CS 205A class_18 notes

Scientific Computing

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CS205 – Class 18 Covered in Class : 1, 2, 3, 4 Readings : Heath 11.1-11.2 Partial Differential Equations 1. There are three types of PDE’s a. Elliptic, Hyperbolic, Parabolic 2. The Laplace Equation is 2 2 2 1 n i i p p pf x = ∇= Δ = = is the model elliptic equation a. For PDE’s we often use subscript notation for derivatives. E.g. in 2D it is xx yy p += , in 1D it is xx p f = . b. Let’s look at a simpler case where f =0. c. In 1D we have 0 xx p = . The solution analytically is a straight line, i.e., p ax b = + . d. To determine the exact formula for p we need certain constraints. Those usually come in the form of boundary conditions, i.e. constraints on the value of p or its derivatives for values of x belonging to the boundary of the domain in which we want to solve. e. We can also have boundary conditions which involve the derivatives of the function p , called Neumann boundary conditions. We note that ( )( ) 01 p pb ′′ == , thus if we supply Neumann conditions for all boundary points then the function p is only determined up to an additive constant. In order to determine the exact function p we must supply Dirichlet conditions for at least one of the boundary points i. Take this plot 0.2 0.4 0.6 0.8 1 1 2 3 4 5 which represents the solution to the 1D Laplace equation. 1. We can get it by specifying two Dirichlet conditions (0) 2 p = and (1) 3 p = . 2. We can also get it by specifying one Dirichlet p(0)=2 and one Neumann p’(0)=1 (or indeed anywhere in 1D i.e. () 1 x pt = for any t). 3. Specifying both Neumann end points i.e. (0) x p and x p is troublesome. a. They need to be the same, because our solution is a line. Similar restrictions occur in multi-D. This is called the compatibility condition .
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b. Additionally only can determine p
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CS 205A class_18 notes - CS205 Class 18 Covered in Class:...

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