1. In two dimensions the Laplace Operator has the form Au = дги a²u + მ22 ax2 Perform the all the computations to verify that in terms of polar coordinate 1 x = p cos 0 and y = p sin 0 the Laplace Operator has the for Upp + up + 1 ρ uee. Some of the computations appear on page 69 of Logan and further computations appear in Class Power Point Presentations. 2. Show that satisfies w(x,t) = 1 2c. t x+c(t-s) | f (y, s) dyds 0 x-c(t-s) Wtt = c²wxx + f (x.t) - ∞o < x < ∞o, t> 0 w(x, 0) = 0 wt(x, 0) = 0 -xx<∞ ∞ > x > ∞ - 3. Classify the following partial differential equations a. Uxx+2uxt + Utt - ux + 3u = 0 b. -4uxt2uxx - Utt 8ux +5u0 - c. 3uxx+2uxt + 5utt + 7ux = 0 d. 2uxx+2uxt + 2utt - ux + 3u = 0 4. Let a. Show that 1 00 u(x,t) = √4πDt .00 e (x-y)² 4Dt (y)dy b. If (x) = (cos πx)² 2+x2 Ut - Duxx = 0. find lim u(1,t) t-0+ 4. a. Show that Wnm(x, y, t) = e−d(n²+m²)t awnm/atdawnm = cos nx cos mx satisfies for 0 ≤ x ≤π, 0 ≤ у≤л,О≤πt > 0 subject to the boundary conditions awnm дх JWnm (0, y, t) (л, y, t) = 0 0≤ y ≤π,t> 0 дх awnm JWnm (x, 0, t) = (x,π,t) = 0 0≤ x ≤π,t> 0 ду ду b. Write the solution to Ju/ǝt = 5Au for 0≤x≤π, 0 ≤ y ≤л, t> 0 ди ди (0, y, t) (л, y, t) = 0 0≤ y ≤ n,t> 0 дх дх ди ди (x, 0, t) = -(x,π,t) = 0 0≤x≤π,t> 0 ду ду u(x, 0): = 2 cos x cos y + 4 cos 2x cos 3y - cos 3x cos y 5. Suppose that u(x, y) satisfies Au = 0 on the square 0 ≤ x ≤ 1,0≤ y≤ 1. On the boundary of the square, assume: u(x, 0) = sin 2πx 0 ≤ x ≤1 u(x, 1) = 3 =-x-x3 0 ≤ x ≤ 1 x u (0, y) = 0 0 ≤ y ≤ 1 0 ≤ y ≤ 1 (1, y) = y - y² Find the maximum and minimum values of the solution u(x, y) on the square 0 ≤x≤ 1,0≤ y ≤ 1. 6. Solve the following a. Utt 4uxx = 0 1 u(x, 0) 1+ ex² 7. b. ut(x, 0) = x³ cosx4 Utt - -3uxx = 0 u(x, 0) = 0 ut(x, 0) = -xe-x6 Suppose that is a bounded two-dimensional region and that it has a continuously differentiable boundary, ǝn. Let u(x, y, t) satisfy the following initial boundary value problem. Why is the solution (x, y, t) unique? ut-3Au = 0 x, yЄ, t> 0 u(x, y, t) = 0 x, y Є an, t> 0 Χ,ΥΕΩ u(x, y, 0) = p(x, y) Hint. You have seen problems similar to this on the homework. Assume there are two solutions u₁(x, y, t) and u2(x, y, t). Set (x, y, t) = u₁(x, y, t) -u2(x, y, t). Examine the equation that v(x, y, t) satisfies. You will need to use Green's Identity I, 8. Solve the following: a. b. ut + 4ux = 0 1 u(x, 0) 1+ ex² Ut- 3ux+7u0 u(x, 0) = 0 x² ut(x, 0) = 2+x4 9. Find integrals that give solutions to a. ut-4ux = 0 b. 1 u(x, 0) = 1+x4 ut 7ux+2u = 0 - u(x, 0) = 1 + sin x 1+x4 10. Let r and be polar coordinates. A function h(r, 0) is said to be radially symmetric if the function values depend only on r and not on 0. In дп this case = 0 and there is a function g(r) such that h(r, 0) = g(r). де Find a radially symmetric solution u to the Laplace Equation on the annular ring bounded by the circles r = 1 and r = 2. Given that u satisfies the boundary conditions at u(x, y) = 1 when x² + y² = 1 and u(x, y) = 3 when x2 y2 = eπ. This essentially Logan: Problem 6, Page 71 11. Let D > 0 and suppose that u(x, t) satisfies, ut - Duxx = 0 0≤x≤L, t> 0 ux(0,t) = ux(L,t) = 0 u(x, 0) = (x) 0 ≤ x ≤ L/ Perform the computations that demonstrate that L L [ (q(x))²dx ≥ [ (u(x,t))²dx