MATHS 361: Partial Differential Equation Tutorial 9
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MATHS 361: Partial Differential Equation
Tutorial 9: Distributions and Weak solutions
Please find below a number of questions related to distributions and weak solutions. Attack the questions in which ever order seems most useful to you.
1. Distributions
(a) Write series of functions (or distributions) that will approach δ\ (x−2). (Hint: ϕ\ (2) = limh→0 ϕ(2+h−ϕ(2)) (b) Find the first and second distributional derivatives of f(x) = |x| + sgn(x). Your answers will
likely be defined piecewise, and may include dirac delta functions. (sgn(x) = +1 if x > 0 and sgn(x) = −1 if x < 0, sgn(x) = 0 if x = 0).
(c) Is it possible to have a distribution ξ that < ξ,ϕ >= sgn(ϕ(1) − ϕ(2)). Why/why not? (Look at the properties of distributions, see what you can figure out.)
2. Weak Solutions
Suppose I have the equation u北北 − u = δ(x − 1), with initial conditions u(0) = 0,u北 (0) = 1.
(a) What integral equation/s must u solve in order to to be a weak solution of u北北 − u = δ(x − 1).
(b) What are our boundary conditions at x = 1? How to the values of u and u北 at x = 1 − ϵ relate
to the values at x = 1 + ϵ .
(c) Solve u北北 − u = δ(x − 1) in the region −∞ < x < 1.
(d) Solve u北北 − u = δ(x − 1) in the region 1 < x < ∞ .
2023-06-28
Distributions and Weak solutions