# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 5 The Euler-Type Universal Numerical Integrator (E-TUNI)
### Fundamental Theorem
: _If a function \(\delta^{i}_{j}(t)\) for \(j=1,2,\ldots,n\) is a continuous function and defined over the closed interval \([t_{k},t_{k+1}]\), then there exists at least one internal point \(t_{k}^{x}\) in \([t_{k},t_{k+1}]\) such that, \(\delta^{i}_{j}(t_{k}^{x})\cdot\Delta t=\int_{t_{k}}^{t_{k+1}}\delta^{i}_{j}(t)dt\)._  
The notation used to represent \({}^{k}y^{i}_{j}\) is sufficient to map it uniquely to its respective region of interest, where its associated characteristic secant \(tan_{\Delta t}{}^{k}\alpha^{i}_{j}\) is confined. Thus, it can be stated that for the state \({}^{k}y^{i}_{j}=y^{i}_{j}(t_{0}+k\cdot\Delta t)\), we have a single characteristic secant associated with it and given by \(tan_{\Delta t}{}^{k}\alpha^{i}_{j}\). The existence of a unique secant associated with each point \({}^{k}y^{i}_{j}\) is an immediate consequence of the uniqueness theorem (Theorem 1). Consider also \(y^{i}_{j}=y^{i}_{j}(t)\) for \(j=1,2,\ldots,n\) a particular trajectory of a family of solutions of the system of differential equations \(\dot{y}=f(y)\) passing through \(y^{i}_{j}(t_{0})\) at time \(t_{0}\), i.e., initializing within a finite domain of interest \([y^{min}_{j}(t_{0}),y^{max}_{j}(t_{0})]^{n}\). In [29; 30], by definition, the secant curve between two points \({}^{k}y^{i}_{j}\) and \({}^{k+1}y^{i}_{j}\) belonging to the curve \(y^{i}_{j}(t)\) for \(j=1,2,\ldots,n\) is the straight line segment that joins these two points. Thus, the tangents of the secants between the points \({}^{k}y^{i}_{1}\) and \({}^{k+1}y^{i}_{1}\), \({}^{k}y^{i}_{2}\) and \({}^{k+1  
the general solution of this dynamical system, given by, \(y^{i}_{j}(t)=g^{i}_{j}(t)\) and \(\dot{y}^{i}_{j}(t)=\delta^{i}_{j}(t)\) are previously known for \(j=1,2,\ldots,n\); \(i=1,2,\ldots,\infty\) and \(t\in[t_{o},t_{f}]\)._  
Proof. : Let the first-order non-linear and autonomous dynamical system be given by \(y^{i}_{j}(t)=f_{j}(y^{i}_{1},y^{i}_{2},\ldots,y^{i}_{n})=f_{j}(y^{i})\) for \(j=1,2,\ldots,n\) and \(i=1,2,\ldots,\infty\). If the solution of the dynamical system is known and equal to \(y^{i}_{j}(t)=g^{i}_{j}(t)\), then the function \(f_{j}(y^{i})\) can be replaced by \(\delta^{i}_{j}(t)=h^{i}_{j}(t)\), i.e., \(\dot{y}^{i}_{j}(t)=h^{i}_{j}(t)\). In this way, we can write that \(\int_{h^{i}_{j}}^{k+1}\!\!y^{i}_{j}\,dy^{i}_{j}(t)=\int_{t_{k}}^{t_{k+1}}\!\!h^ {i}_{j}(t)dt\). Note that the indices \(i,j,\) and \(k\) uniquely map the secant we are interested in and in accordance with Figure 1. This last integral can still be simplified as \({}^{k+1}\!\!y^{i}_{j}-\,{}^{k}\!\!y^{i}_{j}=\int_{t_{k}}^{t_{k+1}}\!\!h^{i}_{j} (t)dt\). Thus, applying the mean integral value theorem, in this last expression, we obtain \({}^{k+1}\!\!y^{i}_{j}-\,{}^{k}\!\!y^{j}_{j}=h^{i}_{j}(t^{x}_{k})\cdot\Delta t= \dot{g}^{i}_{j}(t^{x}_{k})\cdot\Delta t\) where \(t^{x}_{k}\in[t_{k},t_{k+1}]\) and for \(h^{i}_{j}(t)=g^{i}_{j}(t)\). On the other hand, applying the differential mean value theorem on the function \(\dot{y}^{i}_{j}(t)=g^{i}_{j}(t)\) for \(j=1,2,\ldots,n\) in the interval \([t_{k},t_{k+1}]\) we obtain that \(\dot{g}^{i}_{j}(t^{*}_{k})=\frac{{}^{k+1}\!\!y^{i}_{j}-\,{}^{k}\!\!y^{i}_{j}}{ \Delta t}\)\(\stackrel{{\rm def}}{{=}}tan_{\Delta t}{}^{k}\!\!a^{j}_{j}\). Thus, by Property 1 we have that \(\dot{g}^{i}_{j}(t^{*}_{k})=\dot{g}^{i}_{j}(t^{x}_{k})=tan_{\Delta t}{}^{k}\!\!a^ {j}_{j}\) for \(j=1,2,\ldots,n\). Therefore, \({}^{k+1}\!\!y^{i}_{j}=tan_{\Delta t}{}^{k}\!\!a^{j}_{i}\cdot\Delta t+\!\!y^{i}_ {j}\) for \(j=1,2,\ldots,n\) or, in vector form, it turns out that \({}^{k+1}\!\!y^{i}=tan_{\Delta t}{}^{k}\!\!a^{i}\cdot\Delta t+{}^{k}\!\!y^{i}\). Note that the functions \(\dot{y}^{i}_{j}(t)\) and \(y^{i}_{j}(t)\), used in the previous demonstration, need to be continuous and differentiable because of Theorems 2 and 3, but they do not need to be known. Their existence is enough. What needs to be known are the points \({}^{k}\!\!y^{i}_{j}\).