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Solving 2D Laplace Equation with Boundary Conditions
- Thread starter dzi
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In summary, the conversation discusses the use of separation of variables and trial solutions to find an analytical solution for a 2D equation with boundary conditions. The person seeking help has already solved the equation numerically but needs an analytical solution for comparison. They later discover that their error was in their Matlab program, not in their calculations.
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CompuChip
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Try separation of variables:
[tex]U(x, y) = \xi(x) \eta(y)[/tex]
then find trial solutions for both, work out the product and impose the boundary conditions.
That is the (most common, probably not the only) way to do it. If you want us to help you locate your error or give you a complete answer, you'll have to show some work.
[tex]U(x, y) = \xi(x) \eta(y)[/tex]
then find trial solutions for both, work out the product and impose the boundary conditions.
That is the (most common, probably not the only) way to do it. If you want us to help you locate your error or give you a complete answer, you'll have to show some work.
- #3
dzi
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dear CompuChip..
i solved equation numerical with finite diferencial method..
and also needed analytical sollution for comparison..
finaly few minutes ago i found my error..
huhh..
it wasn't placed in calculation like i was thinkig,
but in my Matlab program which i used for drawing sollution of equation..
i appreciate on quick answer,
thanks..
i solved equation numerical with finite diferencial method..
and also needed analytical sollution for comparison..
finaly few minutes ago i found my error..
huhh..
it wasn't placed in calculation like i was thinkig,
but in my Matlab program which i used for drawing sollution of equation..
i appreciate on quick answer,
thanks..
FAQ: Solving 2D Laplace Equation with Boundary Conditions
What is the Laplace equation and why is it important in scientific research?
The Laplace equation is a partial differential equation that describes the steady-state behavior of a physical system. It is important in scientific research because it allows for the prediction of the behavior of a system based on its boundary conditions, making it useful in fields such as physics, engineering, and mathematics.
What are the boundary conditions in the 2D Laplace equation?
The boundary conditions in the 2D Laplace equation are the values of the dependent variable (such as temperature or electric potential) at the boundaries of the system. These values are known and used to solve for the behavior of the system within the boundaries.
How is the 2D Laplace equation solved?
The 2D Laplace equation is solved using mathematical methods such as separation of variables, the method of images, or the use of Green's functions. These methods involve breaking down the equation into simpler parts and using mathematical techniques to find a solution.
What are the applications of solving the 2D Laplace equation with boundary conditions?
The applications of solving the 2D Laplace equation with boundary conditions are numerous and diverse. It is used in fields such as fluid dynamics, electrostatics, heat transfer, and signal processing to model and predict the behavior of systems. It is also used in computer simulations to design and improve various technologies.
What are the limitations of the 2D Laplace equation with boundary conditions?
While the 2D Laplace equation is a useful tool in scientific research, it does have limitations. It can only be applied to systems that are in steady-state, meaning they are not changing over time. It also assumes that the system is linear, meaning that the behavior of the system is directly proportional to its input. Additionally, it may not accurately predict the behavior of complex systems with non-uniform boundaries or strong nonlinearities.