Mathematics Asked by Bob Pen on December 3, 2020

If you are given a nonlinear PDE

$$frac{partial c}{partial t} = Dnabla^2c + alpha c , , ,, vec{r} in Omega, ,, , tgt 0 , , , , (1)$$

where $D, alpha$ are constants.

And then you are introduced a function defined by:

$$phi(vec{r}, t) = c(vec{r}, t)e^{-alpha t} , , , , (2)$$

where $c$ satisfies $(1)$.

You are asked to **derive the PDE for the function $phi$**

How does one solve this? My attempt is given below, but I have no proper idea of what I am doing.

I find a way to express $c$ in terms of $phi$:

$$c = phi e^{alpha t} , , , , (3)$$

Then I just insert $(3)$ into $(1)$ to get:

$$frac{partial(phi e^{alpha t})}{dt} = Dnabla^2phi e^{alpha t} + alpha phi e^{alpha t} , , , , (4)$$

Is this the solution, or are there more steps to the problem? If someone could give me a guiding hand I would appreciate it very much.

$$e^{alpha t}frac{partialphi}{partial t}+ alpha phi e^{alpha t} = Dnabla ^2 phi e^{alpha t} + alpha phi e^{at} $$

Dividing by $e^{alpha t}$

$$frac{partialphi}{partial t}+ alpha phi = Dnabla ^2 phi + alpha phi $$

Taking $- alpha phi$ on both sides

$$frac{partialphi}{partial t} – Dnabla ^2 phi = 0 $$

$$frac{partialphi}{partial t} – Dfrac{partial^2phi}{partial x^2} = 0 $$

Your new solution is right. You may recognize $phi=c e^{-alpha t}$ from the method of integrating factors for solving the ODE in $t$, $$ frac{dc}{dt}(t) - alpha c(t) = f(t).$$ Your exercise is essentially an application of this to the new setting of a certain PDE. It somehow worked out because $nabla^2 phi = (nabla^2 c)e^{-alpha t}$, by which I mean that this change of variables simplifies the equation from $(partial_t - alpha -Dnabla^2)c=0$ to the same equation (for $phi$) but with $alpha=0$. Since knowing $phi$ tells you everything about $c$, it suffices to study the $alpha=0$ case.

This is not what happens if $alpha=alpha(x)$ was a function of $x$; going through the same motions, if we set $phi(x,t) = c(x,t)e^{-a(x)t}$, we get instead $$e^{alpha(x)t}partial_t phi =(partial_t-alpha(x)) (phi e^{alpha(x)t}) = Dnabla^2(phi e^{alpha(x)t}) = e^{alpha(x)t}Dleft[nabla^2 phi + 2nablaalphacdotnabla phi + phi nabla^2 alpha + phi|nabla alpha|^2right]$$ which does not lead to a simpler equation.

Correct answer by Calvin Khor on December 3, 2020

2 Asked on December 6, 2021 by 666user666

1 Asked on December 6, 2021

dirac delta distribution theory noise poissons equation stochastic pde

1 Asked on December 6, 2021 by ajit-kumar

improper integrals integration laplace transform real analysis

4 Asked on December 6, 2021 by tnxy

1 Asked on December 6, 2021 by user287133

2 Asked on December 6, 2021 by yan-qin

1 Asked on December 6, 2021 by sirrahe73

0 Asked on December 6, 2021

measure theory problem solving sequences and series solution verification summation

3 Asked on December 6, 2021

1 Asked on December 6, 2021 by jeppe-stig-nielsen

1 Asked on December 6, 2021

complex numbers hermitian matrices linear algebra vector spaces

1 Asked on December 6, 2021

2 Asked on December 6, 2021 by cand

algebra precalculus calculus derivatives exponential function factorial

2 Asked on December 6, 2021

1 Asked on December 6, 2021

1 Asked on December 6, 2021 by hai-smit

calculus functional analysis functions maxima minima real analysis

2 Asked on December 6, 2021

1 Asked on December 6, 2021 by cronus

Get help from others!

Recent Questions

- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?
- Does Google Analytics track 404 page responses as valid page views?

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir