On solving PDE using separating the variable.

In summary, the conversation discusses the equality of two formulas, with one formula being dependent on x and the other on t. It is stated that for these two formulas to be equal, they must both be equal to a single constant. The conversation also touches on the possibility of these two formulas being equal for specific values of x and t, which would also be equal to the same constant. The concept of an arbitrary constant is mentioned, implying that these two formulas can be equal for more than one value of x and t.
  • #1
A Dhingra
211
1
hi..

with refrenence to http://www.math.uah.edu/howell/MAPH/Archives/Old_Notes/PDEs/PDE1.pdf
page 7,
“Observe” that the only way we can have
formula of t only= formula of x only​
is for both sides to be equal to a single constant.

here I do understand that for these to being equal requires them to be constant, because changing either t or x would affect either side of the formula breaking the inequality, so to be equal they are supposed to be a constant. (please correct me if i go wrong anywhere..)
Now what i am not sure about is that these two formulas can also be equal to each other for a specific value of x and t, is this anything to do with the constant that they will be equal to??

It is said that the constant say k is arbitrary, suggesting these two sides are equal for more than one value of x and y... how is this said?

(I gave the link so that i don't have to explain what i am asking, because i don't know the correct terms and that might turn confusing.. no copyright violation intended)
 
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  • #2
A Dhingra said:
hi..

with refrenence to http://www.math.uah.edu/howell/MAPH/Archives/Old_Notes/PDEs/PDE1.pdf
page 7,
“Observe” that the only way we can have
formula of t only= formula of x only​
is for both sides to be equal to a single constant.

here I do understand that for these to being equal requires them to be constant, because changing either t or x would affect either side of the formula breaking the inequality, so to be equal they are supposed to be a constant. (please correct me if i go wrong anywhere..)
Now what i am not sure about is that these two formulas can also be equal to each other for a specific value of x and t, is this anything to do with the constant that they will be equal to??
I'm not sure what you are asking here. If X(x)= T(t) for all x and t, then, yes, they must be equal to the same constant for all x and t, which, of course, includes any "specific value of x and t".

It is said that the constant say k is arbitrary, suggesting these two sides are equal for more than one value of x and y... how is this said?
(You mean "t" not "y", right?) What do you mean "for more than one value of x and t"?? You just said they were equal for all x and t. That surely includes more than one!

(I gave the link so that i don't have to explain what i am asking, because i don't know the correct terms and that might turn confusing.. no copyright violation intended)
 
Last edited by a moderator:
  • #3
HallsofIvy said:
I'm not sure what you are asking here. If X(x)= T(t) for all x and t, then, yes, they must be equal to the same constant for all x and t, which, of course, includes any "specific value of x and t".
Okay i got it..
If X(x) = T(t) for all x and t, and there is no relation or interdependence between x and t other than this, then X(x)= constant just as T(t). And this constant is constant for all values of x and t... Thanks for the quick help.
 

What is the process of solving a Partial Differential Equation (PDE) using the method of separating variables?

The method of separating variables is a technique used to solve PDEs by breaking down the equation into simpler ordinary differential equations (ODEs) that can be solved separately. This involves assuming a solution in the form of a product of single-variable functions, and then substituting this solution into the PDE. This allows for the separation of variables, which can then be solved individually.

What are the advantages of using the method of separating variables to solve PDEs?

The method of separating variables is advantageous because it allows for the solution of complex PDEs by breaking them down into simpler ODEs that can be solved using known techniques. It also provides a systematic approach to solving PDEs, making it easier to follow and understand the solution process.

What types of PDEs can be solved using the method of separating variables?

The method of separating variables can be used to solve linear, homogeneous PDEs with constant coefficients. These include the heat equation, wave equation, Laplace's equation, and others. Nonlinear and nonhomogeneous PDEs cannot be solved using this method.

What are the limitations of using the method of separating variables to solve PDEs?

The method of separating variables is limited to certain types of PDEs, as mentioned above. It also may not always provide a complete solution, as some PDEs may have additional boundary or initial conditions that need to be satisfied. In some cases, it may also be difficult to determine the appropriate initial or boundary conditions for the separated ODEs.

Are there any other methods for solving PDEs?

Yes, there are several other methods for solving PDEs, such as the method of characteristics, Fourier transform, and numerical methods. Each method has its own advantages and limitations, and the choice of method depends on the specific PDE and its properties.

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