# 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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### Unique solution of an integral equation in $L^1[0,1]$

Let $h\in L^1[0,1]$. Prove that there is a unique solution (almost everywhere) of the following integral equation: $$f(x)=h(x)+\frac{1}{2}\int_0^x\log(1+f(y)^2)dy$$ The idea is to use the fixed-point ...
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### Derivation of an integral equation

I have the following system $$\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=f, \text{ for } x \in (0,1)$$ with boundary conditions $u_x(0)=0$ and $u(1)=0$. For $a(x)>0$, and $b(x)=\frac{1}{a(x)}$, I ...
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### Integral equation: existence

Let $H$ and $h$ be smooth functions of one and two variables respectively. Consider equation $$H(x) = \int_\Bbb Rh(x,y)f(y)\mathrm dy \qquad \forall x\in \Bbb R.$$ When does it have a solution, ...
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### Solve this integral equation using Laplace transform

Solve this integral equation using Laplace transform $$f(x)=x^2 + \int_{0}^{x}f^{\prime}(x-t) e^{-at} dt ,f(0)=0$$ Please Help see mu answer below Thank you for your participation
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### Find the Laplace transform for this function

Find the Laplace transform for this function $$f(x)=(1+2ax)x^{-\frac{1}{2}}e^{ax}$$ Please, help me see my answer below Thank you for your participation
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### Solve this integral equation using Fourier transform

Solve this integral equation using Fourier transform $$\int_{-\infty}^{\infty} \frac{f(t)}{(x-t^2)+a^2} dt= \frac{\sqrt{2} \pi}{x^2 + b^2}$$ for $b> a > 0$ Please Help see my answer ...
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### Solving a Fredholm equation using Neumann series technique

It is a very simple kind of Fredholm equation: $$f(x)=x+\int_0^1(1+xt)f(t)\,dt.$$ I solved it and I know that the answer is $f(x)=-2$. But how can i solve this equation by the Neumann series ...
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### A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt$$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
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### Optimal Control of Integral Equations of the first kind

I have managed to find a couple of papers which deal with the optimal control of systems governed by integral equations of the second kind (e.g. here: https://hal.inria.fr/inria-00473952/file/RR-7257....
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### Solving a weighed integral equation

I am struggling in solving a specific type of equation which simply pops out when we're dealing with weighed functions. I think the general context is worth mentioning. Let's say we have a discrete ...
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### Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it
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### Checking whether the function $u(x)=e^x$ solves the integral equation $u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$

The function being $u(x)=e^x$ and the integral equation is $$u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$$ I can't do the integration and I'm confused about how to deal with the parameter $\lambda$ here....
Can anyone help me in finding a closed-form solution to the integral equation $$x\left(t\right)=1-\lambda \int _{0}^{t}e^{-\alpha \left(t-\tau \right)} \cos ^{2} \left(k\tau \right) x\left(\tau \... 0answers 14 views ### when the solution of a volterra integral equation is a probability distribution? I have a Volterra integral equation looking like this$$F(t) = \int_0^t \lambda(s)ds - \int_0^t{F(s)\lambda(t-s)ds}$$is there any condition I can ask on \lambda for the solution F to be upper ... 0answers 38 views ### Love's equation f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1) Let us consider Love's equation:$$f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$$Is f(x) a two times differentiable function? 0answers 56 views ### Cosine Fourier series solution of semi-major axis nonlinear integral equaton Consider an integral equation$$ \frac{1}{z(t)}=f(t)+\alpha\int_0 ^\infty \cos(ts)z(s)\,ds $$I am required to solve for z(t). I approached this problem by considering the integral on right hand ... 1answer 53 views ### How to find exact solution of this volterra equation? I was working on numerical solution of this equation (by block pulse).$$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]B(t) is standard brownian motion. Author of the ...
General form of 'Fredholm Integral Equation of the First Kind' $f(x) = \int_a^b{K(x,t)\phi(t)} dt$ Where $\phi(t)$ is the unkown My special case is $1 = \int_a^b{k(t)\phi(t)} dt$ A trivial ...