Separation of Variables, but not equal to constant

In summary, Hamidi's equation is separable if and only if ##\frac{\partial q}{\partial t} = \frac{\partial q}{\partial t} \frac{\partial Q}{\partial q}##.
  • #1
Foracle
30
8
TL;DR Summary
Is the following PDE separable
$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$
where ##\frac{\partial q}{\partial t}## can depend on ##q## and ##t##?
Suppose I have 2 variables q and t (time), where q is some reparameterization of x (position) : ##x \to q = x f(t)##.

Suppose I have a partial differential equation :
$$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$
where k = constant
Then I do a separation of variables ## u(q,t) = Q(q)T(t) ##
The differential equation becomes (after some manipulation):
$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$
where I have used the fact that ##\frac{\partial Q}{\partial t} = \frac{\partial q}{\partial t} \frac{\partial Q}{\partial q}##

Now the left hand side is only dependent on ##t##, while the right hand side depends on both ##q## and ##t##. Since both sides still depend on ##t##, can I say that
$$(LHS) = (RHS) = g(t)$$
(LHS = Left hand side, RHS = Right hand side, g(t) is some function of time)

Additional question :
I have seen on a research paper where the author says that for the above equation to be separable, ##\frac{\partial q}{\partial t}## has to be constant so that RHS only depends on ##q##, hence ##(LHS) = (RHS) = constant##.
But since ##q## still depends on time (##q = x f(t)##), doesn't this mean RHS still depends on time and it should be ##(LHS) = (RHS) = g(t)## instead?
 
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  • #2
Foracle said:
Summary:: Is the following PDE separable
$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$
where ##\frac{\partial q}{\partial t}## can depend on ##q## and ##t##?

Suppose I have 2 variables q and t (time), where q is some reparameterization of x (position) : ##x \to q = x f(t)##.

Suppose I have a partial differential equation :
$$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$
where k = constant
Then I do a separation of variables ## u(q,t) = Q(q)T(t) ##
The differential equation becomes (after some manipulation):
$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$
where I have used the fact that ##\frac{\partial Q}{\partial t} = \frac{\partial q}{\partial t} \frac{\partial Q}{\partial q}##

If by [itex]\dfrac{\partial}{\partial t}[/itex] you mean differentiation with respect to [itex]t[/itex] with [itex]q[/itex] held constant, then by definition [tex]
\frac{\partial q}{\partial t} = 0[/tex] and [itex]Q[/itex] depends on [itex]q[/itex] alone.

Now the left hand side is only dependent on ##t##, while the right hand side depends on both ##q## and ##t##. Since both sides still depend on ##t##, can I say that
$$(LHS) = (RHS) = g(t)$$
(LHS = Left hand side, RHS = Right hand side, g(t) is some function of time)

Additional question :
I have seen on a research paper where the author says that for the above equation to be separable, ##\frac{\partial q}{\partial t}## has to be constant so that RHS only depends on ##q##, hence ##(LHS) = (RHS) = constant##.
But since ##q## still depends on time (##q = x f(t)##), doesn't this mean RHS still depends on time and it should be ##(LHS) = (RHS) = g(t)## instead?

You need to provide more context. Are you changing variables from [itex](x,t)[/itex] to [itex](q = xf(t),t)[/itex]? If so, differentiation with respect to [itex]t[/itex] with [itex]x[/itex] held constant is not the same thing as differentiation with respect to [itex]t[/itex] with [itex]q[/itex] held constant, and you need to be clear which of the two you mean when you write [itex]\dfrac{\partial}{\partial t}[/itex].
 
  • #3
pasmith said:
Are you changing variables from (x,t) to (q=xf(t),t)?
Yes, I am doing that.
pasmith said:
you need to be clear which of the two you mean when you write ##\frac{∂}{∂t}##
When I do ##\frac{\partial u(q,t)}{\partial t}##, I am not holding anything constant. So
$$\frac{\partial }{\partial t} = \frac{\partial q}{\partial t} \frac{\partial}{\partial q} + \frac{\partial}{\partial t}$$
The ##\frac{\partial}{\partial t}## on the 2nd term of the RHS holds q constant.
 
  • #4
I have another question that is kind of related.
Untitled.png

I am currently studying this research paper by O. Hamidi. What bothers me is in the equation (5), why did he choose to write ##\tau (x,t) = G(x) + K(t)## instead of the usual separation of variables ##\tau (x,t) = G(x)K(t)##?
Both substitutions give different solution, and both methods seem good to me. Why do people use one over the other?
 

Related to Separation of Variables, but not equal to constant

1. What is the concept of separation of variables?

The concept of separation of variables is a mathematical technique used to solve partial differential equations. It involves breaking down a complex equation into simpler equations that can be solved separately. This allows for a more systematic and efficient approach to finding solutions.

2. How is separation of variables different from solving for a constant?

Separation of variables involves breaking down an equation into simpler equations that can be solved separately, while solving for a constant involves finding a single value that satisfies the equation. In separation of variables, the variables are not equal to a constant, but rather are separated into distinct parts.

3. When is separation of variables used in scientific research?

Separation of variables is commonly used in fields such as physics, engineering, and mathematics to solve partial differential equations. It is often used to model physical systems and predict their behavior, such as in fluid dynamics or heat transfer.

4. What are the benefits of using separation of variables?

One of the main benefits of using separation of variables is that it simplifies complex equations into smaller, more manageable parts. This allows for a more systematic and efficient approach to finding solutions. It also allows for a deeper understanding of the underlying mathematical principles involved.

5. Are there any limitations to using separation of variables?

While separation of variables is a powerful technique, it is not always applicable to all types of equations. It is typically used for linear equations with constant coefficients, and may not work for more complex equations. Additionally, the process of separating variables can sometimes lead to the loss of certain solutions, so caution must be taken when using this technique.

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