How could I compare a SYSTEM of linear second-order partial differential equations with two different functions within them to the heat equation? Please also provide a reference that I can cite in my paper.

Question

In particular, for a paper, I am looking to compare this equation

#ie^(4omegat) (delPhi)/(delt)+(del^2Phi)/(dely^2) = 0#

to the forward heat equation in one dimension,

#(delu)/(delt)-(del^2u)/(delx^2) = 0#,

and the backward heat equation in one dimension,

#(delu)/(delt)+(del^2u)/(delx^2) = 0#,

where #omega# is a constant and #i# is the familiar imaginary unit.

My problem is, anytime I multiply by #i#, it can look like either the forward or backward heat equation, and I can't just have it look like either one arbitrarily...

I tried rewriting #Phi# in terms of parts with real and imaginary coefficients:

#\Phi(y,t) = N ext{exp}(\frac{i\epsilon}{4\omega}e^{-4\omega t})sin(\sqrt{\epsilon}y)#

#= N [cos(\frac{\epsilon}{4\omega}e^{-4\omega t})+isin(\frac{\epsilon}{4\omega}e^{-4\omega t})]sin(\sqrt{\epsilon}y)#

#= stackrel(Phi_{re})overbrace(Ncos(\frac{\epsilon}{4\omega}e^{-4\omega t})sin(\sqrt{\epsilon}y)) + istackrel(Phi_{im})overbrace(Nsin(\frac{\epsilon}{4\omega}e^{-4\omega t})sin(\sqrt{\epsilon}y))#

where #epsilon# and #N# are constants too. I could then write this as:

#= \Phi_{re} + i\Phi_{im}#

However, when I plug it back into the PDE, I get **a system of PDEs with mixed functions...**

#e^{4\omega t}\frac{\partial\Phi_{im}}{\partial t} -\frac{\partial^2\Phi_{re}}{\partial y^2} = 0#

#e^{4\omega t}\frac{\partial\Phi_{re}}{\partial t} + \frac{\partial^2\Phi_{im}}{\partial y^2} = 0#

How can I still compare to the forward and/or backward heat equation? Please help soon, this is due by Friday April 28 for a 15-page paper. I am almost done, except for this.

Classifying these wasn't a problem (they are both parabolic). It's the comparison to the heat equation that's giving me trouble.

Christopher Allers · Accepted Answer

#"See explanation"# #"Maybe my answer is not completely to the point, but i know"# #"about the "color(red)("Hopf-Cole transformation")."# #"The Hopf-Cole transformation is a transformation, which maps"# #"the solution of the "color(red)("Burgers equation")" to the "color(blue)("heat equation")."# #"Maybe you can find inspiration there."#

Isabella Ackerman · Answer

undefined To compare a system of linear second-order partial differential equations with two different functions to the heat equation, you can consider the similarities and differences in their forms, solutions, and physical interpretations. One approach is to examine the coefficients and structures of the equations, as well as the boundary or initial conditions they are subject to. You may also analyze how the solutions of the system evolve over time and space compared to the solutions of the heat equation.

A reference you can cite in your paper is:

"Partial Differential Equations for Scientists and Engineers" by Stanley J. Farlow. This textbook provides comprehensive coverage of various types of partial differential equations, including linear second-order equations, and discusses their applications in physics and engineering. Specifically, Chapter 2 of the book covers the heat equation and its properties, which can serve as a basis for comparison with other linear second-order partial differential equations.