2
votes
1answer
28 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
0
votes
1answer
41 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
0
votes
0answers
29 views

How to Solve This Special Case of Multidimensional Integral Operator?

I'm dealing with an integral equation of the following form: $1 = f(x)\int dy f(y)B(x,y)$ where $B(x,y)$ is a known function, and I want to solve for $f(x)$. If I treat $f(x)f(y)$ as one big unknown ...
2
votes
1answer
162 views

Question about integral equations

Consider the equation $$g(t) = \int_a^b K(t,s)f(s) ds $$ where $g$ and the kernel $K$ are known and $f$ is to be determined. Suppose that the equation has a solution. Under what conditions on the ...
3
votes
1answer
175 views

Multiple Integral Equation

$$f(x) = 2a \int_{0}^{x}{f(t)\;dt} - \left(\frac{b^2}{2}\right)\int_{0}^{1}{|x-t|f(t)\;dt}$$ where $0<a<b$ My task is to solve for $f(x)$. I'm having difficulty solving this integral equation. ...
3
votes
0answers
78 views

Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$ \int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t) $$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
1
vote
0answers
202 views

Unusual Application of Stokes Theorem

How can one get equation 117 on page 21 of the below document by employing Stokes' Theorem to solve the square of the line integral on the left such that the path is closed and infinitesimal, e.g., ...