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Calling Sequence
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Laplace(PDE, F, numberofiterations = ...)
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Parameters
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PDE
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a linear partial differential equation in two independent variables
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F
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the unknown of the PDE
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numberofiterations = ...
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optional - the right hand side is a positive integer limiting the number of iterations used to tackle the PDE
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Description
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The general form of a second order scalar linear PDE in two independent variables is , where is the unknown function, the coefficients are functions of the independent variables and . The method of Laplace (not to be confused with integral transform methods of the same name) is a method which, when successful, will yield the general closed-form solution to such equations, depending upon two arbitrary functions of a single variable.
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The method works by transforming the original PDE into another second order scalar linear PDE with the remarkable property that solutions of can be found from solutions of by differentiations and simple linear algebraic manipulations. In favorable circumstances solutions to equation can be found and this then leads to solutions of the original equation. If solutions to cannot be found, then one may iterate the process to generate a sequence of equations with the property that solutions to can be constructed from solutions to by differentiations and simple linear algebraic manipulations. The third optional argument in the calling sequence to Laplace specifies the number of iterations the procedure will calculate in attempting to arrive at a PDE which can be integrated. The default numberofiterations is 5.
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The specific details of the method of Laplace are easiest to explain for equations of the type (although this special form is not required for the procedure). For an equation of this form, define the Laplace invariants and . One can show that if either or then the PDE can be integrated directly by linear ODE methods. If both and then the PDE can be easily transformed to the wave equation and the solution thus found.
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If then one defines and finds that: [1] also satisfies an equation of the form ; and [2] the equation can be inverted to give . A similar transform can be defined if . See the examples for an explicit computation of these transforms.
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Examples
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The PDE is known to be integrable in steps if .
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For , Laplace returns NULL since the default number of iterations is 5.
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To obtain the solution in this example use the optional argument numberofiterations.
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We analyze here the case to show some of the details of the method. We define a sequence of three PDEs, , and . We wish to solve . The PDEs and are generated by the method of Laplace. We also define three maps which we denote by , and . These are also prescribed by the method of Laplace.
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Let's show that if is a solution to , then is a solution to .
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Also, if is a solution to , then is a solution to .
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Finally, if is a solution to , then is a solution to .
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Now, remarkably, we start with the zero solution to , integrate the equation to find and apply to find :
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So this is the solution to
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A similar sequence of PDEs and transformations can be constructed to find a solution depending on an arbitrary function of y.
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