Solving PDEs involving complex variables

[masnet]

Hello all,
I was wondering if you could share your thoughts regarding how one should go about solving PDEs in which all or some of the variables are complex.
To solve ODEs involving real variables, my favorite method is to take the equations to Laplace domain, then solving the resulting set of linear equations. Taking an Inverse Laplace of the result would give the answer. Now, I don't know whether this method is equally valid, if the variables are complex.
For PDEs of 2 variables (one being complex), one method that I've seen in books, is the use of Method of Characteristics. As I've seen already this method being used for complex variables, I've been using it myself as well.
Now, checking the DE handbooks, I don't see a discussion about whether each method is still valid, if the variable is complex. My question is: what should I look for, or examine, in determining whether or not a method can be applied to complex variables?
My knowledge of complex variables is a bit rusty; hence the current confusion. I appreciate any thought that you could share regarding this.
Thank you in advance.

 

[jvicens]

Hello, thank you for your question. Solving PDEs involving complex variables can be a bit more complicated than solving ODEs with real variables. However, there are some methods that can be applied in this case.

One possible approach is to use the Cauchy-Riemann equations to transform the complex PDE into a system of real PDEs. This method is known as the method of holomorphic functions and is commonly used in fluid dynamics and electromagnetism.

Another approach is to use the method of separation of variables, where the complex variables are separated into real and imaginary parts, and then solving the resulting real PDEs using traditional methods.

The method of characteristics, as you mentioned, can also be used for PDEs involving complex variables. However, care must be taken to ensure that the characteristics are complex-valued and not real-valued.

In terms of determining whether a method can be applied to complex variables, it is important to check if the method relies on any assumptions or properties that are not valid for complex numbers. Additionally, it is important to check if the method is applicable to complex-valued functions and if the resulting solutions are physically meaningful.

I hope this helps to clarify some of your confusion. If you have any further questions, please don't hesitate to ask. Thank you.