Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
MATH 323 DEPARTMENT: Mathematical Sciences FURTHER METHODS OF APPLIED MATHEMATICS Time allowed: Two and a half hours INSTRUCTIONS TO CANDIDATES: This paper contains SEVEN questions, all carrying equal weight. Full marks will be awarded for complete solutions to FIVE questions. Candidates may attempt all questions, but only the best FIVE solutions will be taken into account. Candidates are only permitted to use calculators deemed acceptable, and affixed with an official holographic sticker, by the Department of Mathematical Sciences. Paper Code MATH 323 Page 1 of 6 CONTINUED 1. (a) By substituting y = xn find the general solution to the Euler’s differ- ential equation x2y′′ + 3xy′ + y = 0, x > 0. [4 marks] (b) Use the method of variation of parameters to find the solution to the differ- ential equation x2y′′ + 3xy′ + y = 4 ln x, x > 0 satisfying boundary conditions y(1) = 0, y′(2) = 1 4 . [8 marks] (c) For the differential equation y′′ − 1 x y′ − 4x2y = 0 verify that y1(x) = e −x2 is a solution, find the second solution and write down the general solution. [8 marks] 2. (a) Write down the Euler–Lagrange equation for a function y(x) which ex- tremises the functional J [y] = ∫ b a F (x, y, y′)dx, where y′ = dy dx . Write down the sufficient condition for J [y] to have a minimum at y = y0. [3 marks] (b) For the given functional J [y(x)] = ∫ 2 1 (x2y′2 + 12y2 + 4yx4)dx write down and solve the differential equation for y(x) which extremises J [y] and satisfies the boundary conditions y(1) = −4 and y(2) = 127. [8 marks] (c) Use the sufficient condition to investigate the nature of the extremum of J [y]. [3 marks] (d) Using the result from (b) write down the solution of the Euler-Lagrange equation if the functional is now J [y(x)] = ∫ 2 1 (x2y′2 + 12y2)dx and y(1) = 1, y(2) = 8. Calculate the extremal value of J [y] corresponding to the solution y(x) and compare it with the value of J [y] for y(x) = 7x− 6. Comment on the nature of the extremum. [6 marks] Paper Code MATH 323 Page 2 of 6 CONTINUED 3. (a) Explain how to find a function y(x) : [a, b]→ R for which the functional I[y] = ∫ b a F (x, y, y′)dx has an extremum and for which the constraint equation J [y] = ∫ b a G(x, y, y′)dx = C, C ∈ R and the boundary conditions y(a) = y0 and y(b) = y1 are satisfied. Explain the origin of the three constants required to fulfill y(a) = y0, y(b) = y1 and J [y] = C. [2 marks] (b) The shape of a flexible string of length 10 fixed at two points (−4, 0) and (4, 0) is described by the function y(x). By maximising the functional I(y) = ∫ 4 −4 y(x)dx for J [y] = ∫ 4 −4 √ 1 + y′2dx = 10, show that the area between the string and the x axis is a maximum if the shape of the string is an arc of a circle. Find the radius and the centre of the circle. [13 marks] [Hint: The numerical solution to the equation 10 = 2 √ 16 + c21 tan −1 4 c1 is c1 = 1.88163.] (c) Consider now a string fixed at the same two points as in (b) but in the shape of a semicircle centered on the origin. What is the length of the string and the area A it encloses with the x axis. Compare A with a rectangular shaped string of the same length and comment on the result. [5 marks] Paper Code MATH 323 Page 3 of 6 CONTINUED 4. (a) The functions u(x, y) and v(x, y) satisfy the following system of partial differential equations yux + vy = y sinx yvx + uy = −y cosx. Show that this system is hyperbolic with the characteristics α = x− y 2 2 , β = x+ y2 2 . [4 marks] (b) Use the characteristics to find the Riemann invariants. [6 marks] (c) Find the general solution for u(x, y) and v(x, y) which satisfy the boundary conditions u(x, 0) = sin x and v(x, 0) = cos x. [6 marks] (d) Check that u(x, y) and v(x, y) found in (c) satisfy the system of differential equations and given boundary conditions. [4 marks] 5. (a) State well posedness for initial/boundary value problems. [2 marks] (b) Consider the following initial-boundary value problem: utt = 4uxx + 3t, 0 < x < 2, t > 0, (1) ux|x=0 = 0, ux|x=2 = 0, (2) u|t=0 = cos(2pix), ut|t=0 = 0. (3) Determine the type of the PDE (1). Explain briefly without proof whether the problem (1)-(3) is or is not well posed. [3 marks] (c) Show that for y 6= 0, x 6= 0 the partial differential equation y2uxx − 2xyuxy + x2uyy = 1 xy ( x3uy + y 3ux ) (4) for the function u(x, y) is parabolic with the family of characteristics ξ(x, y) = x2 + y2 = const. [3 marks] (d) Explain briefly when the equations ξ = x2 + y2, η = η(x, y) define a functionally independent coordinate transformation. By setting η = x show that this transformation reduces equation (4) to the canonical form uηη = 1 η uη. [7 marks] (e) Find the solution of equation (4) that satisfies the boundary conditions u(0, y) = e−y 2 , u(1, y) = 0. [5 marks] Paper Code MATH 323 Page 4 of 6 CONTINUED 6. (a) Show that the Joukowski transformation w = u+ iv = ( z + 1 z ) , z 6= 0 is conformal for all z 6= ±1, where u and v are real and z = x + iy is a com- plex variable. Explain what happens in the w-plane with angles between lines intersecting at these points in the z plane. [4 marks] (b) Find u and v in terms of polar coordinates r and θ where x = r cos θ, y = r sin θ. Show that the circles |z| = 2 and |z| = 1 2 are mapped to the same ellipse E1 : 4u2 25 + 4v2 9 = 1 in the w−plane. Explain what happens with the circle |z| = 1. Hence show that the exterior of |z| = 2 is mapped to the exterior of E1. [9 marks] (c) The potential Φ outside the circular region |z| = 2 is given by Φ = Re(A ln z) = A ln r, where A is a real constant. Show that Φ satisfies Laplace’s equation in polar coordinates ∂2Φ ∂r2 + 1 r ∂Φ ∂r + 1 r2 ∂2Φ ∂θ2 = 0. By clearly giving your reasoning, show that the potential outside the ellipse E1 is given by Φ = ARe ln [ w + √ (w2 − 4) 2 ] . Determine the potential at the point w = 6i. [7 marks] Paper Code MATH 323 Page 5 of 6 CONTINUED 7. (a) The Fourier transform of a suitable function f(x) defined on −∞ < x <∞ is F{f(x)} = f¯(ω) = ∫ ∞ −∞ f(x)e−iωxdx. Show that the Fourier transform f¯(ω) of the function f(x) = e−a 2x2 , where a is a real non-zero constant, satisfies the first order differential equation f¯ ′(ω) = − ω 2a2 f¯(ω). [4 marks] (b) By considering the square of f¯(0) show that for f(x) = e−a 2x2 f¯(0) = √ pi a . [4 marks] (c) Hence show that f¯(ω) = √ pi a e− ω2 4a2 [3 marks] (d) The function u(x, t) satisfies the partial differential equation ∂u ∂t = c2 ∂2u ∂x2 for −∞ < x <∞ and t ≥ 0. In addition u satisfies the conditions u(x, 0) = e−a 2x2 , u→ 0 as x→ ±∞. By using a Fourier transform show that u(x, t) is given by u(x, t) = 1 2a √ pi ∫ ∞ −∞ e− ω2 4a2 −c2ω2t+iωxdω. Hence by substituting 1 X2 = 1 a2 + 4c2t or otherwise, show that u(x, t) = 1√ 1 + 4a2c2t e − a2x2 1+4a2c2t .