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Julio1
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Let the PDE $u_{xx}-4u_{xy}+4u_{yy}=0.$ Reduce to the canonical form.Good Morning MHB :). My problem is find the canonical form of the PDE know an variable change. But how I can transform the equation? Thanks.
Julio said:Let the PDE $u_{xx}-4u_{xy}+4u_{yy}=0.$ Reduce to the canonical form.Good Morning MHB :). My problem is find the canonical form of the PDE know an variable change. But how I can transform the equation? Thanks.
evinda said:(Wave)
$u_{xx}-4u_{xy}+4u_{yy}=0 \Rightarrow \left(\frac{\partial^2}{\partial{x}^2} -4 \frac{\partial^2}{\partial{xy}}+4 \frac{\partial^2}{\partial{y}^2} \right) u=0 $
Find the discriminant of $\frac{\partial^2}{\partial{x}^2} -4 \frac{\partial^2}{\partial{xy}}+4 \frac{\partial^2}{\partial{y}^2}=0$ in order to find the characteristic equations.
Julio said:Thanks evinda :).
But why the discriminant? I don't understand this :(.
One question: What's up if does the change $\xi=\xi(x,y)$ and $\eta=\eta(x,y)$?
Partial differential equations (PDEs) are mathematical equations that involve multiple variables and their partial derivatives. They are used to describe the behavior of complex systems that vary in space and time, such as fluid dynamics, heat transfer, and electromagnetism.
The main difference between ordinary differential equations (ODEs) and PDEs is that while ODEs involve derivatives with respect to a single independent variable, PDEs involve derivatives with respect to multiple independent variables. This makes PDEs more suitable for describing systems that vary in multiple dimensions.
PDEs are used in a wide range of scientific and engineering fields, including physics, engineering, biology, economics, and finance. They can be used to model heat transfer, fluid flow, electrostatics, quantum mechanics, and many other phenomena.
There are various numerical and analytical methods for solving PDEs, depending on the specific equation and its boundary conditions. Some common methods include finite difference, finite element, and spectral methods. Analytical solutions can also be obtained for certain simplified PDEs.
One of the main challenges in solving PDEs is that they often have no closed-form analytical solution, so numerical methods must be used. Additionally, PDEs can be highly nonlinear and may have multiple solutions, making it difficult to find the most accurate solution. Another challenge is that PDEs can be computationally intensive, requiring powerful computers and advanced algorithms to solve them efficiently.