show that the solution which satisfies u(0, t) = T1 and u(L, t) = T2 and the initial condition u(x, 0) = f(x), is u(x, t) = T1 + (T2 T1)x L + 1X n=1 bn sin ⇣n⇡x L ⌘ exp  n 2⇡2 L2 t where bn = 2 L Z L 0 sin ⇣n⇡x L ⌘ h f(x) T1 (T2 T1)x L i dx. Dirac delta function 4. What is the Dirac delta function and how can it be defined? Why is it useful in physical problems? Integral transforms 5. The Fourier sine and cosine transforms of a piecewise smooth function f(x) defined on the interval 0  x <1 are given by S[f(x)] = S(!) = r 2 ⇡ Z 1 0 f(x) sin(!x) dx C[f(x)] = C(!) = r 2 ⇡ Z 1 0 f(x) cos(!x) dx. The inverse transforms are given by S1[S(!)] = f(x) = r 2 ⇡ Z 1 0 S(!) sin(!x) d! C1[C(!)] = f(x) = r 2 ⇡ Z 1 0 C(!) cos(!x) d!. Assuming f(x) ! 0 and f 0(x) ! 0 as x ! 1, prove the following properties of the Fourier sine and cosine transforms (hint: use integration by parts) (a) S[f 0(x)] = !C[f(x)] (b) C[f 0(x)] = q 2 ⇡f(0) + !S[f(x)] (c) S[f 00(x)] = q 2 ⇡!f(0) !2S[f(x)] (d) C[f 00(x)] = q 2 ⇡f 0(0) !2C[f(x)] 6. A semi-infinite rod (0  x < 1) of thermal conductivity  initially at a uniform temperature of 0 is subject to a constant steady heat flow of = @u@x (into the rod) at one end (x = 0). Using the Fourier cosine transform with respect to position x on the rod, show that the temperature distribution at later times t is given by u(x, t) = 2 ⇡ Z 1 0 1 exp[!2t] !2 cos(!x) d! Page 2 7. At steady state the heat distribution in a semi-infinite metal plate (0  x < 1, 0  y <1) is described by the 2D Laplace equation @2u(x, y) @x2 + @2u(x, y) @y2 = 0. A small section of one edge of the plate 0  x  l is held at a temperature of T0, whilst all other points along the same edge are held at 0. The x = 0 edge of the plate is insulated, preventing any heat flow, so @u(x, y)/@x|x=0 = 0. Using the Fourier cosine transform show that the temperature distribution is given by u(x, y) = T0 ⇡  arctan ✓ l + x y ◆ + arctan ✓ l x y ◆ . You may use the integral below without proof:Z 1 0 exp[bx] x sin(ax) dx = arctan ⇣a b ⌘ . 8. Fourier transforms: Assuming the Fourier transform is defined according to U(k) = 1p 2⇡ Z 1 1 u(x)eikx dx, determine the Fourier transforms (taken with respect to the specified variables) of the following PDEs: (a) @nu(x) @xn = 0 (with respect to x) (b) @2u(x, y) @x2 + @2u(x, y) @x@y 5@u(x, y) @y = 0 (with respect to x and y) (c) @2u(x, t) @x2 + 1 v2 @2u(x, t) @t2 = 0 (with respect to x) (d) @2u(x, t) @x2 + 1 v2 @2u(x, t) @t2 = 0 (with respect to x and t) (e) ~ 2 2m @2u(x, t) @x2 = i~@u(x, t) @t (with respect to x and t) (f) ~ 2 2m @2u(x, t) @x2 + v(x)u(x, t) = i~@u(x, t) @t (with respect to x and t where v(x) is a potential function) (g) r · u = 0 (with respect to x, y and z) (h) r⇥ u = 0 (with respect to x, y and z) 9. Water evaporates from a large lake. On a still day, the concentration of water vapour above the lake reaches a steady state condition. Assuming the problem can be mod- elled as a two dimensional problem the water vapour concentration can described by u(x, z) where x is a coordinate along the surface of the lake and z describes the height above the lake. Page 3 (a) Write down a PDE for u(x, z) describing the steady state distribution. (b) Since the domain in the x direction is infinite we can approach solution of this PDE using a Fourier transform with respect to x. What form does your PDE take under this transformation? (c) Local temperature variations along the surface of the lake (e.g. due to clouds casting shadows at some points producing less heating from the sun) means the local evaporation rate di↵ers. As a result the concentration of water vapour at the surface of lake is given by u(x, 0) = c(x). Using your results so far show that for 1 < x <1 and 0  z <1 u(x, z) = 1 ⇡ Z 1 1 zf(s) z2 + (s x)2 ds. (d) Find u(x, z) if c(x) = c0 for a  x  a and 0 outside (i.e. a finite sized lake). Green’s functions (non-examinable) 10. What is a Green’s function? How are they useful for solving PDEs? 11. A string of length l is fixed at the ends and is initially at rest. It is struck at its mid-point x = l/2, such that a momentum of p is imparted to the string. Determine the motion of the string. Hint: Solve the 1D wave equation with the initial conditions u(x, 0) = 0, @u(x, t) @t t=0 = 8<: 0 0 < x < l/2 v0 l/2 < x < l/2 + 0 l/2 + < x < l where v0 = p/(2⇢) is the initial velocity over a small length of the string of length 2 near x = l/2 and ⇢ is the linear mass density. Then take the limit ! 0. 12. Consider a thin glass rod of length L = 10 cm. When rubbed with a silk cloth a triboelectric static charge of Q = 8 ⇥ 1010 C is built up. Write down a di↵erential equation governing the electrostatic potential V (r) at position r. Determine the electrostatic potential assuming that V (r !1) = 0. 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