# Questions tagged [integral-equations]

This tag is about questions regarding the integral equations. An integral equation is an equation in which the unknown function appears under the integral sign. There is no universal method for solving integral equations. Solution methods and even the existence of a solution depend on the particular form of the integral equation.

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### Degenerate/ separable kernel

Why e^tx is non- degenerate kernel, While e^(x-t) is degenerate or separable? Iam confused because in both cases after expansion we get results in powers of x and t and according to the definition of ...
1answer
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### Continuity of the following integral.

Let $f(x,y):[0,1]^2 \rightarrow R$ be a continuos function, and $F(x)$ defined as $$F(x)= \int^{1}_{0} f(x,y)1_{(y\leq x)} dy.$$ where $1_{(y\leq x)}= 1$ if $y\leq x$ and $1_{(y\leq x)}=0$ if $y> x$...
1answer
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### Solve PDE. Is this the right solution? $\frac{\partial u^{2}}{\partial x \partial y} + \frac{\partial u}{\partial y} + x + y + 1 = 0$

This is the equation: $\frac{\partial u^{2}}{\partial x \partial y} + \frac{\partial u}{\partial y} + x + y + 1 = 0$ I also have a solution but I don't know how it removed -1 in front of the arrow
1answer
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### Can we find a closed form expression for the solution of $y(t)=y_0-ct+\alpha\int_0^t(t-s)y(s)^p\:{\rm d}s$?

Let $\alpha>0$, $p>1$, $T>0$, $c\ge0$ and $y_0>0$. Are we able to find an explicit form for the solution $y:[0,T]\to\mathbb R$ of $$y(t)=y_0-ct+\alpha\int_0^t(t-s)y(s)^p\:{\rm d}s$$ for ...
1answer
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### Does the problem $\operatorname{div}(\vec{p}) = f$ on $\Omega$ with $\vec{p}\cdot\vec{\nu}=0$ on $\partial \Omega$ admits non-conservative solutions?

Define $V:=\{\vec{p} \in L^2(\Omega;\mathbb{R}^n) \mid \operatorname{div}(\vec{p}) \in L^2(\Omega)\}$ where $\Omega$ is a non-empty connected bounded open subset of $\mathbb{R}^n$ with $C^2$ boundary ...
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### Find the control for this system of ODE minimizing the energy.

Suppose I have the system of ODE $$x'(t)=Ax(t)+Bu(t)$$ $$x(0)=x_0$$ $t$ is defined on an interval $I$ of $\mathbb{R}$ containing $0$, for $x(t)=(x_1(t),...x_n(t))^T$, $x_i : I \to \mathbb{R}$ ...
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### Solving a functional equation with inverse

The following question has bothered me for a long time. Any suggestions will be highly appreciated. I need to find a monotone positive function $f(x)>0$ and $f'(x)<0$ (whose inverse $f^{-1}$ is ...
2answers
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### Probability that a Particle which moves Unit distance in a Random direction on each step will be inside the Unit Sphere after $n$ steps

The following integral equation arises while calculating the probability that, a particle which starts at the origin and moves a unit distance in a random direction on each ‘move’, will be within the ...
0answers
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### Finding the characteristic number and the eigenfunctions of a homogeneous Fredholm integral equation

The kernel of the homogeneous Fredholm integral equation is given as: $$K(x,t) = \begin{cases} x(t-1), \text{ } 0 \leq x \leq t \\ t(x-1),\text{ } t\leq x\leq 1 . \end{cases}$$ The given options ...
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### Reference request to understand the “Linear integral Equation” from Rainer Kress's book

I am reading "Linear integral Equation" by Rainer Kress. Mean while he is using many terms from differential geometry and several variable calculus for example volume elements in multi ...