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18-494: [REDACTED] Look up dde in Wiktionary, the free dictionary. DDE may refer to: D.D.E. (band) , a Norwegian rock band Delay differential equation , a type of differential equation Deep-dose equivalent , a measure of radiation absorbed by the body Dichlorodiphenyldichloroethylene , a chemical that results from the breakdown of DDT Doctrine of double effect ,

36-664: A τ + 1 ) + a ∫ s = τ t ( a ( s − τ ) + 1 ) d s = ( a τ + 1 ) + a ∫ s = 0 t − τ ( a s + 1 ) d s , {\displaystyle {\begin{aligned}x(t)=x(\tau )+\int _{s=\tau }^{t}{\frac {d}{dt}}x(s)\,ds&=(a\tau +1)+a\int _{s=\tau }^{t}\left(a(s-\tau )+1\right)ds\\&=(a\tau +1)+a\int _{s=0}^{t-\tau }\left(as+1\right)ds,\end{aligned}}} i.e., x ( t ) = (

54-490: A τ + 1 ) + a ( t − τ ) ( 1 2 a ( t − τ ) + 1 ) . {\textstyle x(t)=(a\tau +1)+a(t-\tau )\left({\frac {1}{2}}{a(t-\tau )}+1\right).} In some cases, differential equations can be represented in a format that looks like delay differential equations . Similar to ODEs , many properties of linear DDEs can be characterized and analyzed using

72-617: A t + 1 {\displaystyle x(t)=at+1} , where the initial condition is given by x ( 0 ) = ϕ ( 0 ) = 1 {\displaystyle x(0)=\phi (0)=1} . Similarly, for the interval t ∈ [ τ , 2 τ ] {\displaystyle t\in [\tau ,2\tau ]} we integrate and fit the initial condition, x ( t ) = x ( τ ) + ∫ s = τ t d d t x ( s ) d s = (

90-538: A finite number of eigenvalues in any vertical strip of the complex plane. This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically. In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE: d d t x ( t ) = − x ( t − 1 ) . {\displaystyle {\frac {d}{dt}}x(t)=-x(t-1).} The characteristic equation

108-538: A set of ethical criteria to evaluate the permissibility of acting when one's otherwise legitimate act may also cause an effect one would normally be obliged to avoid Dwight D. Eisenhower , the 34th president of the United States Dynamic Data Exchange , a Microsoft Windows and OS/2 inter-application data communication protocol Escort destroyer , a US Navy classification used between 1945 and 1962 D.De., an abbreviation used for

126-604: A stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay d d t x ( t ) = f ( x ( t ) , x ( t − τ ) ) {\displaystyle {\frac {d}{dt}}x(t)=f(x(t),x(t-\tau ))} with given initial condition ϕ : [ − τ , 0 ] → R n {\displaystyle \phi \colon [-\tau ,0]\to \mathbb {R} ^{n}} . Then

144-417: Is − λ − e − λ = 0. {\displaystyle -\lambda -e^{-\lambda }=0.} There are an infinite number of solutions to this equation for complex λ . They are given by λ = W k ( − 1 ) , {\displaystyle \lambda =W_{k}(-1),} where W k is the k th branch of

162-703: Is different from Wikidata All article disambiguation pages All disambiguation pages Delay differential equation A general form of the time-delay differential equation for x ( t ) ∈ R n {\displaystyle x(t)\in \mathbb {R} ^{n}} is d d t x ( t ) = f ( t , x ( t ) , x t ) , {\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x_{t}),} where x t = { x ( τ ) : τ ≤ t } {\displaystyle x_{t}=\{x(\tau ):\tau \leq t\}} represents

180-463: Is the exponential polynomial given by det ( − λ I + A 0 + A 1 e − τ 1 λ + ⋯ + A m e − τ m λ ) = 0. {\displaystyle \det(-\lambda I+A_{0}+A_{1}e^{-\tau _{1}\lambda }+\dotsb +A_{m}e^{-\tau _{m}\lambda })=0.} The roots λ of

198-615: The Lambert W function , so: x ( t ) = x ( 0 ) e W k ( − 1 ) ⋅ t . {\displaystyle x(t)=x(0)\,e^{W_{k}(-1)\cdot t}.} The following DDE: d d t u ( t ) = 2 u ( 2 t + 1 ) − 2 u ( 2 t − 1 ) . {\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).} Have as solution in R {\displaystyle \mathbb {R} }

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216-791: The United States District Court for the District of Delaware Deepin Desktop Environment , a Desktop Environment used by several Linux Distributions Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title DDE . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=DDE&oldid=1245115063 " Category : Disambiguation pages Hidden categories: Short description

234-490: The characteristic equation . The characteristic equation associated with the linear DDE with discrete delays d d t x ( t ) = A 0 x ( t ) + A 1 x ( t − τ 1 ) + ⋯ + A m x ( t − τ m ) {\displaystyle {\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\dots +A_{m}x(t-\tau _{m})}

252-529: The characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum . Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a spectral analysis more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only

270-516: The initial value problem can be solved with integration, x ( t ) = x ( 0 ) + ∫ s = 0 t d d t x ( s ) d s = 1 + a ∫ s = 0 t ϕ ( s − τ ) d s , {\displaystyle x(t)=x(0)+\int _{s=0}^{t}{\frac {d}{dt}}x(s)\,ds=1+a\int _{s=0}^{t}\phi (s-\tau )\,ds,} i.e., x ( t ) =

288-672: The solution on the interval [ 0 , τ ] {\displaystyle [0,\tau ]} is given by ψ ( t ) {\displaystyle \psi (t)} which is the solution to the inhomogeneous initial value problem d d t ψ ( t ) = f ( ψ ( t ) , ϕ ( t − τ ) ) , {\displaystyle {\frac {d}{dt}}\psi (t)=f(\psi (t),\phi (t-\tau )),} with ψ ( 0 ) = ϕ ( 0 ) {\displaystyle \psi (0)=\phi (0)} . This can be continued for

306-482: The successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically. Suppose f ( x ( t ) , x ( t − τ ) ) = a x ( t − τ ) {\displaystyle f(x(t),x(t-\tau ))=ax(t-\tau )} and ϕ ( t ) = 1 {\displaystyle \phi (t)=1} . Then

324-491: The trajectory of the solution in the past. In this equation, f {\displaystyle f} is a functional operator from R × R n × C 1 ( R , R n ) {\displaystyle \mathbb {R} \times \mathbb {R} ^{n}\times C^{1}(\mathbb {R} ,\mathbb {R} ^{n})} to R n . {\displaystyle \mathbb {R} ^{n}.} DDEs are mostly solved in

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