2. Laplace transform solution of the diffusion equation In class, we solved Ju Ət U ди əx u(x, 0) J²u 8x²¹ → 0(x → ±∞o), kr → 0(x→ ±∞o), f(x) = using Fourier transforms. Here, consider the heat conduction in a half space (0 < x <∞o) for which the boundary at x = 0) is subject to a source of heat for t> 0 J²u dx² KT Ju Ət u (0, t) Uo, u(x, t) → 0, (x→∞0), u(t,0) 0 (4) (5) (6) (7) (8) (9) (10) (11)/nwhere u(r, t) is the temperature anomaly (difference of the temperature from a constant background temperature). Show that I u(r, t) = Uoler fe(2/krt)] '2√krt' (12) where er fc = 1- er f(x) is the complementary error function (see page 454 of Haberman). The Laplace transform pair may be useful: f(t) = erfc(2) f(s) = ¹ exp(-a√/s) 8

. Determine the set of points at which the function is continuous \text { a. } f(x, y)=\frac{1+x^{2}+y^{2}}{1-x^{2}-y^{2}} \text { b. } g(x, y)=\sqrt{x}+\sqrt{1-x^{2}-y^{2}}

Consider the regular Sturm-Liouville eigenvalue problem \left\{\begin{array}{l} -\left(x u_{x}\right)_{x}+\frac{2 u}{x}=\lambda \frac{u}{x} \text { for } 1<x<e^{x}, \\ u^{\prime}(1)=u^{\prime}\left(e^{\pi}\right)=0 . \end{array}\right. ) Determine the eigenvalue problem solved by v(y), where we define v(y)=u(x) \quad \text { with } y=\ln x \text {. } \phi_{n}(x)=\frac{\sqrt{2}}{\sqrt{\pi}} \cos (n \ln x) .

Consider the following ODE with given IC: Y^{\prime}(x)=x^{2} \cos (Y(x))^{2}, Y(0)=1 \text { and answer the following questions: } \text { What is } \frac{\partial f(x, z)}{\partial z} ? In what region of x will the solution exist? c) Find the analytical solution Y(x) and verify where it exists.

O Show that the Cauchy-Euler eigenvalue problem- equation (1) x^{2} \frac{d^{2} u}{d x^{2}}+x \frac{d u}{d x}+k^{2} u=0, \quad y(1)=y\left(e^{\pi}\right)=0 has eigenvalues given by k = n, n=1,2,... and eigenfunctions u(x) = sin (n ln(x)) (b) Put equation (1) into Sturm-Liouville form and thus identify the weight function[3 marks]w(x). Explain how we know that {sin ((n ln(x)), n = 1,2,...} is orthogonal over [1, e^] with-respect to the weight function found in part (b). (d) Find the corresponding generalised Fourier series for the function f(x): = x.Hint: in the integrals that you will have to solve, change the variable to 0 = ln(x)and use the maths handbook[11 marks]

2. Find the limit \lim _{(x, y) \rightarrow(3,2)}\left(y^{3} x^{2}-4 y^{2}\right)

2 Using equation (5.3b), find the Taylor series around xo= 1 of the function f(x) = e^−¹+2x+x² up to and including the term in h². Remember not to use any decimal points in your answer. f(xo + h) ~

A taut wire is held in place between two fixed points at x = 0 and x = 1.

2b Demonstrate how this function works using the domain, and range. (5 pts)

2. Laplace equation (Haberman § 2.5) Consider 8² u 8² u + əx² Əy² u(0, y) u(x, 0) u(1, y) = 0 (0<x< 1, 0<y<3) 0, u(x, 3) = 0 20 sin(x) 30 sin(Ty/3) (6) (8) (9) that describes the steady state temperature distribution in a rectangle subject to Dirichlet boundary conditions. (a) Using separation of variables, solve (6)-(9) for u(x, y). (b) Use Matlab to plot the isotherms (lines of constant u) for your solution in (a). See laplaccexallS23.m on Canvas for the example from lecture. (c) Briefly discuss your solution in the context of the maximum/minimum principle of the Laplace equation.