Solving a Partial Differential Equation

Thread starter YongL
Start date Apr 8, 2008
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Differential Differential equation Partial

In summary, a partial differential equation (PDE) is a mathematical equation involving multiple variables and their partial derivatives, commonly used to describe physical phenomena. The steps for solving a PDE include determining the type of PDE, transforming it into a standard form, choosing an appropriate solution technique, solving the resulting equations, and verifying the solution. PDEs differ from ordinary differential equations (ODEs) in that they involve multiple independent variables and are used to model systems with varying physical properties. Real-world applications of PDEs include heat transfer, fluid dynamics, and electrical circuits. There are various software programs and tools available for solving PDEs, such as MATLAB, Mathematica, and COMSOL Multiphysics, which use numerical methods

Apr 8, 2008

YongL

A partial differential equation.

Last edited: Apr 8, 2008

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Apr 8, 2008

Roberto Bramb

It is the Laplace equation in cylindrical coordinates with symmetry about y-axe.
You can solve it by variable separation, once given the boundary condition:
phi(x,y)=X(x)Y(y)

X''+X'/x+cX=0 (in x, 0-order Bessel equation)
Y''-cY=0 (in y)
c=arbitrary positive/negative real constant

Apr 9, 2008

YongL

Thanks, roberto, i got it

1. What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple variables and their partial derivatives. It is used to describe physical phenomena such as heat transfer, fluid dynamics, and electromagnetic fields.

2. What are the steps for solving a PDE?

The steps for solving a PDE are: 1) Determine the type of PDE (e.g. elliptic, parabolic, hyperbolic), 2) Transform the PDE into a standard form, 3) Choose an appropriate solution technique (e.g. separation of variables, method of characteristics), 4) Solve the resulting equations, and 5) Verify the solution satisfies any boundary or initial conditions.

3. What is the difference between a PDE and an ordinary differential equation (ODE)?

A PDE involves multiple independent variables and their partial derivatives, while an ODE involves only one independent variable and its derivatives. PDEs are used to model systems with varying physical properties, while ODEs are used to model systems with constant physical properties.

4. What are some real-world applications of PDEs?

PDEs have numerous applications in physics, engineering, and other scientific fields. Some examples include modeling heat transfer in a solid, predicting fluid flow around an object, and simulating the behavior of electrical circuits.

5. Are there any software programs or tools available for solving PDEs?

Yes, there are many software programs and tools available for solving PDEs, such as MATLAB, Mathematica, and COMSOL Multiphysics. These programs use numerical methods to solve PDEs and allow for visualization of the solution.