Delay differential equation

A general form of the time-delay differential equation for $x(t)\backslash in\; \backslash mathbb\{R\}^n$ is :$\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}x(t)=f(t,x(t),x\_t),$ where $x\_t=\backslash \{x(\backslash tau):\backslash tau\backslash leq\; t\backslash \}$ represents the trajectory of the solution in the past. In this equation, $f$ is a functional operator from $\backslash mathbb\{R\}\backslash times\; \backslash mathbb\{R\}^n\backslash times\; C^1$ to $\backslash mathbb\{R\}^n.\backslash ,$

Examples

Solving DDEs

::$\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}x(t)=f(x(t),x(t-\backslash tau))$

with given initial condition $\backslash phi:[-\backslash tau,0]\backslash rightarrow\; \backslash mathbb\{R\}^n$. Then the solution on the interval $[0,\backslash tau]$ is given by $\backslash psi(t)$ which is the solution to the inhomogeneous initial value problem

::$\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}\backslash psi(t)=f(\backslash psi(t),\backslash phi(t-\backslash tau))$,

with $\backslash psi(0)=\backslash phi(0)$. This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically.

Example

::$x(t)=a\backslash int\_\{s=0\}^t\; \backslash phi(s-\backslash tau)\backslash ,\{\backslash rm\; d\}s+C$,

i.e., $x(t)=at+1$, where we picked $C=1$ to fit the initial condition $x(0)=\backslash phi(0)$. Similarly, for the interval $t\backslash in[\backslash tau,2\backslash tau]$ we integrate and fit the initial condition to find that $x(t)=at^2/2+t+D$ where $D=(a-1)\backslash tau+1-a\backslash tau^2/2$.

Reduction to ODE

:Introduce $y(t)=\backslash int\_\{-\backslash infty\}^0x(t+\backslash tau)e^\{\backslash lambda\backslash tau\}\backslash ,\{\backslash rm\; d\}\backslash tau$ to get a system of ODEs ::$\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}x(t)=f(t,x,y),\backslash quad\; \backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}y(t)=x-\backslash lambda\; y.$

:is equivalent to ::$\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}x(t)=f(t,x,y),\backslash quad\; \backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}y(t)=\backslash cos(\backslash beta)x+\backslash alpha\; z,\backslash quad\; \backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}z(t)=\backslash sin(\backslash beta)\; x-\backslash alpha\; y,$

:where ::$y=\backslash int\_\{-\backslash infty\}^0x(t+\backslash tau)\backslash cos(\backslash alpha\backslash tau+\backslash beta)\backslash ,\{\backslash rm\; d\}\backslash tau,\backslash quad\; z=\backslash int\_\{-\backslash infty\}^0x(t+\backslash tau)\backslash sin(\backslash alpha\backslash tau+\backslash beta)\backslash ,\{\backslash rm\; d\}\backslash tau.$

The characteristic equation

The roots λ of 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 a some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues to the right of any vertical line in 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: :$\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}x(t)=-x(t-1).$ The characteristic equation is :$-\backslash lambda-e^\{-\backslash lambda\}=0.\backslash ,$ There are an infinite number of solutions to this equation for complex λ. They are given by :$\backslash lambda=W\_k(-1)$, where $W\_k$ is the kth branch of the Lambert W function.

Notes

References

External links

Category:Differential equations