Method of separation of variables for Laplace

The Method of Separation of Variables for Laplace is a mathematical way to separate variables into functions that are independent so those sets can be solved efficiently and accurately. This method is mostly used in the continuous space but can also be applied in discrete space if there are enough degrees of freedom. The parameters involved with this method must be differentiable functions which can easily be shown with a change in parameter graph due to integrating by parts. Also, the functions must be analytic in all the regions of interest.

What is  Separation of Variables?

With the separation of variables, the sets of equations are lumped into a single set of equations and solved at the same time with one iteration. This is extremely useful in solving certain differential equations where they are excessively difficult to solve using other techniques such as the Runge-Kutta method in cases where there are high degree polynomials involved and the derivatives do not change with respect to time or space. 

Benefits:

Using this method, we can easily solve differential equations but it does come with some disadvantages as well such as increased computational cost since we need to multiply out all the unknown functions as well as integrating them. The method is also more complex when compared to other methods because of the fact that we must also use a change in parameter graph to integrate by parts. In some simple cases, it would be purely a matter of following the order of integration and not working out the derivatives at all which would then save us time.

Additive Separation:

This method of separation is based on adding up the order of integration. This method is similar to the Runge-Kutta method in that it also involves using a set of finite difference approximations. The main difference between this method and the Runge-Kutta method is that we are only solving for one unknown function at a time in this case. In this case, we can take advantage of an easier possible solution by attempting to integrate by parts and get our polynomial out as fast as possible while still getting the final result.

Multiplicative Separation:

To solve for Laplace’s equation via multiplier method, we must first assume a particular solution and differentiate it to find the derivative. We then differentiate this derivative with respect to the space variables and multiply that result by the appropriate constant in order to get the original unknown function. If we follow this procedure, we will get our solution if a particular solution is assumed. 

Application of The Method of Separation of Variables for Laplace:

To give a better understanding of the method of separation of variables for Laplace, we will now tackle a few examples to highlight how this method works.

Example 1: Method of Separation of Variables for Laplace:

Find the solution to the following oscillation problem by separating variables and integrating as one iteration.  Δx = 0.4; T = 0.04; h = 1.0.

Solution:

From the components of the solution

We can see that by taking the partial derivative of y with respect to x, we then get

Since these functions are independent, we now need to integrate them as a single entity so that we only need to do it once instead of doing it for every value in the interval. We get 

We finally get our solution to our oscillation problem which is shown below in the graph with black dots.  You can see how for small values of x and small values of t, the solution is oscillating around Point A but as soon as we reach large values, it goes away from Point A and starts oscillating around Point B. This can be due to the fact that the derivative of h with respect to x is not constant with respect to x.

To check this, we will now eliminate h in our solution. We are left with

By substituting this back into our original equation, we get 

We see that as expected, the oscillation problem becomes much easier to solve when we eliminate h in our solution. Since T is a constant and Δx is a constant but h is now a function of time as opposed to space, it doesn’t matter if we integrate again or not because if all the derivatives are independent, then performing another integral won’t affect our final answer regardless of whether or not we integrate again or not.

Conclusion:

The method of separation of variables is an important method that can be used to solve a variety of problems. It can be used to find solutions for differential equations in which there are more than one variable involved. This method relies on the assumption that the different variables are subject to a separate set of laws and that the laws governing each variable can be solved separately. This assumption is usually correct, but sometimes it is not. Thus, it requires a bit more work than other methods but can produce very good results if you follow the procedure properly. Once you have established your initial guess, you can also modify this initial guess by removing terms or variables at will.