Consider the integro-differential equation

\begin{equation} \kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1} \end{equation}

such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\mathbb{R}^+$, and $J_1$ is first order the Bessel function of first kind. Letting \begin{equation} \mathcal{S}:=\sum_{i=0}^n J_1\left(x_{n\Delta t}-x_{i\Delta t}\right)e^{-\epsilon\left(n\Delta t-i\Delta t\right)},\tag{2} \end{equation}

Equation $(1)$ may be expressed as the system \begin{equation} x_{\Delta t(n+1)}=\lim_{\Delta t\rightarrow 0}\frac{2\Delta t^2 \mathcal{S} + x_{n\Delta t}\left(2\kappa+1\right)-\kappa x_{\Delta t(n-1)}}{\kappa+1},\tag{3} \end{equation} such that $x_0=0$ and $x_1\in\mathbb{R}$. This is particularly useful as a numerical method when solving the first equation. However, can a recurrence relation be derived from the above? Setting $\Delta t=1$ and letting $\mathcal{S}$ be constant simply produces $$x_{n}=K_1\left(\frac{\kappa}{1+\kappa}\right)^n+2\mathcal{S}\left(n-1-\kappa\right)+K_2,$$ for constants $K_1,K_2$. Does a more general relation exist for the function $\mathcal{S}$ and $0\leq\Delta t<1$?



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