Student[NumericalAnalysis][Euler] - numerically approximate the solution to a first order initial-value problem using Euler's method
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Calling Sequence
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Euler(ODE, IC, t=b, opts)
Euler(ODE, IC, b, opts)
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Parameters
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ODE
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equation; first order ordinary differential equation of the form
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IC
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equation; initial condition of the form y(a)=c, where a is the left endpoint of the initial-value problem
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t
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name; the independent variable
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b
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algebraic; the point for which to solve; the right endpoint of this initial-value problem
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opts
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(optional) equations of the form keyword=value, where keyword is one of numsteps, output, comparewith, digits, or plotoptions; options for numerically solving the initial-value problem
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Description
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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 Euler command computes an approximate value of y(b) using the classical forward Euler method.
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If the second calling sequence is used, the independent variable t will be inferred from ODE.
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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.
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The Euler command is a shortcut for calling the InitialValueProblem command with the method = euler option.
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Options
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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.
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comparewith = [[method_1, submethod_1], [method_2, submethod_2]]
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If either method lacks applicable submethods, the corresponding submethod_n entry should be omitted.
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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.
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The number of digits to which the returned values will be rounded (using evalf). The default value is 4.
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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.
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Controls what information is returned by this procedure. The default value is solution:
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output = solution returns the computed value of at = b;
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output = Error returns the absolute error of at = b;
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output = plot returns a plot of the approximate (Euler) solution and the solution from one of Maple's best numeric DE solvers; and
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output = information returns an array of the values of , Maple's numeric solution, the approximations of as computed using this method and the absolute error between these at each iteration.
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The plot options. This option is used only when output = plot is specified.
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Notes
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To approximate the solution to an initial-value problem using a method other than Euler's Method, see InitialValueProblem.
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Download Help Document
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