Paper 1, Section II, E

Further Complex Methods
Part II, 2011

(i) By assuming the validity of the Fourier transform pair, prove the validity of the following transform pair:

q^(k)=0eikxq(x)dx,Imk0,q(x)=12πeikxq^(k)dk+c2πLeikxq^(k)dk,0<x<,\begin{gathered} \hat{q}(k)=\int_{0}^{\infty} e^{-i k x} q(x) d x, \quad \operatorname{Im} k \leqslant 0, \\ q(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i k x} \hat{q}(k) d k+\frac{c}{2 \pi} \int_{L} e^{i k x} \hat{q}(-k) d k, \quad 0<x<\infty, \end{gathered}

where cc is an arbitrary complex constant and LL is the union of the two rays arg k=π2k=\frac{\pi}{2} and argk=0\arg k=0 with the orientation shown in the figure below:

The contour LL.

(ii) Verify that the partial differential equation

iqt+qxx=0,0<x<,t>0,i q_{t}+q_{x x}=0, \quad 0<x<\infty, t>0,

can be rewritten in the following form:

(eikx+ik2tq)t[eikx+ik2t(kq+iqx)]x=0,kC.\left(e^{-i k x+i k^{2} t} q\right)_{t}-\left[e^{-i k x+i k^{2} t}\left(-k q+i q_{x}\right)\right]_{x}=0, \quad k \in \mathbb{C} .

Consider equation (2) supplemented with the conditions

q(x,0)=q0(x),0<x<q(x,t) vanishes sufficiently fast for all t as x\begin{aligned} &q(x, 0)=q_{0}(x), \quad 0<x<\infty \\ &q(x, t) \text { vanishes sufficiently fast for all } t \text { as } x \rightarrow \infty \end{aligned}

By using equations (1a) and (3), show that

q^(k,t)=eik2tq^0(k)+eik2t[kg~0(k2,t)ig~1(k2,t)],Imk0\hat{q}(k, t)=e^{-i k^{2} t} \hat{q}_{0}(k)+e^{-i k^{2} t}\left[k \tilde{g}_{0}\left(k^{2}, t\right)-i \tilde{g}_{1}\left(k^{2}, t\right)\right], \operatorname{Im} k \leqslant 0

where

q^0(k)=0eikxq0(x)dx,Imk0\hat{q}_{0}(k)=\int_{0}^{\infty} e^{-i k x} q_{0}(x) d x, \quad \operatorname{Im} k \leqslant 0

Part II, 20112011 \quad List of Questions

[TURN OVER

g~0(k,t)=0teikτq(0,τ)dτ,g~1(k,t)=0teikτqx(0,τ)dτ,kC,t>0.\tilde{g}_{0}(k, t)=\int_{0}^{t} e^{i k \tau} q(0, \tau) d \tau, \quad \tilde{g}_{1}(k, t)=\int_{0}^{t} e^{i k \tau} q_{x}(0, \tau) d \tau, k \in \mathbb{C}, t>0 .

Use (1b) to invert equation (5) and furthermore show that

eikxik2t[kg~0(k2,t)+ig~1(k2,t)]dk=Leikxik2t[kg~0(k2,t)+ig~1(k2,t)]dk,t>0,x>0\int_{-\infty}^{\infty} e^{i k x-i k^{2} t}\left[k \tilde{g}_{0}\left(k^{2}, t\right)+i \tilde{g}_{1}\left(k^{2}, t\right)\right] d k=\int_{L} e^{i k x-i k^{2} t}\left[k \tilde{g}_{0}\left(k^{2}, t\right)+i \tilde{g}_{1}\left(k^{2}, t\right)\right] d k, t>0, x>0

Hence determine the constant cc so that the solution of equation (2), with the conditions (4) and with the condition that either q(0,t)q(0, t) or qx(0,t)q_{x}(0, t) is given, can be expressed in terms of an integral involving q^0(k)\hat{q}_{0}(k) and either g~0\tilde{g}_{0} or g~1\tilde{g}_{1}.