# Questions tagged [integro-differential-equations]

An integro-differential equation is an equation involving both the integrals and derivatives of a function. The solution to an integro-differential equation is a function which satisfies the original equation.

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### What is the word for an algebraic equation that includes other functions?

A question about vocabulary/semantics: An algebraic/polynomial equation is an equation that only has polynomial terms, e.g. $x^2+5=0$ (but $\log(x) + x^2 = 5$ is not because of the log). A ...
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### Solving a volterra integro-differential equation

I've encountered a problem where I have to solve a volterra integro-differential equation of the following format. I tried different approaches but the exponential term makes the life miserable. Any ...
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### How to show $F$ is uniquely determined, given initial conditions?

Assume $a,b,c,d\in (0,\infty)$, $n\in \mathbb{N}$, and $g:\mathbb{R}\to\mathbb{R}$ is a continuous, decreasing function. I would like to show that given initial conditions the following equality ...
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### Eigenvalue Problem for Fredholm (Generalised?) Integro-Differential Equations

Consider the following problem: For $\Omega\subset\mathbb{R}^2$ a bounded domain, find $(\lambda, f(x))\in\mathbb{R}\times L^2(\Omega)$ such that \begin{align*} Lf(x) & = \lambda Kf(x) && ...
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### Integro-Differential Equations

I was attempting to solve the following integro-differential equation using convolutions. My answer also had a convolution which did not seem right and was wondering if someone would check my process. ...
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### Solving an lineare integro Differential equation

My integro differential equation reads: $\dot{f}(t)=-\int_0^t\mathrm{d}s\text{ }g(t-s)f(s)$ with $g(t)=\frac{\text{exp}(i\gamma t)}{(1+it)^2}$ with $t\ge 0$, $\gamma>0$. I tried to solve it with ...
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### Linear integro differential operator

I have stated reading Linear integral equation.\ It is mention that for a function $u:\mathbb{R}^{N}\rightarrow\mathbb{R}$ and $x\in\mathbb{R}$ consider the linear integrodifferential operator ...
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### Integro-differential equation

I found the following differential equation: $$\frac{\partial}{\partial t}f\left(x,t\right)=xf\left(x,t\right)+g\left(x\right)\int_{0}^{1}f\left(x',t\right)x'\mathrm{d}x'$$ to solve for $f(x,t)$. ...
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### How to rewrite the Fredholm equation into second IVP

I have the Fredholm integro-differential equation $$u'(x)= u(x) - \frac{x}{2} + \frac{1}{x+1} - \ln(x+1) + \frac{1}{\ln^2 2} \int_0^1 \frac{x}{t+1} u(t)dt, \quad u(0)=0.$$ and I know the exact ...
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### Convert a Fredholm equation to second order IVP

I have the Fredholm integro-differential equation $$u'(x)= u(x) + \int_0^1 \frac{x}{t+1} u(t)dt, \quad u(0)=0.$$ I need to find the corresponded second order IVP. My attemp is: Differentiating in ...
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### Solving a particular integro-differential equation: $g(x) = \int_1^x \frac{F(u)}{\sqrt{x^2-u^2}} \, du$

As the title says: I'm interested in the following integro-differential equation. Let $g:(1,\infty) \to [0,1]$ be given, and assume $g$ is smooth. I want to find functions $F:[1,\infty) \to [0,1]$ ...
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### Solving partial integro-differential equation with symmetry

I want to solve below equation: $$\partial_t u(\vec{\rho},t)=\nabla^2u - \nabla \cdot (u U)$$ where $$U = \int_a^L \frac{u \vec{\rho}}{(\rho^2+a^2)^{3/2}} \rho d\rho d\phi$$ $\phi$ is angle ...
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### Show an integral satisfies the Blasius equation [closed]

Define $H(\zeta)=\int^\zeta_0F(\mu)d\mu$. Show that $H(\zeta)$ satisfies $2v\frac{d^3 H}{d \zeta^3}+H\frac{d^2 H}{d \zeta^2}=0$ I'm not sure how to approach this. Using the fundamental theorem of ...
$$\phi(u)=\int_0^{\infty} \lambda e^{-\lambda t}\int_0^{u+ct}f(x)\phi(u+ct-x)dxdt~~~(1)$$ Substituting $s = u + ct$ in the equation 1, \phi(u)=\dfrac{1}{c}\int_u^{\infty} \lambda e^{-\lambda (s-u)/...
I want to solve a integro-differential equation numerically. The equation is given by : $\dot{c}(t)=-\int_0^t \mathrm{d}t_1f(t-t_1)c(t_1)$ Hereby, $f(t-t_1)$ will be given a realisation of some ...