quZz said:Are a and b constants? If so, you need to change the x-scale: x = x' * b/a and get the standard Poisson's equation.
PDE stands for partial differential equation. It is a type of mathematical equation that involves multiple variables and their partial derivatives. PDEs are commonly used in physics, engineering, and other scientific fields to model complex systems.
Separation of variables is a method used to solve PDEs by breaking the equation into simpler parts that can be solved individually. However, not all PDEs can be solved using this method. Some PDEs are too complex or do not have separable solutions.
Yes, there are other methods for solving PDEs, such as using numerical methods, integral transforms, or series solutions. The method that is most appropriate for a specific PDE depends on its complexity and the desired level of accuracy.
One approach is to look for symmetries or patterns in the PDE, which may suggest a possible solution method. It can also be helpful to consult with colleagues or reference materials, as well as to break the PDE down into simpler parts and solve them individually.
Yes, it is not uncommon to come across PDEs that cannot be solved by separation of variables. Many real-world systems are highly complex and do not have simple mathematical solutions. This is where numerical and other methods become essential tools for solving PDEs.