# 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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### Homogeneous Fredholm Equation of Second Kind

I'm trying to show that the eigenvalues of the following integral equation \begin{align*} \lambda \phi(t) = \int_{-T/2}^{T/2} dx \phi(x)e^{-\Gamma|t-x|} \end{align*} are given by \begin{align*} \...
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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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### Applied Mathematics Book on Integro-Differential Equations

I'm interested in teaching a course on integro-differential equations and their applications. I was wondering if anyone could suggest a decent book on the subject. I'm currently looking at "Nonlocal ...
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### Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds,$$ with $K$ given. ...
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### Solve y′−∫x0y(t)dt=2 [duplicate]

How i solve this? I have not idea how to approach this differential equation. y′−∫x0y(t)dt=2 I get the probably familiar DE: y′′−y=0. please someone can solve this exercise
On several papers, I found the following model for a multiple integral equation: $$g(s)=\int\limits_{\Omega} h(s,t)f(t)\,\mathrm{d}t$$ where $s,t \in \mathbb{R}^3$, and $\Omega \subseteq \mathbb{R}^... 0answers 30 views ### How do I solve this integral equation$x(t)-\mu \int_a^bcx(\xi)d\xi=v(t)$? Also, how the corresponding Neumann series to this equation help obtain a convergence condition for a general integral equation like $$x(t)-\mu\int_a^bk(t,\xi) x(\xi) d\xi=v(t).$$ I think the second ... 0answers 25 views ### Collocation method for integral equation with monotone increasing kernel Is it possible to approximate the solution ($f(x)$) of this type of integral equation if the kernel ($k(x,t)$) is a strictly monotone increasing function?$f(x-T)= g(x)+\int_0^{\infty } k(x,t) f(t) \,...
Consider the Volterra integral equation $$f(t) = g(t) +\int_0^t K(t,s) f(s) ds$$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...