Efficiently Solve PDE Systems: Expert Tips and Solutions | Help Needed

In summary, the conversation involves a person seeking help with a system of partial differential equations. They have found a solution to one of the equations, but are unsure if it is the most general one. They ask for clarification and someone suggests a simplification that involves an arbitrary function. The person thanks them for the answer but clarifies that they need the solution to depend on only one variable due to dealing with (anti)holomorphic functions. The responder suggests making the arbitrary function constant to meet this condition.
  • #1
L0r3n20
36
2
Hi all! I'm stuck with a system of PDE. I'm not sure I want to write it here in full, so l'll write just one of them. I've found a solution to this equation but I'm not sure it's the most general one since when I plug this solution into the other eqs, I get a trivility condition for the coefficients
[tex] 2\bar{k}^1\left(\bar{s},\bar{t},\bar{u}\right)-2 k^1\left(s,t,u\right)+\left(s-\bar{s}\right)\left(\partial_s k^1\left(s,t,u\right) + \bar{\partial}_{\bar{s}}\bar{k}^1\left(\bar{s}, \bar{t}, \bar{u}\right)\right) =0 [/tex]
Can someone help?
 
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  • #2
Not sure I've understood the equation. Is this simplification valid:
2u(x) - 2v(y) + (y-x)(∂v/∂y + ∂u/∂x) = 0
?
If so:
∂v/∂y + ∂u/∂x = 2(u-v)/(x-y)
Consider (u, v+v') is also a solution. So
∂v'/∂y = 2v'/(y-x)
v' = (y-x)2f(x)
where f is an arbitrary function of x.
Does that help?
 
  • #3
Thank you for the answer. Ok that was useful, at least a bit. In fact, although it is correct, I need a u and a v depending ONLY from one of the two variables (x and y). In fact I'm dealing with (anti)holomorphic functions and I need them to respect the holomorphicity condition.
 
  • #4
L0r3n20 said:
Thank you for the answer. Ok that was useful, at least a bit. In fact, although it is correct, I need a u and a v depending ONLY from one of the two variables (x and y). In fact I'm dealing with (anti)holomorphic functions and I need them to respect the holomorphicity condition.
OK, so make f(x) constant. You can do likewise for (u+u', v).
 
  • #5
Even thought I set f(x) constant I get the x-dependence from y-x, right?
 

1. What is a PDE system?

A PDE (partial differential equation) system is a set of equations that involve multiple variables and their partial derivatives. These equations are often used to describe physical phenomena in fields such as physics, engineering, and mathematics.

2. How do you solve a PDE system?

The process of solving a PDE system involves identifying the type of PDE (e.g. elliptic, parabolic, hyperbolic), choosing an appropriate method (e.g. separation of variables, numerical methods), and applying boundary and initial conditions to obtain a solution.

3. What are some applications of PDE systems?

PDE systems have numerous applications in various fields, including fluid dynamics, heat transfer, electromagnetism, and quantum mechanics. They are also used in the development of mathematical models for complex systems and in computer simulations.

4. Are there any challenges or limitations when dealing with PDE systems?

Yes, there are several challenges and limitations when working with PDE systems. These include the nonlinearity of some PDEs, the difficulty of finding exact solutions, and the high computational cost of solving PDE systems numerically.

5. Can PDE systems be used to model real-world problems?

Yes, PDE systems are commonly used to model and solve real-world problems in various fields. They are especially useful for describing phenomena that involve multiple variables and complex interactions, making them a powerful tool for scientific research and engineering applications.

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