# Tagged Questions

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. (Handbook of Mathematics - ...

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### Recommended Textbook for Integral Equation [on hold]

I am doing a self reading in preparation for the courses I have next semester of which Integral Equation is part of it. I keep on seeing very strange notations in the materials given to me by my ...
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### Solution of an integral equation

Consider a simple Wiener-Hopf integral equation of the first kind with unknown function $\phi(x)$ for $x\geq 0$: $$f(x)=\int_0^\infty \phi(y)\min\{x,y\}\,\mathrm{d}y$$ where $f(x)=x-a$ and $a \geq 0$...
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### Is it possible to solve these simultaneous equations with integrals?

I have the following two equations: \begin{align} 2-3\int_{-\infty}^{y_0} f(x_0,y)\,\mathrm{d}y +\varepsilon x_0=0\\ 2-3\int_{-\infty}^{x_0} f(x,y_0)\,\mathrm{d} x+\varepsilon y_0=0 \end{align} where ...
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### Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y')^2 - y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ ...
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### Solving Volterra integral equation

I would like to solve $4u(t)+\int_0^t\sin(t-s)u(s)ds=5t, \ t\geqslant 0$. Any ideas on how to approach this equation?
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### Solution of the following Fredholm integral of the second kind

$H(s,x)=\int_0^{\infty } \frac{e^{(-s-1) (u+x)} \left(2 e^{(s+2) u+s x}+s\right) }{2 s}H(s,u) \, du+2 e^{-(1+s) x}$ Is there any chance to obtain the solution ($H(s,x)$) of this equation? I managed ...
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### Integral equation: $x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t$ [closed]

Would you please find the function $f$ such that $$x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t \quad ?$$ Thank you.
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### For which $a, b,$ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx$ essentially determine $f(x)$

For which values of $a, b,$ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx$ essentially determine $f(x)$. This is a generalization of Solve ...
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### Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
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### integral equation and laplace transform

Solve the following integral equation $u(x)= \cos x - \int_{0}^{x} (x-y)cos(x-y)u(y) dy$ I applied Laplace transforms to the above integral equation and so the initial equation is written as: ...
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### Direct numerical solutions for first kind Volterra integral equations

For clearly deliver my purpose, I rewrite this question. Consider first kind Volterra integral equations $$\int_0^t k(t,s)f(s)ds=g(t) \quad 0\leq t\leq T$$ where $k(t,s)$ is continuous but not ...