Solving PDEs involving complex variables

In summary, there are a few different methods that can be used to solve PDEs involving complex variables, such as the method of holomorphic functions, separation of variables, and the method of characteristics. It is important to carefully consider the properties and assumptions of each method when applying them to complex numbers.
  • #1
masnet
14
0
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.
 
Physics news on Phys.org
  • #2


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.
 

1. What are complex variables and how are they used in solving PDEs?

Complex variables are numbers that have both real and imaginary components. They are used in solving PDEs as they allow for the representation of physical quantities that cannot be described by real numbers alone. In particular, complex variables are used to represent solutions to PDEs that involve oscillatory or exponential behavior.

2. What is the difference between a complex-valued function and a real-valued function?

A complex-valued function is a function that takes complex numbers as inputs and outputs complex numbers. On the other hand, a real-valued function takes real numbers as inputs and outputs real numbers. In the context of solving PDEs, complex-valued functions are used to represent solutions to PDEs that involve complex variables.

3. How do we solve PDEs involving complex variables?

Solving PDEs involving complex variables typically involves using techniques such as separation of variables, the method of characteristics, or the method of finite differences. These methods allow us to break down the PDE into simpler equations that can then be solved using standard techniques for complex variables.

4. What is the importance of boundary conditions in solving PDEs involving complex variables?

Boundary conditions are crucial in solving PDEs involving complex variables as they help determine the specific solution to the PDE. In particular, boundary conditions for complex variables can involve specifying the behavior of the solution along complex curves or specifying the values of the solution at certain complex points.

5. Can PDEs involving complex variables be solved numerically?

Yes, PDEs involving complex variables can be solved numerically using methods such as finite difference methods, finite element methods, or spectral methods. These methods discretize the complex variable domain and approximate the solution to the PDE using a finite number of complex points. However, the convergence and stability of these numerical methods can be more challenging to analyze compared to PDEs involving only real variables.

Similar threads

I Understanding the change of variables for PDEs
    • Differential Equations
Replies
2
Views
2K
I PDEs: Diffusion Equation Change of Variables
    • Differential Equations
Replies
1
Views
2K
I PDEs: Laplace's Equation over a Parallelogram
    • Differential Equations
Replies
10
Views
3K
I Real ODE yields real solution through complex numbers
    • Differential Equations
Replies
3
Views
2K
A FEM for Solving PDEs: Beginner Textbooks & Fluid Flow
    • Differential Equations
Replies
5
Views
2K
Prob/Stats Material on complex random variables and exotic probabilities
    • Science and Math Textbooks
Replies
1
Views
681
MHB Change variable to solve equation
    • Differential Equations
Replies
14
Views
2K
I Heat Equation: Solve with Non-Homogeneous Boundary Conditions
    • Differential Equations
Replies
4
Views
1K
Mathematical Methods Book That Uses SI Units
    • Science and Math Textbooks
Replies
9
Views
1K
MHB Matrix Methods for Difference Problems
    • Differential Equations
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top