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Questions tagged [delay-differential-equations]

Questions about delayed differential equations.

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Using delay block in Simulink to account for flow rate in a fluid loop

I am trying to model a fairly simple cooling system loop where coolant flows over a battery to remove heat, then flows into a large reservoir where the coolant is mixed, coolant then flows out of the ...
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44 views

Approximating $\max \big\{\frac{x_\tau}{x}\big\}$ as $x$ and $x_\tau \rightarrow 0^+$

I have the following delay system: $$x'(t) = g(t,\tau,x,x_\tau)$$ Given that $g(\cdot)$ is smooth and bounded, $x(t)$ is bounded in a non-negative region. What are some possible ways to obtain an ...
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2answers
230 views

Periodic or non periodic function plotting

I am currently working on plotting a function and figuring out if its periodic or not. The function is as follows:$$x_n=\beta\cdot x_{n-1}+\alpha\gamma\cdot\operatorname{sgn}(x_{n-3})+\alpha(1-\gamma)\...
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how to find Stability of N=0 and characteristic equations corresponding to positive steady states

I have spend my enough time to figure out that how i can find the stability of below delayed logistic equation at N=0 and how I can find characcteristic euation corresponding to positive steady states....
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4answers
278 views

Solve differential equation $-f'(x)= a_1 f(a_2 x+a_3)$ with $f(0)=1$.

How to solve the following differential equation \begin{align} -f'(x)= a_1 f(a_2 x+a_3), \end{align} where $f(0)=1$. I looked around I think this falls under the category of discrete delayed ...
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26 views

Using Euler Eqns to separate a quadratic into real and imaginary components

I'm working on a research paper on Delay Differential Equations and have arrived at the same characteristic equation for the linearised system: \begin{align} \lambda^2+p\lambda+r+(s\lambda+q)e^{-\...
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1answer
33 views

Delayed differential equation is positive for all $t>0$

if any one can help me how to prove this theorem. Prove that for the IVP $$ \left\{ \begin{aligned} x′(t)&=cx(t)[1−x(t−r)]\\ x(\mu)&=\phi(\mu),\quad \mu \in [−r,0] \end{aligned} \right\...
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1answer
263 views

Prove that there is a unique continuous solution to the following integral equation.

I am trying to prove that there is a unique continuous solution to the integral equation $$F(\alpha) = \int_{0}^{\alpha}F\left(\frac{t}{1-t}\right)\frac{dt}{t}; \qquad F(\alpha)=1 \text{ for } \alpha\...
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3answers
56 views

Existence of a real valued function satisfying $f'(x)=f(x+1)$ where $f(x)\ne0$

I am wondering about the existence of a real valued function that satisfies $f'(x)=f(x+1)$ other than the trivial solution $f(x)=0$. I thought a likely solution would be a repeating (perhaps ...
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1answer
154 views

How to solve following linear differential-difference equation?

How to solve following linear differential-difference equation? $$\frac{da_{n}(t)}{dt}=i k na_{n}(t)+G\left\{n(n-1)a_{n-2}(t)-a_{n+2}(t) \right\},~n=0,1,2,\ldots~~~~~(1)$$ where, k and G is a ...
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1answer
18 views

How to calculate the Delay Formula (clarify the formula)

I would like to understand how their respective answers were reached, this is a question in my current class and everyone in the class has A-Level mathematics experience - I have asked my tutor to ...
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0answers
33 views

Show monotonicity of solution of Delayed Differential Equation with respect to a parameter

Short Description of the General Question Suppose we have some Delayed Differential Equation (DDE) which depends on a parameter $a$, $x_a'(t)=f(a,x_a(t),x_a(t-s))$ for some fixed $a$ and $s$. I would ...
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2answers
55 views

A challenging delay differential equation with both delay and advanced argument.

I've been working with this kind of delay differential equation \begin{equation} \begin{split} &f(x)' = f(x-1)^2 - f(x) \cdot (f(x+1) + 1), \\ &f(x) = f_0>0, \,\,\, 0\leq x < 1. \end{...
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finite difference method to the delayed damped wave equation

I want to do the numerical simulations in MATLAB for the following partial delay differential equations with Dirichlet boundary condition. I need to know the simplest method to do that. \begin{...
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0answers
31 views

What is the best numerical method to solve a system of semi-linear parabolic differential equations with time delays and Neumann boundary conditions?

I want to do the numerical simulations in MATLAB for a system of partial delay differential equations with a diffusion term and Neumann boundary conditions.I need to know the simplest method to do ...
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1answer
33 views

Linearizing the following equation

I'm working through a section in a book I'm reading about delay differential equations (Semi-discretization for time delay systems, Springer), and the authors are discussing the following equation $$\...
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0answers
59 views

delay partial differential equations

In an effort to solve a delay partial differential equation $$\partial_t f(t,x)= \Phi(x) f(t,x)+\Psi(x) f(t,x-\alpha),$$ with $$f(0,x)=1,\hspace{0.3cm} f(t,0)=1$$ Where $\alpha$ is the delay ( a real ...
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0answers
25 views

A system similar to a linear system

Let us consider the system $$\eqalign{ & {\alpha _{1,1}}f(t) + {\alpha _{1,2}}f(t + \tau ) + {\beta _{1,1}}g(t) + {\beta _{1,2}}g(t + \tau ) = 0 \cr & {\alpha _{2,1}}f(t) + {\alpha _{2,2}...
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0answers
101 views

Solving a Delay Differential Equation with the Method of Steps

I am trying to verify if my solution is correct for the following Delayed Differentiable Equation of $y'(t) = 3y(t - 2)$ with the history function $h(t) = 1$ for $t \leq 0$. On the interval $0 \leq ...
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1answer
32 views

Existence of a surjective map between the two space $X$ and $C([a,b],X)$

Let $X$ a Hilbert space and $C\left(\left[a,b\right],X\right)$ the space of Continuous functions from $\left[a,b\right]$ into $X$. My question is as follows: is there exists any surjectif map from $C\...
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0answers
51 views

construction of Lyapunov functionals for DDE

Consider the system of delay differential equations $$ \displaystyle x_i^{\prime}(t)=x_i(t)\left[b_i+\sum_{j=1}^{n}a_{ij}x_j(t)+\sum_{j=1}^{n}b_{ij}x_j(t-\tau_{ij})+\sum_{j=1}^{n}c_{ij}\int_{-\infty}^{...
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2answers
50 views

Bounding Lyapunov functionals

In a calculation that I'm doing, I end up with the following derivative \begin{equation} V^{\prime}(t)=-a[x^2(t)+x^2(t-h)]+2b(t)x(t)x(t-h), \label{eqn1} \end{equation} where $V(t)$ is a functional, $a&...
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0answers
26 views

Stability of delayed linear time invariant differential equation

When considering a delayed differential equation of the form $$ \dot{x}(t) = \sum_{n=1}^N A_n\,x(t-\tau_n) \tag{1} $$ A solution to $(1)$ is assumed to be able to be written as $$ x(t) = e^{M\,t}\,...
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1answer
126 views

$f'(x) = f(x-1)$ then $f$ is not bounded

Let $f:\mathbb{R} \rightarrow \mathbb{R}$. Then consider the following delay equation : $$f'(x) = f(x-1)$$ Let $S$ be the set of solution ot this equation. Then I would like to prove that : $\...
4
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3answers
160 views

How to solve this particular difference equations?

Considere a real polynomial function $P_n\left(x\right)$ of two variable, when $n$ is a discrete variable and $x$ a continuous variable where $P_0\left(x\right)=1$ and satisfies the following ...
4
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2answers
104 views

Solving differential equations of the form $\theta''(t)+\theta'(t)+\theta(t-\delta)=0$

TLDR: Hi, is there an exam friendly* method to solve (without using numerical methods) differential equations of the form $$\theta''(t)+\theta'(t)+\theta(t-\delta)=0$$ I have been searching for '...
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0answers
103 views

Prove a function is completely continuous

I must show that: Let $f(\phi)$ = $F(\phi(0), \phi(-r))$, where $F: R^{2n} -> R^{n}$ is continuous , and thus the above becomes: $x'(t) = F(x(t), x(t-r))$. Show that $f$ is completely continuous. ...
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111 views

Simplify $\int\limits_{-\infty}^X\exp[-(Ae^x+Bx+Cx^2)]\mathrm dx$ and $\sum\limits_{n=0}^\infty\frac{(-A)^n}{n!}e^{Dn^2-Kn}$

This integral has arisen in a pricing formula for an exotic financial option. I have scoured books on integrals, series, and products that might help (e.g., Gradshteyn and Ryzhik; volumes 1-2 of ...
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1answer
50 views

Is this DDE system possible?

I need to solve system of two differential delayed equations, but I have some problems. I think that second equation is not mathematicaly correct or possible because of position of time derivatives, ...
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1answer
56 views

Stability and Hopf Bifurcation of $y''(t)+qy'(t)-v^2\sin(y(t))=-py(t-r)$

The Question The equation $$y''(t)+qy'(t)-v^2\sin(y(t))=-py(t-r)$$ describes a damped pendulum with a delayed negative feedback restoring force. Where $y(t)$ is the deviation from the "up" state, ...
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1answer
61 views

Solution to the one-dimensional wave equation

We are told according to http://mathworld.wolfram.com/WaveEquation1-Dimensional.html (and other sources as well) that the general solution the the 1-D wave equation; $$\frac{\partial^2 \psi}{\partial ...
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1answer
562 views

Linearizing a delay differential equation at an equilibrium point.

I am confused about the general procedure to linearize a delay differential equation (DDE) at an equilibrium point. I was given the following two examples but I do not know how to get from DDE to ...
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1answer
185 views

Reference request for an introduction to delay differential equations.

I am currently taking a course in delayed differential equations (DDEs) and I am finding the instructor's notes too narrow/hard to follow. I am looking for for good textbooks/papers in different ...
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1answer
106 views

Determine the stability of the solution $x=0$ of the equation $mx''(t)+bx'(t)+qx'(t-r)+kx(t)=0$

The Problem Discuss the stability of the solution $x(t)=0$ of the equation $$mx''(t)+bx'(t)+qx'(t-r)+kx(t)=0$$ where $m>0,b\geq 0,q\geq 0,$ and $k\geq 0$ are all constants, by constructing an ...
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3answers
97 views

Solving $x'(t)=x(t-1)$ using Laplace transform!

I'm trying to solve $$x'(t)=x(t-1)$$ where $x(t)=1$ for $x\in[-1,0]$. I first need to show that $s\bar{x}(s)-1= \frac{1}{s}(1-e^{-s})+e^{-s}\bar{x}(s)$ where $\bar{x}(s)=\int_{0}^{\infty}x(t)e^{-...
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1answer
89 views

Prove that the solution to a delayed differential equation is positive for all $t>0$.

Prove that for the IVP \begin{cases} x'(t)=cx(t)[1-x(t-r)] \\ x(\mu)=\phi(\mu) & \mu \in [-r,0] \end{cases} for every $\phi\in C([-r,0],\mathbb{R})$ with $\phi(0)>0$, has a unique solution ...
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2answers
170 views

All the solutions of $f'(x)=f(x+\pi/2)$

Consider the following equation (with $f \in C^{\infty}(\mathbb{R})$): $$f'(x)=f(x+\pi/2)$$ This equation is satisfied by $f(x) = A\cos(x) +B\sin(x)$, for any $A,B \in \mathbb{R}$. Question: What ...
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1answer
73 views

Linear Functional Differential equation

I have a recursive equation which turns into this linear functional DE. $\Big[1 + a - az\Big]G(z) + \Big[bz - bz^2\Big]G^{\prime}(z) = \Big[a + (1-a)z\Big]G(1-c+cz), G(1) = 1$ where I assume $G(z)=\...
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0answers
92 views

Solve a differential equation by roots of characteristic equation

I read in a paper that solution of a differential equation $$\dot{x}(t) = -cx(t-h)+f(t)$$ where $x$ is a scalar with zero initial condition (for all $t\leq 0$) can be written as $$x(t) = \int_0^t f(\...
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1answer
71 views

ODE with functional argument

I am trying to find a closed form solution for this ODE: $(a_0+a_1x)f(x)+(b_0x+b_1x^2)f^\prime (x) = (c_0+c_1x)f(d_0+d_1x)$. It is not a simple ODE due to the $f(d_0+d_1x)$ term. I also found a ...
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1answer
38 views

Reference help: asymptotic reducibility of systems

I was reading a paper about the reducibility of linear functional differential equations with quasi periodic coefficients. The paper talks about the asymptotic reducibility of systems. I don't the ...
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0answers
65 views

Solving a “partial” delay differential equation with non-constant delay

I would like to solve an equation of the form $$\varphi_t(t, s) + \alpha(s) \varphi(t, s - \beta(t)) = 0$$ but cannot find a single example of how to do so. Maybe I just don't have the right search ...
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0answers
365 views

Solving a non-separable PDE with Mellin or other integral transform

I am trying to solve a PDE of the form $$\xi_t(t, x) + a(t) x^{f(t)} \xi(t, x) + b(t) x^{f(t)+1} \xi_x(t,x) + c(t) x^{f(t)+2} \xi_{xx}(t, x) = 0.$$ I have no idea if this is even possible. There ...
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1answer
362 views

How to solve delayed partial differential equations

The parabolic differential equations with Neumann conditions and history functions have the following general form : $$\left\{\begin{array}{lc} \dfrac{\partial u(t,x)}{\partial t}=d_1\triangle u(t,x)...
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1answer
62 views

delayed differential equations

I would like to solve this differential equation that is similar to the delayed differential equation here: How to solve differential equations of the form $f'(x) = f(x + a)$ this is the DE: f'(x)...
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0answers
38 views

Confusion with differential equation's periodic solution

It is known that if $0<\epsilon<<1$ then differential equation with delay $$\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0$$ has periodic solution. In one book author ...
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0answers
73 views

A property of eigenvalues of the generators of a strongly continuous semigroup

I am trying to read a book on Linear Delay Differential Equation, but I am getting stuck with some passages in some important proofs. I tried looking for alternative versions in Internet but couldn't ...
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0answers
139 views

Proof of Coppel's Inequality for the matrix “measure”.

I am trying to understand the matrix measured proposed by W. Coppel in "Stability and Asymptotic Behavior of Differential Equations" in 1965, but I cannot find a pdf of this paper online, so if anyone ...
1
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1answer
134 views

Euler Scheme of Delay Differential Equation

Given an ode $x' = f(t)$. Then a basic Euler discretization scheme yields $$ x_{n+1} = x_n + h f(t_n).$$ Now suppose you have a delay differential equation, say $x' = f(t-\tau)$, does it make sense ...
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1answer
71 views

Can we approximate delayed-differential equations with higher-order-ordinary-differential-equations?

I noticed that the most simple numerical approximation of a higher order-differential equation has the same form as the numerical approximation of a delayed first-order differential equation. This ...