# Tagged Questions

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### Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$\displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s$$ I tried solving it by collocation ...
1answer
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### Existence of integral equation solution

I am trying to prove a differentiable solution in some open interval about the origin for the equation: $$u(x) + u(x)^2 + \int_0^x (1+\cos(x+u(y))) dy = 0$$ I have been trying to prove it as a ...
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### Spectrum of $Tu=\int^1_0 (x+y)u(y)dy$

Given the operator $$Tu(x)=\int^1_0 (x+y)u(y)dy$$ on $L^2(0,1)$, find the spectrum of $T$. For all eigenvalues, find their multiplicities and the eigenfunctions. The kernel is Hilbert Schmidt ...
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### Contraction mapping principle application

I'm to prove that the following equation has a unique solution: $$f(x) = \int_0^1 e^{-sx} \cos(\alpha f(s)) ds.$$ (Here, $\alpha \in (0,1)$.) The form of the exercise screams to apply the ...
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### 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 ...
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### A question on convergence of solution of an integral equation.

In Pipkin's "A Course on Integral Equations", on page 24 problem 2, he asks us to find out whether or not iteration will converge uniformly for an integral equation of the second kind, i.e $u=f+Ku$ on ...
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### Volterra integral equation with variable boundaries

$$\phi (x)=x+\lambda \int_{a}^{x}(x-y)\phi (y)dy$$ I'm also Trying to solve this integral equation like she does Solving an integral equation with a separable kernel. and I also have some doubts ...
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### applications integrals

I have the following problem to solve: $\int_{0}^{1}K(x,y)\phi(y)dy$ where: $K(x,y)=x(1-y), 0\leq x\leq y\leq 1$ and $K(x,y)=y(1-x), 0\leq y\leq x\leq 1$ already tried using the methods suggested ...
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### Integral eigenvectors and eigenvalues

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
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### Find the eigenvalues and eigenvectors of an integral operator

I need to find the eigenvalues e eigenvectors of this integral. $$\int_{0}^{1} K(x,y)\phi (y)dy,$$ where $K(x,y)=x(1-y),\; 0 \le x\le y \le 1$ and $K(x,y)=y(1-x),$ $0\le y\le x \le 1$ I ...
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### Matrix inversion of an analytical function

Following problem: I have a function $f(x_1,x_2)$ and Im looking for the inverse $finv(x_1,x_2)$ of the function which is defined through: $\int f(x_1,y)\cdot finv(y,x_2) d y =\delta(x_1,x_2)$ ...
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