A How Can Taylor Series Expansion Help Solve This Nonlinear Differential Equation?

[MelissaM]

ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo

i have this equation... i was thinking of taylor series expansion to solve it and make it easier...

any ideas on how to solve?

 

[Student100]

ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo

i have this equation... i was thinking of taylor series expansion to solve it and make it easier...

any ideas on how to solve?

Kinda hard to read what you wrote since it looks like it dropped your powers and possible subscripts, is this what you meant? $$\rho C_p(\frac{\partial T}{\partial t})+k(\frac{\partial^2T}{\partial x^2}) = e^{-\sigma t^2}e^{- \lambda x^2}\psi_0$$

I know how I would try to solve any PDE I encounter anymore, and it rhymes with mathematica. So I can't help you there. Where do you get when you try series expansion?

Care to give anymore context on the problem itself? Looks almost familiar.

 

[MelissaM]

Kinda hard to read what you wrote since it looks like it dropped your powers and possible subscripts, is this what you meant? $$\rho C_p(\frac{\partial T}{\partial t})+k(\frac{\partial^2T}{\partial x^2}) = e^{-\sigma t^2}e^{- \lambda x^2}\psi_0$$

I know how I would try to solve any PDE I encounter anymore, and it rhymes with mathematica. So I can't help you there. Where do you get when you try series expansion?

Care to give anymore context on the problem itself? Looks almost familiar.

Yes, this is exactly what the equation looks like.

I've never used mathematica... Context of problem: we have a pulsed laser that is heating up a pexiglas sheet, we want to determine by how much does the temperature rise per pulse. The exponentials refer to the heat source (laser) in gaussian form as a function of time and space.

that's what I understood.

As for taylor series (I'm not sure of my way of thinking, but I dropped all terms with power > 2), this is what I got:

ρC_p (∂_tT) - k(∂_xxT) = λσ(xt)² - σt²-λx²

i dropped the last two terms on the right hand side, so I'm left with: ρC_p (∂_tT) - k(∂_xxT) = λσ(xt)²

 

[MelissaM]

As for taylor series (I'm not sure of my way of thinking, but I dropped all terms with power > 2), this is what I got:

ρC_p (∂_tT) - k(∂_xxT) = λσ(xt)² - σt²-λx²

i dropped the last two terms on the right hand side, so I'm left with: ρC_p (∂_tT) - k(∂_xxT) = λσ(xt)²

coming to think about it, using taylor series is pointless because the integral of a gaussian in known...

I still need some guidance on how to approach it though

 

[MelissaM]

Kinda hard to read what you wrote since it looks like it dropped your powers and possible subscripts, is this what you meant? $$\rho C_p(\frac{\partial T}{\partial t})+k(\frac{\partial^2T}{\partial x^2}) = e^{-\sigma t^2}e^{- \lambda x^2}\psi_0$$

I know how I would try to solve any PDE I encounter anymore, and it rhymes with mathematica. So I can't help you there. Where do you get when you try series expansion?

Care to give anymore context on the problem itself? Looks almost familiar.

can you please share how you solve it on mathematica?

now I'm thinking of using separation of variables (obviously) with Fourier transform.

 

[the_wolfman]

You can solve this equation using separation of variables.

I'll also point out that this equation is linear.

 

[MelissaM]

You can solve this equation using separation of variables.

I'll also point out that this equation is linear.

ha, i meant partial diff equation (tried to fix the title didn't know how to)

So separation of variables and Fourier transform? Or just separation of variables?

 

[S.G. Janssens]

Context of problem: we have a pulsed laser that is heating up a pexiglas sheet, we want to determine by how much does the temperature rise per pulse. The exponentials refer to the heat source (laser) in gaussian form as a function of time and space.

You can solve this equation using separation of variables.

I'll also point out that this equation is linear.

ha, i meant partial diff equation (tried to fix the title didn't know how to)

So separation of variables and Fourier transform? Or just separation of variables?

So, to summarize: The problem concerns the inhomogeneous linear heat equation on a symmetric bounded spatial domain, say for $x \in [-L,L]$ where $2L$ is the length of the glass sheet. Does that make sense to you?

(It is inhomogeneous because there is the source term $\rho(t,x) := e^{-\sigma t^2}e^{-\lambda x^2}\psi_0$, where I assume $\psi_0$ is a constant. This source term separates into a time-dependent and a space-dependent part. Also, I assume the domain is bounded because you were talking about a glass sheet. Probably you would like to model this as a two-dimensional rectangle at some point, but it appears that you wish to solve the one-dimensional problem first.)

Since the domain is bounded, it is natural to use Fourier series. (Not Fourier transform.) However, you cannot apply separation of variables (at least, not directly) because of the source term $\rho$. In order to proceed and have the problem fully specified, it is now useful to know about your boundary conditions. Based on the physics, what conditions do you impose on $T(t,\pm L)$ and/or, possibly, the spatial derivatives $T_x(t,\pm L)$?

 

[MelissaM]

So, to summarize: The problem concerns the inhomogeneous linear heat equation on a symmetric bounded spatial domain, say for $x \in [-L,L]$ where $2L$ is the length of the glass sheet. Does that make sense to you?

(It is inhomogeneous because there is the source term $\rho(t,x) := e^{-\sigma t^2}e^{-\lambda x^2}\psi_0$, where I assume $\psi_0$ is a constant. This source term separates into a time-dependent and a space-dependent part. Also, I assume the domain is bounded because you were talking about a glass sheet. Probably you would like to model this as a two-dimensional rectangle at some point, but it appears that you wish to solve the one-dimensional problem first.)

Since the domain is bounded, it is natural to use Fourier series. (Not Fourier transform.) However, you cannot apply separation of variables (at least, not directly) because of the source term $\rho$. In order to proceed and have the problem fully specified, it is now useful to know about your boundary conditions. Based on the physics, what conditions do you impose on $T(t,\pm L)$ and/or, possibly, the spatial derivatives $T_x(t,\pm L)$?

Yes, you are 100% right. I used Fourier series, and it worked out.
The boundary conditions I used are: T(0,t)=T(L,t)=0 and T(x,0)=0.

And after performing Fourier series, i got a homogeneous differential equation where i used separation of variables.. At the end i got an integral dependent on 2 variables, that i will solve in a bit...

I think that should do it... I saw a method called eigenfunction expansion, I've never heard of it. Please share thoughts :)

 

[Chestermiller]

@MelissaM The differential equation you wrote is incorrect. There has to be a minus sign in front of the k, not a + sign. In addition, it might be worthwhile approximating the right hand side of the equation by a Dirac delta function in time, at x = 0. This would correspond to an infinitely high pulse of heat introduced at x = 0 between time t = 0 and time t = 0⁺ (so that a finite amount of heat is introduced). There is definitely an analytic solution to this problem for T. See Conduction of Heat in Solids by Carslaw and Jaeger.

 

[MelissaM]

@MelissaM The differential equation you wrote is incorrect. There has to be a minus sign in front of the k, not a + sign. In addition, it might be worthwhile approximating the right hand side of the equation by a Dirac delta function in time, at x = 0. This would correspond to an infinitely high pulse of heat introduced at x = 0 between time t = 0 and time t = 0⁺ (so that a finite amount of heat is introduced). There is definitely an analytic solution to this problem for T. See Conduction of Heat in Solids by Carslaw and Jaeger.

you are absolutely correct there is a minus sign (i solved it with a minus sign)... and if i put dirac delta function doesn't that mean that the solution is in terms of a green's function?

 

[Chestermiller]

you are absolutely correct there is a minus sign (i solved it with a minus sign)... and if i put dirac delta function doesn't that mean that the solution is in terms of a green's function?

Actually, I didn't express myself properly in my previous post. What I'm really saying is that the temperature profile should be a Dirac delta function with respect to x at time zero (representing the concentrated heat pulse at x = 0), and the differential equation should have a zero on the right hand side. The heat diffuses away from x = 0 at subsequent times, with the area under the curve of T vs x being constant (constant amount of heat). The form of the solution should be $$T-T_0=\frac{c}{\sqrt{t}}f\left(\frac{x^2}{\alpha t}\right)$$

 

[S.G. Janssens]

I like Mr. Miller's suggestion of using a spatial Dirac as an initial condition, but now I realize I am a bit confused by what the OP wrote here (emphasis is mine):

Context of problem: we have a pulsed laser that is heating up a pexiglas sheet, we want to determine by how much does the temperature rise per pulse. The exponentials refer to the heat source (laser) in gaussian form as a function of time and space.

If you have a pulsed laser, wouldn't it make more sense to replace the decaying $e^{-\sigma t}$ term by something periodic in time?

 

[Chestermiller]

I like Mr. Miller's suggestion of using a spatial Dirac as an initial condition, but now I realize I am a bit confused by what the OP wrote here (emphasis is mine):

If you have a pulsed laser, wouldn't it make more sense to replace the decaying $e^{-\sigma t}$ term by something periodic in time?

The pulsed solution would be the linear superposition of a sequence of pulses, with appropriate time offsets.

Another approximation would be to average the pulses with time and apply the heat over a small finite span of x.