Solving PDEs: Separation of Vars., Method of Characteristics

In summary, the conversation discusses the methods of solving PDEs, particularly linear ones. The two main methods mentioned are separation of variables and method of characteristics. It is mentioned that there are also transforms such as Laplace and Fourier, but not many analytical ways of solving PDEs. The conversation also asks for other methods and types of PDEs that exist, with the response stating that in real-life applications, PDEs are usually solved numerically rather than analytically. The challenges of fitting boundary conditions to complex shapes and the vast subject of numerical solutions for PDEs are also mentioned.
  • #1
Abraham
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I've taken a first semester course on PDEs. Basically all we learned was separation of variables and method of characteristics. I understand that there are transforms out there, such as laplace and fourier. However, it looks like there aren't many analytical ways of solving PDEs. Mind you, I'm only talking about linear PDEs. I know nothing of nonlinear ones.

Can anyone tell me what other methods there are to solve PDEs?

Anyone with more experience, can you tell me what other types of PDEs are out there? I know the heat, laplace, and wave eq.

Thx
 
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  • #2


In "real life" applications, PDEs are almost always solved numerically not analytically.

Even when there are general analytic solutions, it is usually impossible to fit the boundary conditions to "real" regions in space which are not simple rectangles, circles, etc. For example, imagine trying to solve something as "simple" as the wave equation analytically inside a region of 3D space shaped like a real automobile (including the seats, passengers, etc), to decide the most effective places to put sound insulation, loudspeakers for audio equipment, etc.

Numerical solution of PDEs is almost as big a subject area as studying the PDEs themselves. There is a lot more to it than the simple finite difference methods that you find in basic "computational methods" courses.
 

1. What is the "Separation of Variables" method for solving PDEs?

The separation of variables method is a technique used to solve partial differential equations (PDEs) by separating the dependent variables into simpler functions. This allows the PDE to be rewritten as a set of ordinary differential equations (ODEs), which can then be solved using integration techniques.

2. How does the "Separation of Variables" method work?

The separation of variables method involves assuming that the solution to the PDE can be written as a product of two or more simpler functions, each dependent on only one of the independent variables. These functions are then substituted into the original PDE, resulting in a set of ODEs. The ODEs can then be solved separately and combined to obtain the solution to the original PDE.

3. When is the "Method of Characteristics" used to solve PDEs?

The method of characteristics is typically used to solve first-order PDEs, specifically those that are linear and have constant coefficients. It is often used in problems involving transport phenomena, such as heat conduction and fluid flow.

4. How does the "Method of Characteristics" work?

The method of characteristics involves finding a set of characteristic curves, which are curves that satisfy the PDE and are used to determine the solution. These curves are found by setting up a system of equations using the PDE and the initial/boundary conditions. The solution is then obtained by following the characteristics to the point of interest.

5. What are the advantages and disadvantages of using "Separation of Variables" vs. the "Method of Characteristics" to solve PDEs?

The separation of variables method is a more general approach that can be used for a wider range of PDEs, but it may not always be possible to find suitable separated solutions. The method of characteristics is more specialized, but can provide a more direct and efficient solution in some cases. It is often a matter of trial and error to determine which method is most appropriate for a given problem.

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