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MA4XJ Integral Equations
创建日期：2024-08-30 14:09:58
所属分类：数学mathematics
标签： MA4XJ
Integral Equations
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MA3XJ/MA4XJ Integral Equations

Notes” for how to classify integral equations and do Q1! Important questions in this classification are: Is this a
1D, 2D, ... integral equation? Is this a linear or nonlinear integral equation? If it is linear, is it first
kind or second kind, homogeneous or inhomogeneous, Fredholm or Volterra?
1. Classify the following integral equations:
(a) (y(x) + 1)2 = e x + R 0 1 (x − t)y(t)dt, for 0 ≤ x ≤ 1;
(b) y(x) = R 0 x sin(x − t)y(t)dt, for 0 ≤ x ≤ 3;
(c) tan x = R1−1 (x − t) 2 y(t)dt, for −1 ≤ x ≤ 1;
(d) e x = y(x) + R − π π ln |x − t|y(t)dt, for −π ≤ x ≤ π;
(e) y(x) = cos(x) + R π0cos(x − t) cos(y(t))dt, for 0 ≤ x ≤ π;
(f) cos(x) = R 0 x xty(t)dt, for 0 ≤ x ≤ π;
(g) 0 = y(x) + R − π π ln |x − t|y(t)dt, for −π ≤ x ≤ π.
2. Find the solutions y ∈ C[0, 1], if any, of the integral equations with separable kernels:
(a) y(x) = 1 + R 0 1 x 2 t 2 y(t)dt, 0 ≤ x ≤ 1;
(b) y(x) = sin x + R 0 1 xty(t)dt, 0 ≤ x ≤ 1.
3. Generalising 1(a), given λ ∈ C with λ ≠ 0, consider the integral equation
(a) Show that there is exactly one solution y ∈ C[0, 1] to this equation if λ ≠ 1/5, and find a formula for this solution;
(b) Show that there is no solution y ∈ C[0, 1] if λ = 1/5;
(c) Does the above integral equation have any solutions if λ = 0?
4. Define the integral operator K : C[−2, 2] → C[−2, 2] by
Define ψ(x) = x + 3i, for −2 ≤ x ≤ 2.
(a) Obtain an explicit formula for the function Kψ .
(b) Obtain an explicit formula for the function K2 ψ := K(Kψ).
(c) Generalising (a) and (b), show that if, for some constants a,b ∈ C, ϕ(x) = a + bx, for
−2 ≤ x ≤ 2, then
5. Show that if y ∈ C[0, 1] satisfies
then
6. [This is a variant on the previous question, with almost the same solution, but showing a slightly stronger result.] 
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