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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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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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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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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How to solve the nonlinear partial integro-differential equation by the finite difference method?

How to solve the following nonlinear partial integro-differential equation? Suppose the following equation: $m \ddot{v}+c_{1} \dot{v} + D\left(v^{\prime \prime \prime}+v^{\prime} v^{\prime \prime 2}+...
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Proving the existence and uniqueness of partial integro-differential equations

If I want to solve a partial differential equation (for simplicity I will take this example) of this type: $$\begin{align} u_t(x,t)&=g(x,t)+\int_{0}^{x}u(x,y)dy \tag{1}\label{1}\\ u(x,0)&=b(...
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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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Parity of a simple linear differential equation's solution [duplicate]

Is it possible to solve this linear integral equation defined on $[-a,a]$? The boundary condition has two possibilities: 1) $y(-a)=y(a)=b$ and 2) $-y(-a)=y(a)=c$. $$y''(x)=\int_{-a}^{a}{ d x' \frac{y(...
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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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Is there a closed form expression of the parallel DGLAP evolution equation?

The parallel DGLAP evolution equation is given by $$t\frac{\partial f(x,t)}{\partial t} = \frac{\alpha_s(t)}{2\pi} \int_x^1 \frac{d\hat{x}}{\hat{x}} P_{qq}(\hat{x})f\left(\frac{x}{\hat{x}},t\right)$$ ...
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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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Proving a identity on the variation of parameters formula using supplementary projection matrices

I'm solving the non-autonomous system $$x´=A(t)x+f(t,x)$$ with initial condition $x(t_1)=\theta_1$,the variation of parameters formula says that the solution is given by $$x(t)=Y(t)Y^{-1}(t_1)\...
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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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Do The Probabilities of Dispersal Kernels Change Over Time?

I'm beginning to study dispersal kernels from an economics background (studying bioeconomics) and we want to incorporate dispersal kernels into an integrodifference equation in discrete time and on a ...
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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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1answer
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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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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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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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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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60 views

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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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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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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51 views

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 ...