MATH0065
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MATH0065
Answer all questions .
1.
(a) Use the method of separation of variables to find a series solution to the one-dimensional heat equation
ut = κu北北 , on 0 < x < 1, t 2 0,
subject to boundary conditions u(0, t) = u北 (1, t) = 0 and initial condition u(x,0) = f (x). Any constants which appear in your solution should be defined in terms of integrals involving f (x).
(b) The stochastic process Xt = y +^2Bt , where 0 < y < 1, is subject to an absorbing boundary condition at x = 0 and a reflecting boundary condition at x = 1. Write down the Fokker-Planck equation satisfied by p(x, t), the probability density of finding Xt at x. What are the associated boundary conditions and initial conditions?
(c) Find p(x, t) and show, in the special case when y = , that
p(x, t) = ^2 & αk sin exp ╱ - 、,
where
1 if mod(k, 4) = 0, 1
αk = .
(d) Use your answer to (c) to find a series expression for P (y, t), the probability that the process starting at y has still not been absorbed by time t. (If you were not able to solve (c), partial credit will be given for finding P ( , t).) (20 marks)
2.
(a) Write down the definition of the Itˆo integral
t
f (Bs , s) dBs .
0
Take care to define your notation. Use the definition you have written to
explain why
.╱ 0 t f (Bs , s) dBs 、= 0.
(b) Consider the Ornstein-Uhlenbeck process
dXt = -aXt dt + b dBt , X0 = x0 ,
where a > 0 and b are real constants. Find.(Xt ), .(Xt(2)) and.(XsXt ) when 0 < s < t.
(c) A population Xt of bacteria evolves according to
dXt = rXt dt +^2σXt dBt , X0 = y .
Find the expected time (y) for the population to first decay to XT = 1 or to first grow to XT = x+ , where 1 < y < x+ . You may assume that r σ . (20 marks)
3.
(a) A stochastic matrix is given by
Mt = 2t -Wt t 、
where Bt and Wt are independent Brownian processes and α > β > 0 are positive constants. Find.(Mt ) and.(Mt(2)).
(b) The stochastic processes Xt and Yt satisfy the following stochastic differ- ential equations,
dXt = Yt dt + a dBt , X0 = x0 .
dYt = -Xt dt + b dWt , Y0 = 0.
where Bt and Wt are independent Brownian processes, and a, b and x0 are real constants. Consider St = Xt(2) + Yt2 . Use Itˆo’s formula to find an expression for the increment dSt . Find.(St ).
(c) Suppose (Xt , Yt ) in (b) describe the coordinates of a particle in a plane. What is the nature of the particle motion when the noise is absent (a = b = 0)? Briefly describe the expected effect of the noise.
(d) Determine distinguished limits of the differential equation ε + 4f3 + 3f2 = 0,
where f = f(x; ε) and ε is a small parameter, 0 < ε < 1. Hence, or otherwise, determine the limiting form of the differential equation under the scaling transformation f = εa F, x = ε8 z, where the parameters α, β satisfy the relation
α2 + β2 - 2(α + β) + = 0.
You do not need to solve any differential equations. (20 marks)
4.
(a) Consider the boundary-value problem
ef\\ + (2北 + 3)f\ - 3f = 0, f (0; e) = 0, f (1; e) = ^5,
for the function f = f (北; e) on 北 e (0, 1). Here e is a small parameter, 0 < e < 1, and the prime denotes the derivative with respect to 北.
(i) Assuming that the boundary layer is at 北 = 0, derive the leading-order term in the outer asymptotic expansion of the solution.
(ii) Derive carefully an order-of-magnitude estimate for the scaling of the independent variable in the boundary layer and obtain the leading- order term in the boundary-layer solution.
(iii) Using matching in the intermediate limit, determine any remaining constants of integration and construct a uniformly valid solution of the boundary-value problem.
(b) Consider now the differential equation
ef\\ + (2北 + m(e))f\ - 3f = 0,
on 北 > 0. Here 0 < m(e) < 1. The boundary layer is at 北 = 0. Derive an estimate for the boundary layer thickness in terms of e for various choices of the parameter m(e) and hence obtain the governing equations for the boundary-layer solution at leading order. You do not need to solve the boundary-layer equations. (20 marks)
5.
(a) The region ABC in the complex z-plane, z = x + iy, is the right-angled triangle as shown in the diagram. The vertex C is at the origin and A is at z = 1. The angle at A is απ . Using the Schwarz-Christoffel transformation, derive a conformal map from the upper half-plane w to the interior of the triangle ABC in the form
u z = a
— 1
ds
(s + 1)y ← (s - s0 )y2
+ c,
where the constants a, c, γ1 , γ2 , s0 are to be determined. Explain how the branches of the multi-valued functions appearing in this formula are chosen.
(b) Consider the Laplace equation V2 T = 0 for the temperature T = T (x, y) on the triangle ABC in the (x, y)-plane. On the sides of the triangle the temperature is given: T = 0 on AB and BC, and T = 1 on CA. Determine the temperature in the w-plane.
(c) Using answers to parts (a) and (b), show that in the vicinity of the point A in the z-plane the temperature distribution may be approximated as
θ + π(α - 1)
απ
using polar co-ordinates centered at z = 1 so that z = 1 + rei9 , π - απ < θ < π, and the radial distance is small, r < 1.
Explain briefly how this local result for the temperature distribution can be verified by solving Laplace’s equation in a corner region 0 < r < o, π - απ < θ < π, using cylindrical polars and seeking a self-similar (separable) solution with the boundary conditions T = 0 at θ = π - απ , and T = 1 at θ = π . (20 marks)
2023-04-13