# Tagged Questions

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### 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, ...
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### 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 $T>0$. ...
Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by $$\frac{\partial}{\partial t} ... 2answers 266 views ### how to solve a singular integral equation involving the kernel 1/x Dear all, Suppose we know that f(x) is nonnegative and HÃ¶lder continuous at zero with exponents 1/2. We also know that$$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$... 1answer 603 views ### Writing an integral equation as a partial differential equation Hello, Can anyone help me see how one can get from the following integral$$\lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha ...
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 ...