# Using separation of variables in solving partial differential equations

• chwala
In summary, the speaker shares their personal approach for solving PDEs using separation of variables, which involves setting ##U(x,t) ##= ##X(x) T(t)## and using two spatial boundary conditions. They also mention their intention to use this approach for future PDE problems on the forum.
chwala
Gold Member
Homework Statement
Kindly see attached;
Relevant Equations
separation of variables, in essence breaking down a pde into two ordinary differential equations.
I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second guessing on the 'letters used' in the textbook notes ...as attached below;

This is (my approach) steps to the working...
##U_t ##= ##k U_{xx}##
Let ##U(x,t) ##= ##X(x) T(t)##
Therefore,
##XT^{'}##=## kX^{''}T##
##\frac{T^{'}}{kT} ##=##\frac{X^{''}}{X} ##=##-λ##
From here we have one initial condition,i.e ##u(x,0)##=##f(x)## and two spatial boundary conditions, i.e ##u_x(0,t)=0## and ##u_x(L,t)=0##
Therefore using the Boundary conditions, we shall have,
##u_x##= ##X'T##
##0##=##X'(0)T(t)## and also, ##0= X'(L)T(t)##
If ## T(t)=0##, then we shall have a trivial solution and therefore for a non- trivial solution to be realized we need to have,
##X'(0)=0## →##X'(L)=0##
and therefore our problem is reduced to,
##T'+λkT=0## and ##X^{''} + λX=0##
##U_x(0)=0##
##U_x(L)=0##
I intend to use this approach in problems related to pde...regards,

Last edited:
This was not a question, rather my way of handling the pde's ...the intention was to keep you informed on my way of solving future pde's on the forum. I guess it should not be regarded as unanswered thread...

## 1. How does separation of variables work in solving partial differential equations?

Separation of variables is a method used to solve partial differential equations by breaking down the equation into simpler, ordinary differential equations. This is done by assuming the solution can be written as a product of two functions, one dependent on only one variable and the other dependent on the remaining variables. This allows the equation to be solved in stages, with each stage solving for one variable at a time.

## 2. What types of partial differential equations can be solved using separation of variables?

Separation of variables can be applied to linear partial differential equations with constant coefficients, as well as some non-linear equations with special properties. It is commonly used to solve equations involving heat transfer, diffusion, and wave propagation.

## 3. What are the advantages of using separation of variables in solving partial differential equations?

One advantage of separation of variables is that it can simplify complex partial differential equations into a series of ordinary differential equations, making them easier to solve. It also allows for a more systematic approach to solving equations, breaking them down into smaller, more manageable parts.

## 4. Are there any limitations to using separation of variables in solving partial differential equations?

Yes, there are some limitations to this method. It can only be applied to a certain class of equations, and not all equations can be solved using this technique. It also may not always provide the most accurate solution, as it relies on making certain assumptions about the form of the solution.

## 5. Can separation of variables be used to solve partial differential equations in higher dimensions?

Yes, separation of variables can be extended to solve partial differential equations in higher dimensions, although it becomes more complex and may not always be feasible. It is more commonly used in two or three dimensions, but can be applied to higher dimensions as well.

Replies
11
Views
1K
Replies
4
Views
619
Replies
1
Views
561
Replies
5
Views
550
Replies
1
Views
760
Replies
2
Views
2K
Replies
7
Views
953
Replies
7
Views
776
Replies
5
Views
1K
Replies
10
Views
1K