Laplace's Equation and Seperation of Multivariable Differential Equation

In summary, if you have a differential equation that involves both r and \theta, you can split it into two equations by assuming that the solution can be written as a product of two functions: one function that is ONLY a function of r and another function that is ONLY a function of \theta. You then insert this into your differential equation and solve it.
  • #1
dduardo
Staff Emeritus
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Can someone explain how to separate a multivariable differential equation into two independent differential equations? I'm having an issue solving for the potential in spherical co-ordinates in terms of r and theta.
 
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  • #2
Let's say you have a differential equation that involves both [tex]r[/tex] and [tex]\theta[/tex]. The way separation of variables works is that you assume that the solution can be written as a product of two functions: one function that is ONLY a function of [tex]r[/tex] and another function that is ONLY a function of [tex]\theta[/tex]. Thus, you guess the solution

[tex]u(r, \theta) = R(r)\cdot \Theta(\theta)[/tex]

You then insert this into your differential equation. Derivatives with respect to [tex]r[/tex] will only affect the [tex]R(r)[/tex] function (the [tex]\Theta(\theta)[/tex] is not affected) and vice versa with derivatives with respect to [tex]\theta[/tex].

After you have determined how your differential equation looks when you have inserted this solution, you can rearrange the differential equation so that it looks something like

[tex]F(R''(r), R'(r), r) = G(\Theta''(\theta), \Theta'(\theta), \theta)[/tex] (1)

Here I have assumed that your differential equation was second order in both [tex]r[/tex] and [tex]\theta[/tex]. To put it in this form usually requires dividing both sides of the equation by [tex]R(r)\cdot \Theta(\theta)[/tex] but it can also involves other kinds of multiplications as well. The whole point of writing as I did in (1) is so you have one side of the differential equation that is completely independent of [tex]\theta[/tex] and the other side that is completely independent of [tex]r[/tex].

When this is true, the only possibility is that each side of the differential equation is equal to constant. This is true because the ONLY function of say [tex]\theta[/tex] that does not depend on [tex]\theta[/tex] is a constant.

At this point you are done...you now have the following:

[tex]F(R''(r), R'(r), r) = constant = G(\Theta''(\theta), \Theta'(\theta), \theta)[/tex]

So you have TWO differential equations of a single variable. In this process however you must realize that separation of variables only works in some cases and in the cases where it does work you must ask the question "Is this solution unique?" But for the problems you are doing in E&M you probably don't have to worry about this.
 
  • #3
Thanks for the help. After I got both differential equations I was able to apply cauchy-euler to solve them.
 

1. What is Laplace's Equation?

Laplace's Equation is a second-order partial differential equation that describes the distribution of a physical quantity (such as temperature or electric potential) in a given region, based on the values at the boundary of that region.

2. What is Separation of Variables in Multivariable Differential Equations?

Separation of Variables is a method used to solve partial differential equations, like Laplace's Equation, by breaking down the equation into simpler, one-variable equations that can be solved separately and then combined to find a solution to the original equation.

3. Why is Laplace's Equation important in science and engineering?

Laplace's Equation is used to model a wide range of physical phenomena in areas such as fluid dynamics, electromagnetism, and heat transfer. It is also a fundamental tool in solving boundary value problems and has many practical applications in engineering and physics.

4. How is Laplace's Equation solved using Separation of Variables?

To solve Laplace's Equation using Separation of Variables, we assume that the solution can be written as a product of separate functions of each variable. These functions are then substituted into the original equation, which results in a set of ordinary differential equations that can be solved using standard techniques. Finally, the solutions are combined to obtain the complete solution to the original equation.

5. What are some limitations of Laplace's Equation and Separation of Variables?

Laplace's Equation and Separation of Variables are limited in their applicability to linear, homogeneous problems. They are not suitable for nonlinear problems or problems with non-constant boundary conditions. Additionally, they cannot be used to solve problems that involve discontinuous or singular functions.

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