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.
1
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0answers
28 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,
$$Τ_t^α u(x,t)=g(x,t)+u(x,t)+\int_0^t \int_0^x (f(y,s))^p/y^{1-α} s^{1-...
3
votes
0answers
34 views
Reference: Good introduction to integro-differential equations
Does anyone know of a good introduction to integro-differential equations, including their theory, solution, and numerical solutions. I have looked through my books on ODEs, dynamical systems, and ...
2
votes
2answers
88 views
Solving Integral Equation by Converting to Differential Equations
Consider the problem
$$\phi(x) = x - \int_0^x(x-s)\phi(s)\,ds$$
How can we solve this by converting to a differential equation?
0
votes
0answers
6 views
Volterra asymptotic
For a system of equations of the form,
$$
\frac{d}{dt}\mathbf{y}(t) = \int_0^t\mathbf{K}(t-\tau)\mathbf{y}(\tau) d \tau
$$
Can it be shown that in the long time, , limit, the solution is given by,
$$
\...
0
votes
1answer
13 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
votes
0answers
28 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:
$$\ddot x(t) + \int_0^t f(\tau) \dot ...
0
votes
0answers
15 views
Numerical method vs topological method for periodic solutions
I'm working with a scalar functional differential equations that is T-periodic in time. The precise form is somewhat complicated. I'm interested in T-periodic solutions and already have an existence ...
0
votes
2answers
42 views
Differential equation $2x^4yy'+y^4 = 4x^6$
I have differential equation $2x^4yy'+y^4 = 4x^6$
How to find real parameter $m$ for which, when we introduce substitution $y=z^m$, given equation becomes first order homogeneous differential ...
2
votes
1answer
41 views
Integro-differential equation including a convolution of the first derivative.
I am having difficulty finding the right approach to solving the following differential equation,
$$
y''(t)+\int_t^Tg(s-t)y'(s)\,ds=f(t),
$$
with the boundary conditions,
$$y(0)=y_0\,,\quad y(T)=0.$$
...
0
votes
1answer
23 views
How to deal with an integro-differential equation of this form - fixed points?
I've encountered an integro-differential equation of the following form:
$$
\frac{dx(t)}{dt} = \int_0^t ds\ f_{1}(s) - \int_0^t ds\ f_{2}(s) x(t - s)
$$
The functions $f_{1}(t)$ and $f_{2}(t)$ are ...
0
votes
0answers
26 views
numerically solve integro-differential equations with forward time
guys do you know some methods that allows to numericaly solve efficiently (maximising both time and stability ) equations like this
$ \frac{\partial^2 \psi}{\partial^2 t}-\xi * \frac{\partial^2 \psi}...
1
vote
0answers
36 views
Does differentiating an integro-differential equation results in equivalent stability of the solution?
Consider the following integro-differential equation:
$$\dot{x}(t)=ax(t)+b\int_0^tx(\tau)\text{d}\tau,$$
where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we derive the above equation and ...
0
votes
0answers
42 views
Numerical approaches for coupled nonlinear PDEs with time-dependent domain
I am trying to solve numerically a system of non-linear PDEs of the following form:
$$ 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{...
3
votes
2answers
58 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:
$$
x'(t)=f(t,x(t)) + \int_0^t g(x(s)...
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votes
0answers
85 views
differential equation with integral function
Could you please help me with any information which would allow me to get an explicit solution for this equation?
It is an implicit solution to an optimization problem involving resource allocation ...
1
vote
0answers
52 views
Numerically solving a system of linear integro-differential equations in Matlab
Given the following system of linear integro-differential equations
$$
\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
51 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
votes
0answers
44 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
137 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.
...
0
votes
0answers
52 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
votes
1answer
47 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 ...
0
votes
0answers
38 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
votes
1answer
44 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
44 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
124 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
64 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
vote
0answers
49 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
43 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,
$$\phi(u)=\dfrac{1}{c}\int_u^{\infty} \lambda e^{-\lambda (s-u)/...
1
vote
0answers
119 views
Solving integro-differential equation - numerically
I want to solve a integro-differential equation numerically.
The equation is given by :
$\dot{c}(t)=-\int_0^t \mathrm{d}t_1f(t-t_1)c(t_1)$
Hereby, $f(t-t_1)$ will be given a realisation of some ...
2
votes
0answers
52 views
How to solve this damped integro-differential equation?
I need to solve the following Integro-Differential Equation.
$$
\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}
$$
in ...
3
votes
3answers
90 views
How to find $f$ which satisfies $f'(x)=f(x)+\int_{0}^{2}f(t)dt$ and $f(0)=\frac{4-e^2}{3}$?
$$f'(x)=f(x)+\int_{0}^{2}f(t)dt \hspace{1cm} with \hspace{1cm} f(0)=\frac{4-e^2}{3}.$$How to find $f(x)$? I tried integrating on both sides but I had no idea next.
4
votes
2answers
318 views
How to solve this integro-differential equation?
I came across this integro-differential equation to solve
$$\frac{du(x;t)}{dt}=-\lambda\int_0^xu(\xi;t)\;d\xi\tag{1}$$
under the initial condition $u(x;0)=f(x)$ where $x$ is a parameter, $\lambda$ is ...
2
votes
1answer
59 views
integro differential equation clarification
So I am given the $ODE$:
$$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^{...
2
votes
1answer
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 ...
0
votes
0answers
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 ...
1
vote
2answers
160 views
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 ...
0
votes
1answer
86 views
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....
0
votes
1answer
43 views
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 ...
0
votes
1answer
95 views
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 ...
3
votes
0answers
212 views
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 ...
2
votes
3answers
59 views
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:
$$\frac{F(p)}{p}-pF(p)+f(0)=\frac{1}{p^2}$$
$$\frac{F(p)}{...
0
votes
1answer
46 views
An integro-differential equation arising in solving a 2nd order ODE
I'd like to solve the ODE
$$(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)=(-...
1
vote
0answers
48 views
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):
$$...
1
vote
1answer
117 views
Are integro-differential equations considered dynamical systems?
A definition of the dynamical system is that:
$\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$,
$E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$,
then $\phi_0(...
1
vote
0answers
63 views
Langevin equation vs. Volterra integro-differential equation
What is the difference between a Volterra integro-differential equation and a Langevin equation (with an integral damping term)? Both seem to be of the same form.
It appears to me that the former is ...
0
votes
1answer
82 views
Functional equations vs. Mathematical ZFC models [closed]
Is the solutions of a functional equation uniquely determined by ZFC axioms, or could it requires adittional axioms?
I refer to differential and integro-differential equations principally
0
votes
1answer
128 views
Fourier transform of Integro-differential equation
A function $f(x)$ vanishes at $x \to \pm \infty$ and satisfies this equation
$\frac{df}{dx} + f(x) = \delta(x) - \int^{+\infty}_{-\infty} g(x-y)f(y) \, dy$
How to obtain the fourier transform, $F(k)$...
2
votes
0answers
206 views
How to evaluate Integro-Differential Equation using Laplace convolution?
Can someone please explain how I begin to evaluate the following integro-differential equation?
I know that it involves a convolution, but the $y(τ)$ within the integral is throwing me off.
$$\int_0^...
2
votes
0answers
145 views
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(...
0
votes
1answer
1k views
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)}{\...
