# 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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### Can we solve integral equations for a multivariable function?

Integral equations are equations in which an unknown function appears under an integral sign. Can we solve integral equations when the function is multivariable and the integral is a multiple integral?...
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1 vote
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### Unique solution to a specific Volterra's integral equation of the third kind

Consider an integral equation (Volterra's integral equation of the third kind) $$(d-cx) u(x) = \int_x^b u(y) dy, \qquad x \in [a,b] \qquad (1)$$ where $u:[a,b] \to \mathbb{R}$ is an unknown function ...
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### On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1)$$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
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### Solution of Integral equation via Fourier Transform

Solve for f(t) using Fourier transform: $$\int_{-\infty}^\infty f(s)f(t-s)\,ds - 2\sqrt{2} \int_{-\infty}^\infty e^{-s^2/\pi}f(t-s)\,ds = -\sqrt{2}\pi e^{-\frac{t^{2}}{2\pi}}$$ Now, I get the ...
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### Convergence of a self-consistent multivariable integral equation

I want to solve the following integral equation $$\mathbf v(\mathbf r)=\int_V \mathbf v(\mathbf r^\prime)\cdot \mathbf F(\mathbf r, \mathbf r^\prime) \mathrm ~d^3r^\prime +\mathbf g(\mathbf r)$$ using ...
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### Integral Equation and inverse laplace

When we get the equation in (5.81), we get complex expressions while taking the inverse laplace transform after moving to the next step. Is there any way to get rid of this complex expression?
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### Verify solution to linear Fredholm integral equation of the second kind

Let $\int_a^b C_X(t, s)\psi_k(s)ds = \lambda_k\psi_k(t)$, which corresponds to a homogeneous linear (Fredholm) integral equation of the second kind. Where $C_X(t, s)$ is the covariance function and is ...
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1 vote
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### Consider a continuous function $f(x)$ with $f(1)=0$ [duplicate]

Given that $$I_1=\int_0^1x^2f(x)dx=\frac {1}{3}$$ and $$I_2=\int_0^1(f'(x))^2dx=7$$ Find the value of $$J=\int_0^1f(x)dx$$ In the first equation using a bit of symmetry I substituted $x=1-t$ and after ...
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### How to solve the integral equation $f(x) \int_a^x \int_a^{t} f(\omega) d\omega dt - \left[ \int_a^x f(t) dt \right]^2 = 0$

I tried to solve the integral equation $$f(x) \int_a^x \int_a^{t} f(\omega) d\omega dt - \left[ \int_a^x f(t) dt \right]^2 = 0$$ by taking the first, second, third, and fourth order derivatives but I ...
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### Converting an integral equation to a differential equation

I was recently working on a problem and ended up with an integral equation that I was hoping can be solved or at least be converted to a differential equation. I have no experience in integral ...
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### Show that $y$ satisfies Integral equation.

Let $y$ satisfies the boundary value problem $$y''(x) + \lambda y(x) = 0, 0 < x < 1,$$ $$ay(0) = by(1), by'(0) = ay'(1),$$ where $a, b$ are constants. Show that $y$ satisfies the integral ...
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### Convert the bessel equation $x^{2}*y''+xy'+(\lambda x^{2}-\alpha^{2})y=0$ to integral equation.

Convert the bessel equation $x^{2}y''+xy'+(\lambda x^{2}-\alpha^{2})y=0$ to integral equation, where $0<x<1,$ $y (x)$ is bounded as $x\to 0$, $y(1)=0$, where $\alpha>0.$ Please give some ...
1 vote
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### Solve for the kernel in a system of Volterra equations

I have a system of $n = 1, \dots, N$ Volterra equations with the same kernel $K$: $$f_n(t) = \int_{0}^{t} K(t, s) g_n(s) \mathrm{d}s\,, \quad t \in [0, 1] \,,$$ where $f_n$ and $g_n$ are known and $K$ ...
163 views

1 vote