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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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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 (f(y,s))^p/y^{1-α} s^{1-...
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Reference: Good introduction to integro-differential equations

Does anyone know of a good introduction to integro-differential equations, including their theory, solution, and numerical solutions. I have looked through my books on ODEs, dynamical systems, and ...
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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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Volterra asymptotic

For a system of equations of the form, $$ \frac{d}{dt}\mathbf{y}(t) = \int_0^t\mathbf{K}(t-\tau)\mathbf{y}(\tau) d \tau $$ Can it be shown that in the long time, , limit, the solution is given by, $$ \...
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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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Integro-differential equation with convolution

Given a $\mathcal{C^\infty}$ matrix-valued function $f$ from $\mathbb{R}^+$ to $\mathbb{R}^{n,n}$, I'd like to solve the following integro-differential equation: $$\ddot x(t) + \int_0^t f(\tau) \dot ...
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Numerical method vs topological method for periodic solutions

I'm working with a scalar functional differential equations that is T-periodic in time. The precise form is somewhat complicated. I'm interested in T-periodic solutions and already have an existence ...
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Differential equation $2x^4yy'+y^4 = 4x^6$

I have differential equation $2x^4yy'+y^4 = 4x^6$ How to find real parameter $m$ for which, when we introduce substitution $y=z^m$, given equation becomes first order homogeneous differential ...
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Integro-differential equation including a convolution of the first derivative.

I am having difficulty finding the right approach to solving the following differential equation, $$ y''(t)+\int_t^Tg(s-t)y'(s)\,ds=f(t), $$ with the boundary conditions, $$y(0)=y_0\,,\quad y(T)=0.$$ ...
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How to deal with an integro-differential equation of this form - fixed points?

I've encountered an integro-differential equation of the following form: $$ \frac{dx(t)}{dt} = \int_0^t ds\ f_{1}(s) - \int_0^t ds\ f_{2}(s) x(t - s) $$ The functions $f_{1}(t)$ and $f_{2}(t)$ are ...
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numerically solve integro-differential equations with forward time

guys do you know some methods that allows to numericaly solve efficiently (maximising both time and stability ) equations like this $ \frac{\partial^2 \psi}{\partial^2 t}-\xi * \frac{\partial^2 \psi}...
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Does differentiating an integro-differential equation results in equivalent stability of the solution?

Consider the following integro-differential equation: $$\dot{x}(t)=ax(t)+b\int_0^tx(\tau)\text{d}\tau,$$ where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we derive the above equation and ...
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Numerical approaches for coupled nonlinear PDEs with time-dependent domain

I am trying to solve numerically a system of non-linear PDEs of the following form: $$ u(s,t)\quad \text{and} \quad v(s,t) \quad \text{for } t\geq 0 \text{, } s \in [0,L(t)]$$ such that $$ \begin{...
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For $x'(t)= f(t, x(t)) + \int_0^t g(x(s))\, ds$ show that if $x_1(0) \leq x_2(0)$ that also $x_1(t) \leq x_2(t)$

Note The Question has changed significantly since it was posed, but it is now as follows: Question Suppose we have some integro or delayed differential equation: $$ x'(t)=f(t,x(t)) + \int_0^t g(x(s)...
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differential equation with integral function

Could you please help me with any information which would allow me to get an explicit solution for this equation? It is an implicit solution to an optimization problem involving resource allocation ...
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Numerically solving a system of linear integro-differential equations in Matlab

Given the following system of linear integro-differential equations $$ \frac{d}{d t}B(t)+\int_{0}^{+\infty}C(x,t)dx+A(t)=0,\\ \left[\frac{\partial}{\partial t}+V(x)\right]C(x,t)+B(t)=0,\\ \frac{d}{d ...
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Find the limiting value of a Delayed (or Ordinary) Differential Equation

General Question Suppose we have a function $f(s)$ which satisfies an Ordinary Differential Equation (ODE) $f'(s) = A(s,f(s))$ which can not be solved explicitly. Is there some method to find $\lim_{...
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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 ...
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How to differentiate Survival probability function?

$$\phi(u)=\int_0^{\infty} \lambda e^{-\lambda t}\int_0^{u+ct}f(x)\phi(u+ct-x)dxdt~~~(1)$$ Substituting $s = u + ct$ in the equation 1, $$\phi(u)=\dfrac{1}{c}\int_u^{\infty} \lambda e^{-\lambda (s-u)/...
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Solving integro-differential equation - numerically

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 ...
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How to solve this damped integro-differential equation?

I need to solve the following Integro-Differential Equation. $$ \begin{equation} \frac{da}{dt} = (-i\Delta-\kappa)a + c -\int_{-\infty}^0 e^{-\gamma(t-\tau)}g(\tau)a(t-\tau)d\tau, \end{equation} $$ in ...
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How to find $f$ which satisfies $f'(x)=f(x)+\int_{0}^{2}f(t)dt$ and $f(0)=\frac{4-e^2}{3}$?

$$f'(x)=f(x)+\int_{0}^{2}f(t)dt \hspace{1cm} with \hspace{1cm} f(0)=\frac{4-e^2}{3}.$$How to find $f(x)$? I tried integrating on both sides but I had no idea next.
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How to solve this integro-differential equation?

I came across this integro-differential equation to solve $$\frac{du(x;t)}{dt}=-\lambda\int_0^xu(\xi;t)\;d\xi\tag{1}$$ under the initial condition $u(x;0)=f(x)$ where $x$ is a parameter, $\lambda$ is ...
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1answer
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integro differential equation clarification

So I am given the $ODE$: $$y'(t) + 2e^{-2t}\int_{0}^{t} e^{2u} y(u) du=e^{-t}\sin(t), y(0)=0$$ and I'm supposed to find $y(t)$ So if I move the exponential inside, I get: $$y'(t) + 2\int_{0}^{t}e^{...
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131 views

Solving an integro-differential equation

I have a set of coupled integro-differential equations: $$ \frac{dx_i(t)}{dt}=-x_i(t)+f_i(\mathbf{x}(t))+\sum_j{\partial_{x_j}f_i(\mathbf{x}(t))\int_{0}^{t}dt'f_j(\mathbf{x}(t'))e^{-(t-t')} }$$ The ...
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115 views

Using this trick to solve an integro-differential equation

I have a set of coupled integro-differential equations: $$ \frac{dx_i(t)}{dt}=-x_i(t)+f_i(\mathbf{x}(t))+\sum_j{\partial_{x_j}f_i(\mathbf{x}(t))\int_{0}^{t}dt'f_j(\mathbf{x}(t'))e^{-(t-t')} }$$ The ...
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Reference books on numerical methods for PDE and integro differential equations

Can you recommend a few good reference books and textbooks on numerical analysis of partial differential and integro partial differential problems that do not assume much knowledge of numerical ...
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Solving Ordinary Integro-delayed differential equation

I have an ordinary integro-differential equation of the form $$y''(t)+C_1y(t)+C_2\int_{0}^{t}f(t-\tau)y''(\tau)d\tau +C_3\int_{0}^{t}f(t-\tau)y(\tau)d\tau=0$$ where $C_1$,$C_2$ and $C_3$ are constants....
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Differential integro-equation

So I'm stuck with the following problem: $b'(t)$ + $\int_{0}^{t} (t-q)b(q)dq = t$ $b(0)=0$ The book calls this an integro-differential, but I can't really understand how to solve it, currently I've ...
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Method for solving a differential equation of a particular form

I have a differential equation of the form $$ \frac{\text{d}y(x)}{\text{d}x} - c_1z_1(x)x^2 \int_x^{c_2} \frac{z_1(x)}{x^2 z_2(x)} y \,\text{d}x = -z_3(x) $$ where $y, z_1, z_2$ and $z_3$ are ...
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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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Find f(x) $\int_0^x f(u)du - f'(x) = x$

Find f(x) $$\int_0^x f(u)du - f'(x) = x$$ I was not given f(0) which makes it difficult for me to find f(x). This is what I have thus far: $$\frac{F(p)}{p}-pF(p)+f(0)=\frac{1}{p^2}$$ $$\frac{F(p)}{...
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An integro-differential equation arising in solving a 2nd order ODE

I'd like to solve the ODE $$(1+\Phi_{x})\,\Phi_{xx}=(-1+\sqrt{1+\Phi^2})\,\Phi,$$ where $\Phi=\Phi(x)$, $x\in(-1,1)$. It can be written as $$ \frac{d}{dx}\Big(\Phi_{x}+\frac{1}{2}\Phi_{x}^2\Big)=(-...
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How to classify this integro-differential equation?

I have a system of three coupled integral equations for three unknowns $j(t), \bar{x}(t)$ and $\lambda(t)$ to be solved between $t=0$ and $t=T$ (b.c. are $\bar{x}(0)=x_0$ and $\bar{x}'(T)=0$): (1): $$...
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117 views

Are integro-differential equations considered dynamical systems?

A definition of the dynamical system is that: $\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$, $E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$, then $\phi_0(...
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Langevin equation vs. Volterra integro-differential equation

What is the difference between a Volterra integro-differential equation and a Langevin equation (with an integral damping term)? Both seem to be of the same form. It appears to me that the former is ...
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1answer
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Functional equations vs. Mathematical ZFC models [closed]

Is the solutions of a functional equation uniquely determined by ZFC axioms, or could it requires adittional axioms? I refer to differential and integro-differential equations principally
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128 views

Fourier transform of Integro-differential equation

A function $f(x)$ vanishes at $x \to \pm \infty$ and satisfies this equation $\frac{df}{dx} + f(x) = \delta(x) - \int^{+\infty}_{-\infty} g(x-y)f(y) \, dy$ How to obtain the fourier transform, $F(k)$...
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How to evaluate Integro-Differential Equation using Laplace convolution?

Can someone please explain how I begin to evaluate the following integro-differential equation? I know that it involves a convolution, but the $y(τ)$ within the integral is throwing me off. $$\int_0^...
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Newton's method on a surface

I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - r')f(u(...
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Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial I(t,\omega)}{\...