Introduction to Partial Differential Equations
Exercise Sheet 5

Ross Ogilvie      30th September, 2024
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13.
Is this an applied math course?

In economics, the Black-Scholes equation is a PDE that describes the price V of an (European-style) option under some assumptions about the risk and expected return, as a function of time t and current stock price S. The equation is

∂𝑉 ∂𝑡 + 1 2σ2S22V S2 = 𝑟𝑉 𝑟𝑆∂𝑉 ∂𝑆,

where r and σ are constants representing the interest rate and the stock volatility respectively. Describe the order of this equation, and whether it is elliptic, parabolic, and/or hyperbolic.
(3 points)

14.
Around and around

Consider the unit circle C = {x2 + y2 = 1} 2. In this question we will evaluate the integral

Cy𝑑𝜎 (∗)

in two different ways, demonstrating that it does not depend on the choice of parametrisation.

(a)
Consider a regular parameterisation Φ of a subset A and a continuous function f on A. Why (or under what conditions) is the integral unchanged by removing a point from A: (1 bonus point)
Af𝑑𝜎 =A{a}f𝑑𝜎.

Therefore in order to compute the submanifold integration () it is enough to use parameterisation that cover all but finitely many points of C.

(b)
Consider the regular parametrisation Φ : (0,2π) C given by t(cost,sint). Compute the integral () using this parametrisation. (2 points)
(c)
Consider upper and lower halves of the circle: U1 = {(x,y) Cy > 0} and U2 = {(x,y) Cy < 0}. There are obvious parametrisations Φi : (1,1) Ui given by Φ1(x) = (x,+1 x2) and Φ2(x) = (x,1 x2). Compute () using these parametrisations.
(2 points)
(d)
Compute this integral using the divergence theorem. (2 points)
15.
The Proof is Left as an Exercise for the Reader

Using Definitions 2.4 and 2.7, prove Lemma 2.9: The following properties hold for a,b and f,g C(A).

(i)

Linearity: A𝑎𝑓 + 𝑏𝑔𝑑𝜎 = aAf𝑑𝜎 + bAg𝑑𝜎. (1 point)

(ii)

Order Preserving: if f g on A then Af𝑑𝜎 Ag𝑑𝜎. (1 point)

(iii)

Triangle Inequality: | Af𝑑𝜎| A|f|𝑑𝜎. (1 point)

(iv)

Transformation: If P : n n is a euclidean motion (translation, reflection, rotation) and s + is a scaling factor then Af𝑑𝜎 = sk(𝑠𝑃)1[A]f (𝑠𝑃)𝑑𝜎. (3 points)

16.
The Black Spot

Consider the plane 2, a disc Br = {x2 + y2 r2} and the function g(x,y) = ln(x2 + y2).

(a)
By calculating g N for the outward pointing normal N = 1 r(x,y), show that the value of the integral
Brg N𝑑𝜎

does not depend on the radius r. (2 points)

(b)
Can you explain this fact using the divergence theorem?
Hint: Apply the divergence theorem to the region Ar,R = BR Br¯ for two different radii r < R. (3 points)