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StLiBVP

# StLiBVP - Mathematics 421 Essay 3 Boundary Value Problems...

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Mathematics 421 Essay 3 Boundary Value Problems Spring 2008 0. Introduction A second order differential equation has a general solution containing two parameters. Typically these parameters are the values of the solution and its first derivative at a single point. Under suitable conditions, the theory predicts that such data leads to a unique solution. However, some natural questions lead to the value of the function at two different points being specified. In such questions, the function restricted to the interval between those points is the main object of interest, so these questions are called boundary value problems . If the equation is linear and homogeneous, and the given boundary values are zero, then a unique solution could only be the zero function. However, there is no uniqueness theorem for boundary value problems . Indeed, certain homogeneous equations with zero boundary values have non-trivial solutions. 1. The main example A typical example is d 2 y dx 2 C y D 0 I y.0/ D 0 I y.L/ D 0 for fixed L and a parameter . If < 0 , write D ˛ 2 . Then, the general solution of the differential equation is y D ae ˛x C be ˛x . The condition at x D 0 requires b D a , so the solution is a multiple of sinh x . This function is strictly increasing, so the condition at x D L allows only the zero function. If D 0 , the solution is y D a C bx , the condition at x D 0 gives a D 0 , and again the solution is a multiple of the increasing function y D x , and only b D 0 allows y.L/ D 0 . If > 0 , write D ˛ 2 . Then, the general solution of the differential equation is y D a cos ˛x C b sin ˛x , and the condition at x D 0 gives a D 0 . However, if ˛ D n =L , giving D n 2 2 =L 2 , all multiples of sin ˛x satisfy the condition at x D L . The functions that appear in these solutions are exactly the odd functions on L x L appearing in half range Fourier series . 2. Eigenfunctions There is something more general than Fourier series involved here. The boundary condition at each endpoints is allowed to be the requirement that some fixed linear combination of y and y 0 be zero at that point. For example, the boundary value problem d 2 y dx 2 C y D 0 I y.0/ D 0 I y.L/ C y 0 .L/ D 0 has a non-trivial solution of the form sin ˛x when sin ˛L C ˛ cos ˛L D 0 . There are still infinitely many such ˛ , but they are characterized by ˛ D tan ˛L — an equation that has one root in each interval of the form .k 1 = 2 / =L; .k C 1 = 2 / =L . As before, D ˛ 2 , but there is no simple expression for ˛ . It can be shown by the method used in the first example that only these positive values of allow nonzero solutions of the boundary value problem. In these problems, we are considering the effect of the linear operator d 2 =dx 2 on the linear space of functions defined on the interval OE0; LŁ satisfying certain homogeneous conditions at the boundary points x D 0 and x D L . We identified functions taken into multiples of themselves by the operator. In

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