# Tagged Questions

**0**

votes

**0**answers

20 views

### Condition of existence and uniqueness of solution for abel integral equation [migrated]

It is well known that Abel integral equation has a unique continuous solution.
For example,
$$
f(t)=\int_0^t\frac{g(s)}{(s-t)^{\alpha}}ds , 0<\alpha<1
$$
where f(t) is known. Specifically, ...

**3**

votes

**2**answers

351 views

### Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi,
I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.
Fix $T>0$. ...

**6**

votes

**0**answers

264 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} ...

**1**

vote

**2**answers

266 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and HÃ¶lder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**1**

vote

**1**answer

603 views

### Writing an integral equation as a partial differential equation

Hello,
Can anyone help me see how one can get from the following integral
$$\lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha ...

**0**

votes

**1**answer

2k views

### From an integral equation to a differential equation

Hello,
I am wondering whether it is possible to convert the following integral equation to a partial differential equation.
where $J_0(t,x)$ is some given nonnegative function and $\nu>0$ is ...