# Selected Solutions (IEL5-Quiz)

## Question 2

Let ϕ be the solution of the integral equation $\frac12\phi(x)-\int_0^1 e^{x-y}\phi(y)dy=x^2, (0\le x\le1$. Then

1. ϕ(0)=20exp(-1) – 8
2. ϕ(0)=20e-8
3. ϕ(0)=22 – 8e
4. ϕ(0)=22 – 8exp(-1)

### Solution 1

Let $c = \int_0^1 e^{-y}\phi(y)dy \implies \phi(x)=2x^2+2e^x c$. Substituting in the equation, $2x^2+2e^x c = 2x^2+ 2e^x\int_0^1 e^{-y}(2y^2+2e^y c)dy\implies c=\int_0^1e^{-y}(2y^2+ce^y)dy\implies c=2[\frac5e - 2]$

Hence, $\phi(x)=2x^2+4e^x [\frac5e - 2]$

Concluding, ϕ(0)=20exp(-1) – 8.

### Solution 2

For a Fredholm integral equation of 2nd Kind, using $\Gamma(s,t;\lambda)=\sum_p=0^\infty \frac{(-\lambda)^p}{p!}C_p(s,t)/ \sum_p=0^\infty \frac{(-\lambda)^p}{p!}c_p$, where

1. $c_0=1,C_0(s,t)=K(s,t)$
2. $c_p=\int_0^1 C_{p-1}(s,s)ds$
3. $C_p = c_pK(s,t)-p\int_0^1K(s,x)C_{p-1}(x,t)dx$

Here, $c_0=1,C_0(x,y)=e^{x-y},c_1=1,C_1(x,y)=0$. Hence $\Gamma(x,t;\lambda)=\frac{e^{x-y}}{1-2}=- e^{x-y}$

Hence the solution is $\phi(x)=2x^2+2\int_0^1\Gamma(x,y;\lambda)f(y)dy = 2x^2-4\int_0^1e^{x-y}y^2dy$ and you know how to finish the rest.

### Solution 3

Using iterated kernels, $\Gamma(s,t;\lambda)=\sum_{m=1}^\infty \lambda^{m-1}K_m(s,t)$

where $K_1(s,t) = K(s,t),K_m(s,t)=\int K(s,x)K_{m-1}(x,t)dx$

We will get $K_m(s,t) = e^{x-y}$, $\implies \Gamma(s,t;\lambda)=\sum_{m=1}^\infty \lambda^{m-1} e^{x-y}$ $= e^{x-y} \sum_{m=1}^\infty \lambda^{m-1}$ $= e^{x-y} \times (1-\lambda)^{-1}$

Here, $\lambda = 2\implies \Gamma(s,t;\lambda)= - e^{x-y}$ and you know where to take it from here.

### Observation

Use Leibniz Integral Rule (link to blog page quoting the result) – Wikipedia (In particular, a specific case called Fundamental Theorem of Calculus here, since limits are constant) to find derivative of the equation w.r to x. What do you observe? [Hint: you will get a differential equation, as $\phi'(x)-\phi(x)=4x-2x^2$ – try solving it!]

## Question 4

### Method

Solve the corresponding transposed homogenous integral equation, $\phi(x)=\frac2\pi\int_0^\pi \cos(x+t)\phi(t)dt$ and find corresponding solution $\psi(x)$ for $\lambda= \frac2\pi$.

Then, for each option, if $\int_0^\pi f(x)\psi(x)dx$. If it is zero, it has solutions, and if it is not zero, no solutions.

Here, solution, $\psi(x) =c_1\cos x +c_2 \sin x$ and hence options 2,3 and 4. ( $\int_0^\pi (\cos x+\sin x)\cos(3x)dx=0$ and so on).

1. Allison says:

Hii thanks for posting this

Liked by 1 person

1. jessepfrancis says:

Glad that you found it useful! Just published my lecture notes, hoping it will benefit someone!

Like

This site uses Akismet to reduce spam. Learn how your comment data is processed.