Lecture 9 - 9 Boundary Value Problems Collocation We now...

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9 Boundary Value Problems: Collocation We now present a different type of numerical method that will yield the approximate solution of a boundary value problem in the form of a function, as opposed to the set of discrete points resulting from the methods studied earlier. Just like the finite difference method, this method applies to both one-dimensional (two-point) boundary value problems, as well as to higher-dimensional elliptic problems (such as the Poisson problem). We initially limit our discussion to the one-dimensional case. Assume we are given a general linear two-point boundary value problem of the form Ly ( t ) = f ( t ) , t [ a, b ] , y ( a ) = α, y ( b ) = β. (81) To keep the discussion as general as possible, we now let V = span { v 1 , . . . , v n } denote an approximation space we wish to represent the approximate solution in. We can think of V as being, e.g., the space of polynomials or splines of a certain degree, or some radial basis function space (see more below). We will express the approximate solution in the form y ( t ) = n j =1 c j v j ( t ) , t [ a, b ] , with unknown coefficients c 1 , . . . , c n . Since L is assumed to be linear we have Ly = n j =1 c j Lv j , and (81) becomes n j =1 c j Lv j ( t ) = f ( t ) , t [ a, b ] , (82) n j =1 c j v j ( a ) = α, n j =1 c j v j ( b ) = β. In order to determine the n unknown coefficients c 1 , . . . , c n in this formulation we impose n collocation conditions to obtain an n × n system of linear equations for the c j . The last two equations in (82) ensure that the boundary conditions are satisfied, and give us the first two collocation equations. To obtain the other n - 2 equations we choose n - 2 collocation points t 2 , . . . , t n - 1 , at which we enforce the differential equation. As in the previous numerical methods, this results in a discretization of the differential equation. 94
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If we let t 1 = a and t n = b , then (82) becomes n j =1 c j v j ( t 1 ) = α, n j =1 c j Lv j ( t i ) = f ( t i ) , i = 2 , . . . , n - 1 , n j =1 c j v j ( t n ) = β. In matrix form we have the linear system v 1 ( t 1 ) v 2 ( t 1 ) . . . v n ( t 1 ) Lv 1 ( t 2 ) Lv 2 ( t 2 ) . . . Lv n ( t 2 ) . . . . . . Lv 1 ( t n - 1 ) Lv 2 ( t n - 1 ) . . . Lv n ( t n - 1 ) v 1 ( t n ) v 2 ( t n ) . . . v n ( t n ) c 1 c 2 . . . c n = α f ( t 2 ) . . . f ( t n - 1 ) β . (83) If the space V and the collocation points t i , i = 1 , . . . , n , are chosen such that the collocation matrix in (83) is nonsingular then we can represent an approximate solution of (81) from the space V uniquely as y ( t ) = n j =1 c j v j ( t ) , t [ a, b ] . Remark Note that this provides the solution in the form of a function that can be evaluated anywhere in [ a, b ]. No additional interpolation is required as was the case with the earlier methods. 9.1 Radial Basis Functions for Collocation The following discussion will apply to any sufficiently smooth admissible radial basic function. However, other basis functions such as polynomials or splines are also fre- quently used for collocation. In particular, the use of polynomials leads to so-called spectral or pseudo-spectral methods (see Chapter 11).
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