代写INTEGRAL EQUATIONS (MA3XJ) 2024代写C/C++程序

MA3XJ 2023/4 A 800

MA3XJ 2022/3 A 850

INTEGRAL EQUATIONS (MA3XJ)

April/May 2024

SECTION A

1. Classify the following integral equations:

[6 marks]

2. Show that if y ∈ C[0, 1] satisfies the integral equation

then

y(x) = 1 + 6x + 6x2 ,    0 ≤ x ≤ 1.

[16 marks]

3. K : C[0, 4] → C[0, 4] is defined by

for ϕ ∈ C[0, 4].

(a) If ψ(x) := 1, for 0 ≤ x ≤ 4, obtain an expression for the functions Kψ and K2 ψ .    [6 marks]

(b) Calculate ∥K∥ .      [4 marks]

4. Suppose that y ∈ C[0,π] satisfies

Show that y ∈ C2 [0,π] with

y′′ (x) = y′ (x) − y(x),    0 ≤ x ≤ π,

andy′ (0) = y(0) = 1.       [16 marks]

5.   (a) Show that the kernel function k(x,t), defined by

is not weakly singular.      [4 marks]

(b) Let y ∈ C[0, 3] satisfy the integral equation

Show that

|y(x)| ≤ 5ex ,     for 0 ≤ x ≤ 3.                                  (3)

Show that y ∈ C1 [0, 3] and, making use of (3), that

|y′ (x)| ≤ 10ex − 3,     for 0 ≤ x ≤ 3.

[8 marks]

SECTION B

6. Define the integral operator K : C[0, 1] → C[0, 1] by

(a) Using the trapezium rule method with steplength h = 0.2 compute approximations to y(0.2) andy(0.4), where y ∈ C[0, 1] is the unique solution of

y(x) = 1 + Ky(x), 0 ≤ x ≤ 1.

[7 marks]

(b) Show that, for every n ∈ N,

Deduce that ∥Kn ∥ → 0 as n → ∞ .       [13 marks]

7. Suppose that g ∈ C[0, 1] and that

with kn ,ℓn  ∈ C[0, 1], for n = 1, . . . ,N.

(a) Show that if y ∈ C[0, 1] satisfies

then

where p1 , . . . , pN  ∈ C satisfy the equations

and

form,n = 1, . . . ,N.

[7 marks]

(b) Conversely, show that if p1 , . . . , pN  ∈ C satisfy (6) and y ∈ C[0, 1] is defined by (5), then y satisfies the integral equation (4). [9 marks]

(c) Deduce that (4) has a solution y ∈ C[0, 1] if the matrix A is invertible, where A is the N × N matrix A = [amn]m(N),n=1, with

amn  := 2δmn  − kmn ,     form,n = 1, . . . ,N,

and where δmn is the Kronecker delta. [4 marks]

8. Define the integral operator K : C[−1, 1] → C[−1, 1] by

for ϕ ∈ C[−1, 1].

(a) Show that ∥K∥ = 3/2 .    [7 marks]

(b) Suppose that g ∈ [−1, 1] is defined by g(x) = 1 − x, for −1 ≤ x ≤ 1, and that y ∈ C[−1, 1] satisfies

y = g + Ky.

Define a sequence of approximations y0 , y1 , y2 , ... to y by y0  := g and by yn  := g + Kyn−1, for n ∈ N. Show that

for n = 0, 1, . . ..

[13 marks]

9. Suppose that k ∈ C(R) with k(x) = 1 for |x|  ≥ 1. Suppose also that u ∈ C(R) satisfies

where

(a) Show that u(x) = eix + Re−ix, for x ≤ −1, and u(x) = Teix, for

x ≥ 1, and obtain expressions for the constants R and T, in terms of the values of u(x) for |x| ≤ 1.   [9 marks]

(b) In the case that, for some c ∈ (0, 1),

|k2 (x) − 1|  ≤ c,      for x ∈ R,

show that (9) has at most one solution u ∈ C(R).               [11 marks]