The Geometry of Linear Algebra
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2019-12-10T14:36:17-08:00The Geometry of Linear Algebra
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Why do you want to classify solutions?
If we can classify a PDE that we are trying to solve, according to the scheme given below, it will help us extend qualitative knowledge that we have about the nature of solutions of similar PDEs to the current case. Most importantly, the types of initial conditions that are needed to ensure that our solution is unique vary according to the classification--see PDE Theorems. In physics situations, the classification is usually obvious: if there are two tâ€¦text/html2018-01-20T22:40:07-08:00Corinne Manoguebook:files:content:pdethms
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The Main Idea
In physics situations, the classification and types of boundary conditions are typically straightforward: if there are two time derivatives, the equation is hyperbolic and we will need two initial conditions on the entire spatial region to make the solution unique; if there is only a single time derivative, the equation is parabolic and we will need only a single initial condition; if the equation has no time derivatives, the equation is elliptic and the solutions are qualitativâ€¦text/html2018-01-20T22:30:29-08:00Corinne Manoguebook:linalg:pdethms
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Boundary and Initial Conditions for PDEstext/html2018-01-20T22:26:03-08:00Corinne Manoguebook:linalg.float:ch9float
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(ss)1. Important PDEs in Physics (ss)2. Classification of PDEs (ss)3. Boundary/Initial Conditions (ss)4. Separation of Variables--Process (ss)5. Sturm-Liouville Theorytext/html2018-01-20T22:25:14-08:00Corinne Manoguebook:linalg:ch9
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Part I: Linear Algebra
Chapter 9: Partial Differential Equations
* (ss)1. Important PDEs in Physics
* (ss)2. Classification of PDEs
* (ss)3. Boundary and Initial Conditions
* (ss)4. Separation of Variables--Process
* (ss)5. Sturm-Liouville Theorytext/html2018-01-20T11:53:36-08:00Corinne Manoguebook:files:content:sturm
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The Main Idea
When you use the separation of variables procedure on a PDE, you end up with one or more ODEs that are eigenvalue problems, i.e. they contain an unknown constant that comes from the separation constants. These ODEs are called Sturm-Liouville equations. By solving the ODEs for particular boundary conditions, we find particular allowed values for the eigenvalues. Furthermore, the solutions of the ODEs for these special boundary conditions and eigenvalues form an orthogonalâ€¦text/html2018-01-20T11:31:52-08:00Corinne Manoguebook:files:content:pdeimportant
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On this page you will find a list of most of the important PDEs in physics with their names. Notice that some of the equation have no time dependence, some have a first order time derivative, and some have a second order time derivative. This difference is the foundation of an important classification scheme. Also notice that the spatial dependence always comes in the form of the laplacian $\nabla^2$. This particular spatial dependence occurs because space is rotationally invariant.