0
votes
1answer
55 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
4
votes
3answers
197 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
1
vote
2answers
674 views

Solving an integral equations using fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
2
votes
1answer
319 views

An inverse definite integral problem

I am seeking a function $f(x)$ that satisfies this condition: $\int_{0}^{\infty }f(x)x^ndx=\sqrt{n!}$ where n is an integer. I guess that $f$ will contain $e^{-\alpha x^2}$ as one of its factors, ...
1
vote
1answer
219 views

Gelfand-Levitan-Marchenko equation

how can one solve the integral $$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$ q(x)= 2\frac{d}{dx}K(x,x) $$ (2) $$ -y''(x)+q(x)y(x)=0 $$ (3) $$ y(0)=0=y(\infty) $$ $ q(x) $ here is ...
2
votes
2answers
104 views

Can we solve for $a$ in $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds$

For all $x \in \mathbb{R}$, $$b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both ...
1
vote
0answers
328 views

What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?