plot the solution to a first order linear or nonlinear partial differential equation - Maple Help

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PDEtools[PDEplot] - plot the solution to a first order linear or nonlinear partial differential equation

Calling Sequence

PDEplot(PDE, inits, srange, options)

Parameters

PDE

-

first order PDE containing one indeterminate function of n unknowns

inits

-

list of n+1 expressions or equations specifying the parametric form of an n1-dimensional manifold in n+1 dimensions; initial data

srange

-

the range or ranges of the parameters for the initial data

options

-

(optional) equations, described in PDEplot_options

Description

• 

The PDEplot command produces the plot of the solution for a first order, linear or nonlinear partial differential equation (PDE), for given initial conditions.

• 

Given initial data consisting of equations of a manifold in n+1-space depending on n1 parameters (for n=2, this is a curve) through which the required integral hypersurface must pass, the method of characteristics is used to generate the solution, resulting in a system of coupled first order ODEs to which numerical methods can be applied. The default is to use an internal Runge-Kutta fourth order (fixed step size) method. Other dsolvenumeric methods can be chosen by way of the method optional equation. Note, however, that use of these external methods tends to slow considerably the creation of plots.

• 

The first argument, PDE, can be any first order partial differential equation. PDEs involving piecewise functions are handled using hardware arithmetic (evalhf) as of Maple V Release 4. All arguments after the first can be given in any order.

• 

The inits parameter is a list of n+1 elements specifying the parametric form of a curve in n+1-space through which the required integral surface must pass. The elements should depend on n1 parameters. Using s and t as the parameters with n=3, for example, the elements of the list must be expressions such as f1s,t,f2s,t,f3s,t,f4s,t, or equations of the form x2=f1s,t,u=f2s,t,x1=f3s,t,x3=f4s,t, where the fs,t are explicit expressions of s and t that can be evaluated numerically. The two possible inputs are equivalent but if equations are given the order of the elements is irrelevant.

• 

The srange parameter is a list of ranges for the parameters used in the initial data. It should be given as [s1=a1..b1,s2=a2..b2,...].

• 

If n&equals;2, by default the solution surface is displayed as an opaque, colored membrane, with the initial conditions highlighted in black. For 2<n, by default a sequence of surfaces is shown in animation which together map out the required solution hypersurface. For more information, see plots[animate].

• 

Remaining arguments are interpreted as options which are specified as equations of the form option = value. In particular, the style option allows one to plot the points as points only, or to interpolate them using line mode. See plot3d,option for more information.

• 

This function is part of the PDEtools package, and so it can be used in the form PDEplot(..) only after executing the command with(PDEtools). However, it can always be accessed through the long form of the command by using PDEtools[PDEplot](..).

Examples

withPDEtools&colon;

We are given a nonlinear partial differential equation in two independent variables (pde1).

pde1:=xux&comma;yyux&comma;yxy&plus;ux&comma;y&equals;0

pde1:=xux&comma;yyux&comma;yxy&plus;ux&comma;y&equals;0

(1)

Using the unit circle in the x-y plane as an initial curve, we can explore the integral surface of the partial differential equation using PDEplot.

PDEplotpde1&comma;cost&comma;sint&comma;0&comma;t&equals;&pi;..&pi;&comma;ic_assumptions&equals;xux&comma;y&equals;cost

Because of the nonlinearity of the PDE, there are actually two possible solutions, depending on the values of the derivatives on the initial curve. These values are given using the option `ic_assumptions`.

PDEplotpde1&comma;cost&comma;sint&comma;0&comma;t&equals;&pi;..&pi;&comma;ic_assumptions&equals;xux&comma;y&equals;cost

Another example.

pde2:=y2&plus;zx&comma;y2&plus;x2xzx&comma;y2xyyzx&comma;y2zx&comma;yx&equals;0

pde2:=y2&plus;zx&comma;y2&plus;x2xzx&comma;y2xyyzx&comma;y2zx&comma;yx&equals;0

(2)

PDEplotpde2&comma;zx&comma;y&comma;t&comma;t&comma;sin&pi;t0.110&comma;t&equals;0..0.1&comma;numchar&equals;40&comma;orientation&equals;163&comma;56&comma;basechar&equals;true&comma;numsteps&equals;20&comma;20&comma;stepsize&equals;0.15&comma;initcolor&equals;costt&comma;style&equals;surfacecontour

The commands to create the plot from the Plotting Guide are

pde3:=sinyux&comma;y&comma;z&equals;ux&comma;y&comma;z

pde3:=sinyux&comma;y&comma;z&equals;ux&comma;y&comma;z

(3)

ics:=costsins&comma;cosscost&comma;cost&comma;sint&comma;t&equals;0..&pi;&comma;s&equals;0..&pi;

ics:=costsins&comma;cosscost&comma;cost&comma;sint&comma;t&equals;0..&pi;&comma;s&equals;0..&pi;

(4)

PDEplotpde3&comma;ics&comma;numsteps&equals;5&comma;6&comma;stepsize&equals;0.1&comma;axes&equals;none&comma;style&equals;surface&comma;numchar&equals;16&comma;16&comma;orientation&equals;225&comma;66&comma;lightmodel&equals;&apos;light2&apos;

More examples are available on the PDEplot_options page.

See Also

DEtools, DEtools[DEplot], dsolve, dsolve[classical], dsolve[dverk78], dsolve[gear], dsolve[lsode], dsolve[numeric], dsolve[rkf45], PDEplot_options, PDEtools, plot3d, plots[animate]


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