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PDEtools

  

Laplace

  

solves a second order linear PDE in 2 independent variables using the method of Laplace

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Laplace(PDE, F, numberofiterations = ...)

Parameters

PDE

-

a linear partial differential equation in two independent variables

F

-

the unknown of the PDE

numberofiterations = ...

-

optional - the right hand side is a positive integer limiting the number of iterations used to tackle the PDE

Description

• 

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.

• 

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.

• 

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.

• 

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.

Examples

The PDE  is known to be integrable in  steps if .

(1)

(2)

(3)

(4)

(5)

(6)

For , Laplace returns NULL since the default number of iterations is 5.

(7)

To obtain the solution in this example use the optional argument numberofiterations.

(8)

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.

(9)

(10)

(11)

(12)

(13)

(14)

Let's show that if  is a solution to , then  is a solution to .

(15)

(16)

(17)

Also, if  is a solution to , then  is a solution to .

(18)

(19)

Finally, if  is a solution to , then  is a solution to .

(20)

(21)

Now, remarkably, we start with the zero solution to , integrate the equation  to find  and apply  to find :

(22)

(23)

So this is the solution to

(24)

(25)

A similar sequence of PDEs and transformations can be constructed to find a solution depending on an arbitrary function of y.

See Also

casesplit

declare

diff_table

DifferentialAlgebra

PDEtools

 


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