hw1ans - MATH 438 1 Homework 1(due at 11:00 am on Problem 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 438 1 Homework 1 (due at 11:00 am on January 29, 2008) Problem 1. Based on the understanding of the D’Alembert solution and the method of characteristics, solve the following problem for semi-infinite string: u tt- a 2 u xx = 0 , x > , t > ,a ∈ R , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) , x > , u (0 ,t ) = 0 , t > . Solution . Let u ( x,t ) = f ( x- at ) + g ( x + at ). Recall that in D’Alembert’s solution: g ( ξ ) = 1 2 φ ( ξ ) + 1 2 a Z ξ ψ ( s )d s + C, f ( ξ ) = 1 2 φ ( ξ )- 1 2 a Z ξ ψ ( s )d s- C, so that the D’Alembert solution is u ( x,t ) = φ ( x- at ) + φ ( x + at ) 2 + 1 2 a Z x + at x- at ψ ( s )d s. In our case we have a restriction at x = 0, from which we find that u (0 ,t ) dictates: f ( z ) + g (- z ) = 0 , z =- at < , since t > 0. So, for x- at < 0: f ( x- at ) =- g ( at- x ) =- φ ( at- x ) 2- 1 2 a Z at- x ψ ( s )d s- C. Hence, the solution is given by x > at : D Alembert s solution , < x < at : u ( x,t ) =- φ ( at- x ) + φ ( x + at ) 2 + 1 2 a Z x + at at- x ψ ( s )d s. Problem 2. Solve u tt- a 2 u xx = 0 , x > , t > ,a ∈ R , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) , x > , u (0 ,t ) = μ ( t ) , t > . MATH 438 2 Solution . It is enough to consider the case φ ( x ) = 0 and ψ ( x ) = 0 (Why?). In this case the only “disturbance” is coming from the boundary condition at x = 0, thus once can seek for a solution in the form...
View Full Document

{[ snackBarMessage ]}

Page1 / 5

hw1ans - MATH 438 1 Homework 1(due at 11:00 am on Problem 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online