Student[NumericalAnalysis]

 numerically approximate the solution to a first order initial-value problem with the fourth-order Adams-Bashforth-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-Bashforth-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, step4, or step5
 The order of the pair of predictor-corrector formulae used to solve this initial-value problem.
 – step2 = Second-Order Predictor-Corrector
 – step3 = Third-Order Predictor-Corrector
 – step4 = Fourth-Order Predictor-Corrector
 – step5 = Fifth-Order Predictor-Corrector
 By default the Fourth-Order Predictor-Corrector 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 AdamsBashforthMoulton command computes an approximate value of y(b) using one of the Adams-Bashforth-Moulton family of predictor-corrector 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 AdamsBashforthMoulton command is a shortcut for calling the InitialValueProblem command with the method = adamsbashforthmoulton option.

Notes

 • The default predictor formula used (when no submethod option is specified) is the fourth-order Adams-Bashforth difference equation

${w}_{i+1,1}={w}_{i}+\frac{h\left(55f\left({t}_{i},{w}_{i}\right)-59f\left({t}_{i-1},{w}_{i-1}\right)+37f\left({t}_{i-2},{w}_{i-2}\right)-9f\left({t}_{i-3},{w}_{i-3}\right)\right)}{24}$

 and the default corrector equation used is the fourth-order Adams-Moulton difference equation

${w}_{i+1}={w}_{i}+\frac{h\left(9f\left({t}_{i+1},{w}_{i+1,1}\right)+19f\left({t}_{i},{w}_{i}\right)-5f\left({t}_{i-1},{w}_{i-1}\right)+f\left({t}_{i-2},{w}_{i-2}\right)\right)}{24}$

 • To approximate the solution to an initial-value problem using a method other than the Adams-Bashforth-Moulton Method, see InitialValueProblem.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right):$
 > $\mathrm{AdamsBashforthMoulton}\left(\mathrm{diff}\left(y\left(t\right),t\right)=\mathrm{cos}\left(t\right),y\left(0\right)=0.5,t=3\right)$
 ${0.6398}$ (1)
 > $\mathrm{AdamsBashforthMoulton}\left(\mathrm{diff}\left(y\left(t\right),t\right)=\mathrm{cos}\left(t\right),y\left(0\right)=0.5,t=3,\mathrm{output}=\mathrm{plot}\right)$