod options, respectively.
 • digits = posint
 The number of digits to which the returned values will be rounded (using evalf). The default value is 4.
 • numsteps = posint
 The number of steps used for the chosen numerical method. This option determines the static step size for each iteration in the algorithm. The default value is 5.
 Controls what information is returned by this procedure. The default value is solution:
 – output = solution returns the computed value of $y\left(t\right)$ at $t$ = b;
 – output = Error returns the absolute error of $y\left(t\right)$ at $t$ = b;
 – output = plot returns a plot of the approximate (Adams-Moulton) solution and the solution from one of Maple's best numeric DE solvers; and
 – output = information returns an array of the values of $t$, Maple's numeric solution, the approximations of $y\left(t\right)$ as computed using this method and the absolute error between these at each iteration.
 • plotoptions = list
 The plot options. This option is used only when output = plot is specified.
 • submethod = step2, step3, or step4
 The order of the difference formula used to solve this initial-value problem.
 – step2 = Two-Step Method (with a local truncation error proportional to ${h}^{4}$, where $h$ is the step size)
 – step3 = Three-Step Method (with a local truncation error proportional to ${h}^{5}$)
 – step4 = Four-Step Method (with a local truncation error proportional to ${h}^{6}$)
 By default the Four-Step submethod is used.

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

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

 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

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