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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18 views
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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48 views
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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54 views
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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26 views
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:
$$\begin{align}
u_t(x,t)&=g(x,t)+\int_{0}^{x}u(x,y)dy \tag{1}\label{1}\\
u(x,0)&=b(...
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51 views
Existence of solution for a simple linear integral equation
Does this linear integral equation defined on $[-a,a]$ have any nonconstant solution? I am more interested in the exsistence of solution rather than the solution itself. The boundary condition has two ...
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27 views
Parity of a simple linear differential equation's solution [duplicate]
Is it possible to solve this linear integral equation defined on $[-a,a]$? The boundary condition has two possibilities: 1) $y(-a)=y(a)=b$ and 2) $-y(-a)=y(a)=c$.
$$y''(x)=\int_{-a}^{a}{ d x' \frac{y(...
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21 views
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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0answers
9 views
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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0answers
33 views
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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27 views
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 $t\in[t_0, t_1]$:
$$
\frac{d\mathbf{x}(t)}{dt}...
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14 views
Proving a identity on the variation of parameters formula using supplementary projection matrices
I'm solving the non-autonomous system
$$xĀ“=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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61 views
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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8 views
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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1answer
86 views
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
36 views
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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13 views
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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1answer
52 views
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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47 views
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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27 views
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 $\int_a^b(k\sqrt{1+[f'(x)]Ā²} -f(x))dx =0$. Frankly, I ...
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1answer
35 views
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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0answers
43 views
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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0answers
39 views
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,
$$Ī¤_t^Ī± u(x,t)=g(x,t)+u(x,t)+\int_0^t \int_0^x \dfrac{(f(y,s))^p}{y^{1-...
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Reference: Good introduction to integro-differential equations
Does anyone know of a good introduction to integro-differential equations, including their theory, solution, and numerical solutions. I have looked through my books on ODEs, dynamical systems, and ...
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2answers
440 views
Solving Integral Equation by Converting to Differential Equations
Consider the problem
$$\phi(x) = x - \int_0^x(x-s)\phi(s)\,ds$$
How can we solve this by converting to a differential equation?
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1answer
19 views
Asymptotic behaviour of Volterra integrodifferential equation
For an equation of the form,
$$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{y}(t) = \int_0^t \mathbf{K}(t-\tau)\mathbf{y}(\tau)d\tau,$$
Can it be shown that in the long time, $t\to\infty$, limit, the ...
2
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0answers
40 views
Integro-differential equation with convolution
Given a $\mathcal{C^\infty}$ matrix-valued function $f$ from $\mathbb{R}^+$ to $\mathbb{R}^{n,n}$, I'd like to solve the following integro-differential equation:
$$\ddot x(t) + \int_0^t f(\tau) \dot ...
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21 views
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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2answers
51 views
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 ...
2
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1answer
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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1answer
46 views
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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0answers
55 views
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 ...
3
votes
2answers
59 views
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:
$$
x'(t)=f(t,x(t)) + \int_0^t g(x(s)...
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0answers
94 views
differential equation with integral function
Could you please help me with any information which would allow me to get an explicit solution for this equation?
It is an implicit solution to an optimization problem involving resource allocation ...
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0answers
77 views
Numerically solving a system of linear integro-differential equations in Matlab
Given the following system of linear integro-differential equations
$$
\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 ...
2
votes
1answer
81 views
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_{...
2
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0answers
53 views
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) && ...
2
votes
1answer
291 views
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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0answers
54 views
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 ...
0
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1answer
97 views
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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0answers
44 views
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 $f(x,t)$. ...
0
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1answer
48 views
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 ...
1
vote
1answer
60 views
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 ...
3
votes
1answer
132 views
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 $F:[1,\infty) \to [0,1]$ ...
1
vote
0answers
67 views
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 ...
1
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0answers
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 ...
0
votes
1answer
65 views
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)/...
1
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0answers
151 views
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 ...
