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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I need to solve an integral equation with feedback

I've encountered the following equation: $p_t = \pi_t + c \int_{-\infty}^te^{-\gamma(t-\tau)}dp_\tau, \quad 0<\gamma, 0<c<1$. It is claimed that it can be rewritten as follows: $p_t = \pi_t + ...
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Numerically Solving a system of coupled Integro-differential equations with different interval of integral than solving interval

I want to solve a system of coupled Integro-Differential equations in the form below: I'm trying to solve this system of IDEs by using Python's IDESolver package, which takes its inputs as: And ...
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Is there a time-reversal invariant algorithm to numerically solve integro-differential equations?

Suppose I have an integro-differential equation of the form $$y'(x) = f(x,y(x)) + \int_{x_0}^{x} ds\; F(x,s,y(s))$$ With some initial condition $y(x_0) = y_0$. The paper "Note on the Numerical ...
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Proof of convergence for the Monte-Carlo method in the Linear Transport Equation

There exists a well known procedure for simulating the transport equation using the Monte Carlo (MC) approach. I want to understand the mathematical connection (not physical) of this approach to the ...
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Differential equation with convolution terms

Suppose that $q,f $ and $p$ are continuously differentiable functions on $\mathbb R$. Is there a method (in general or even in special cases) for solving $p(x)$ in terms of $f(x)$ and $q(x)$. $(x\dot{...
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Solution of first order integro-differential equation

Consider $$ \frac{dy}{dx} + p(x)y = \int_{0}^{\infty} y(x)dx = R~~(say) $$ $$ y(0) = \int_{0}^{\infty} f(x)y(x)dx.$$ I want to solve above differential equation. Here is my try: $$ y(x) = y(0) e^{-\...
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How can I find an analytical solution for this integro-differential equation?

I want to find analytical solutions of the following integro-differential equation: $\left(A\nabla_{\rho}^2 + B\nabla_z^2\right)f(\vec{r}) = C \int{ g(\vec{r},\vec{r\,'}) f(\vec{r\,'})d\vec{r\,'}}, \...
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No solution to general Volterra integral equation of the second type?

I encountered a (likely simple) Volterra-style integral equation as part of my work. For an arbitrary function $p(s)$ which is square integrable, I have the following equation relating $p(s)$ and its ...
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Dealing with a (partial) integro-differential equation

I am dealing with this (partial) integro-differential equation: \begin{align} \frac{\partial v}{\partial t}(t,x) &= c_{3}v(t,x) + \int_{x}^{1} K(x,\xi) v(t,\xi)d\xi,\\ v(0,x)&= v_{0}(x) \end{...
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How can I solve this system of coupled partial integro-differential equations?

Here's my (biophysical) system: https://i.stack.imgur.com/Kchib.jpg First, the intuition: the disk is like a cell (microorganism) that can swim around inside the (2D) box. $R(x,y,t)$ is the ...
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ODE: $y''y+ax+by+c=0,y=k\pm\sqrt2\int\sqrt{a\int\ln(y)dx-(ax+c)\,\ln(y)-by+K}dx,\int\frac{dy}{\sqrt{K-(ax+c)\,\ln(y)+a\int\ln(y)dx-by}}=k\pm\sqrt2x$

Imagine we had a differential equation like: $$y’’-\frac xy=0$$ Now let’s standardize the signs. Note we do not need a constant for the first term because of the zero product property. We can ...
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Solving differential equation featuring $f(1)$ as a coefficient

Starting with $g(x)=\int_{1/x}^1 g(kx)(2-2k)dk+1,$ after integration by parts of the RHS I get the following DE for the second antiderivative of $g(x),$ say $f(x)$: $x^2f''(x)+(2x-2)f'(1)-2(f(x)-f(1))-...
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Name of generalization of convolution to multiple kernel functions

I am wondering if the following integral has some known properties / name in literature. Consider the double convolution with an intermediate non-trivial product ($h$ is not a constant): \begin{...
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Nonlinear differential equation. [closed]

Can someone give me a hint to solve the next nonlinear diferential equation? \begin{equation} \frac{y'}{y}+\frac{f(t)+g(t)y}{h(t)+k(t)y}=0 \end{equation} in some set where is well defined. I know ...
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Is it possible to approximate this type of differential equation without circular references?

For the differential equation $$\frac{dx^2(t_1)}{dt_1^2}=\int_{-\infty}^\infty{f\left(x(t_1),x(t_2),\frac{dx(t_1)}{dt_1},\frac{dx(t_2)}{dt_2}\right)dt_2}$$ One way to approximate the differential ...
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Integro-Differential equation from my Complex analysis exam

My recent complex analysis exam had the following problem as the last question, which I had a hard time solving. The problem Use the Laplace transform to solve the following differential equation for $...
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Volterra (parabolic) integro-differential equation with spatial derivative inside the integral

Let $W(t, x)$ be a function in a domain satisfying \begin{align} (\mathbb{L}W)(t, x) + f(t, x)\int_0^t W_x(s, x)\mathrm{d}s = g(t, x), \tag{1} \end{align} where $\mathbb{L}$ is a parabolic ...
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Maximum principle for a particular kind of (parabolic) integro-differential equation

Let $W(t, x)$ be a function in the domain $\Omega = \{(t, x): 0 < t < T, a_1(t) < x < a_2(t)\}$, with $a_1, a_2$ be continuous functions. Suppose that W satisfies the dynamics \begin{align}...
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Solve PDE. Is this the right solution? $\frac{\partial u^{2}}{\partial x \partial y} + \frac{\partial u}{\partial y} + x + y + 1 = 0 $

This is the equation: $\frac{\partial u^{2}}{\partial x \partial y} + \frac{\partial u}{\partial y} + x + y + 1 = 0 $ I also have a solution but I don't know how it removed -1 in front of the arrow
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integro- (partial) differential equation

I'm dealing with a certain kind of integro-differential equation. The equation reads as : \begin{equation} \frac{du(t,x)}{dt} = \int_{\Omega} u(t,y)K(x,y) dy \end{equation} for some nice kernel ...
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Malthus Model - Solution Differential Equation

I have this equation of a time dependent Malthus model with a term representing a time dependent immigration: $$N'(t)=r(t)N(t) + m(t)$$ with $r(t)$ and $m(t)$ both continuous and periodic with Period $...
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How to calculate this kind of integrataion

Context of my problem: In a continuous-time scenario, consider a factory with a production function $f(\mathbf{x}(t),\mathbf{y})$, where $\mathbf{x}(t)$ (a vector) represents the skills of its workers,...
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How to derive a Boltzmann-like equation for an opinion formation model?

I'm studying an opinion formation model as follows. Opinions are given by $w \in \mathcal{I} = [-1,1]$. If an individual with opinion $w$ meets an individual with opinion $v$, the interaction is ...
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Differential equation with the Substitution method

I'm trying to solve this equation below : $$ y' = \sin\left(\frac{y}{x}\right) + \frac{y}{x} \tag 1 $$ The first step was to to substitute $\frac{y}{x}$ with $u$ => $ u = \frac{y}{x}$ => $...
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Solving ODE with non-local term

I want to ask a similar question as found here: Solving an ODE with non-local coefficient. Except my ODE is slightly more complicated because the integral is not constant over $x$. Consider this ...
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Asymptotic solution of an integro-differential equation

Motivated by a problem of Lévy-flights in a potential, I am looking for the asymptotics of a function $f(y)$, which obeys the the integro-differential equation $$ e^{2y} f(y) = C \frac{d}{dy}\int_y^\...
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Advection-style first order nonlinear PDE system

Given functions $a(t,x,y) > 0, d(t,x) >0$ and constants $b,c > 0$, I would like to find the three functions $u(t,x),v(t,x) , w(t,x,y)$ that solve the system: \begin{align} u_t - u_x &= -\...
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Solving the integral form of a second-order linear PDE

I am trying to solve for the function $A(x,y)$ given an equation of the form $\frac{\partial}{\partial y}\int_{x_0}^xA(x,y)f_1(x,y)dx + \frac{\partial}{\partial x}\int_{y_0}^yA(x,y)f_2(x,y)dy = A(x,y)...
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Passage from the Cauchy problem to the integral equation

I try understand one paper about inverse spectral problem for $$-y''+q(x)y + \int_0^x M(x-t)y(t)dt = \lambda y, \quad 0 < x < \pi, \quad y(0)=y(\pi)=0$$ There is statement that Cauchy problem of ...
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Is there a regular method to solve some types of integro-differential equations?

So I just got out of my first exam on differential equations. One point had an integral equation which we were supposed to turn into a first order linear differential equation. The thing is I wasn't ...
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Sturm-Liouville Equation with a measure involved

I'm working with the book "Continuous Martingales and Brownian Motion", it is used frequently the next result (which is in the appendix): But I'm not sure how they get to the equation $(+)$,...
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Survival probability integro-differential equation question

For the Classical Risk model, the survival probability, $\phi(u)$, satisfies the integro-differential equation: $$\phi(u)=1-\psi(u)=\int\limits_o^\infty \int\limits_o^{u+ct}\lambda e^{-\lambda t}\phi(...
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Help to solve linear 1st order integro-PDE

I am wondering if there are good methods (analytical, numerical) to solve equations of the form $$ f(\mathbf{x})\frac{\partial u(\mathbf{x},t)}{\partial t} + \mathbf{a}(\mathbf{x})\cdot \nabla u(\...
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Simple uniqueness conditions for integro-differential

Let $h:\mathbb{R}^3 \to \mathbb{R} $. Consider the following integro-differential equation: $$g'(t)=\int_0^t h(t,g(t),g(z))\;dz $$ With initial condition $g(0)=0$, is a unique solution for $g:[0,\...
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How to solve non linear integro-differential equation

I am trying to find a solution for this equation: $$f(t)-kt+\int_0^tg(t-\tau)\frac{df^2}{d\tau}d\tau=0$$ I know the $g(t)$ function and I have initial conditions $f(0)$ and $f'(0)$, but I really don't ...
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Existence of solution for a simple linear integral equation

Does this linear integral equation defined on $[-a,a]$ have any nonconstant solution? I am more interested in the exsistence of solution rather than the solution itself. The boundary condition has two ...
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How to estimate numerical bin for Integro-differential equation system?

I try to solve an integro-differential equation system numerically (with LSODA.) The system is following: ( $'$ is the derivation w.r.t. $x$) $y'(x,s_{1}) = f(x,y(x,s_{1}),I[y(x,s)])\\ y'(x,s_{2}) = ...
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General solution of the matrix equation $\dot{F}(t)=L(t) F(t)$

Does there exist a solution to the matrix differential equation $$\dot{F}(t)=L(t) F(t)$$ for a general matrix $L(t)$?
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Approximating integral term in integro-differential equations

I am trying to find an approximate solution to the following integro-differential equation for the $n$-dimensional vector $\mathbf{x}(t)$ in some interval $t\in[t_0, t_1]$: $$ \frac{d\mathbf{x}(t)}{dt}...
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Adiabatic elimination of variables from coupled first-order differential equations

I have asked a related question on the physics stack exchange website, but have realised that this question is actually more about rigorous maths than physics. Suppose I have the following set of ...
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A difficult game theory riddle

Suppose a master hires $n$ servants who work for him. At the end of the day, the service come to him and request a wage for their service. They are able to request a wage up to $\$1$, but no more. ...
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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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Solving $x^2y'' -5xy' +8y=24$

I tried solving $x^2y''-5xy' +8y=24$ using variation of Parameters and I keep getting the wrong answer The correct Answer is $y=C_1x^2+C_2x^4+3$ according to my textbook So I first got $y_c=C_1x^2+...
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On partial integro-differential equation

How to solve the following partial integro-differential equation \begin{eqnarray}u_t(x,t)=\int\limits_{\mathbb{R}}u(y,t)dy\\ u(x,0)=u_0(x) \in L^1(\mathbb{R}) \end{eqnarray} Is there any well-...
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How to solve this differential equation involving an integral

I am working on a problem involving finding a function for which the area under the curve is equal to the arc length. I therefore come up with $\int_a^b(k\sqrt{1+[f'(x)]²} -f(x))dx =0$. Frankly, I ...
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Analytical solution of equation where variable is an integral of itself

I have arrived at the equation below to describe the dynamics of a system. As you can see, state variable $y$ is equal to an integral of itself. $$y(t)=k\int_0^t\frac{1}{y(\tau)}d\tau$$ I'd like to ...
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When is the solution to this integro-differential equation 0?

Consider the integro-differential equation $$ \frac{df(t)}{dt} = - \int_0^t ds \ g(s) f(t-s) $$ subject to the initial condition $f(0)=0$ and where $g$ is a known function. My Question: Is the ...
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Proving the existence and uniqueness of partial integro-differential equation

I am working on a type (shown below) of nonlinear partial integro-differential equation with conformable fractional derivative, $$Τ_t^α u(x,t)=g(x,t)+u(x,t)+\int_0^t \int_0^x \dfrac{(f(y,s))^p}{y^{1-...
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Solving Integral Equation by Converting to Differential Equations

Consider the problem $$\phi(x) = x - \int_0^x(x-s)\phi(s)\,ds$$ How can we solve this by converting to a differential equation?
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Asymptotic behaviour of Volterra integrodifferential equation

For an equation of the form, $$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{y}(t) = \int_0^t \mathbf{K}(t-\tau)\mathbf{y}(\tau)d\tau,$$ Can it be shown that in the long time, $t\to\infty$, limit, the ...
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