Nonlinear ordinary differential equations: A simple approach

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Many problems of physics, biology, and engineering are modeled in the form of nonlinear ordinary differential equations. As mathematicians have said, life is non-linear, in which we face many problems. These types of equations take the following form


\begin{align*}\tag{E} \dot{u}(t)=f(t,u(t)),\end{align*}


where $f:I\times \Omega\to \mathbb{R}^d$ is a continuous function, $I$ an interval in $\mathbb{R}$ and $\Omega$ is an open set of $\mathbb{R}^d$. When $(t,x)\in I\times \Omega,$ then $t$ represents the “time” and $x$ an element of the “workspace”. For example, if we analyze the evolution of cell division in an area of ​​the body; this one is exactly $ \Omega $. Note that $u $ represents the number of cells in the region $ \Omega, $ while the equation (E) represents the way of producing these cells. Thus, the number of cells increases when the time $ t $ passes.


Three types of solutions to nonlinear ordinary differential equations


Solutions of differential equations depend on the length of the interval in which the evolution of the experiment occurs. For instance, during the experiment for a brief time ($t\in J\subset I$), we already obtain some required information and results, so we have a “local solution”. But to obtain more information on the mathematical model we need to continue our experiment and extend the time interval, so we obtain a piece of large and sufficient information on the solution; we then talk about “maximal solutions”. On the other hand, it is important to let the experiment working for a large time (on $I$), then we talk about “global solutions”.


Interpretations of solutions of differential equations on health domain


Let’s go back to the example of cell division. When you want to test the blood to find out if it has cancer or not; this is done using an algorithm based on a mathematical model governed by nonlinear differential equations. Indeed, during a short period of blood analysis, the algorithm gives us a result (local solution). This can be negative if we have normal cell division, which means that cells are produced smoothly.


Otherwise, we have a positive result and that means that the division produces a large number of cells; it is a cancerous infection. In this case, we are in front of two possibilities. If the cell division speed is high, then we arrive at a maximum time $ T $ which is the one where the body cannot support this large number of cells (maximum solution); it is the end of life. We note that this time $T$ is calculated by doctors (this can be some months).


Now, if the speed of division is not really high, then we have the chance to fight cancer again. In this case, your doctor will give you medicines that can help you stay healthy for a very long time, which is the overall (or global) solution.


The mathematical definition of the solution of ordinary differential equations


Mathematically, a local solution of the equation (E) is a couple $(J,u)$, where $J$ is a subinterval of $I$ and $u: J\to \Omega$ is a continuously differentiable function that satisfies the equations. Also, we say that a solution $(J_2,u_2)$ extends another solution $(J_1,u_1)$ if $J_1\subset J_2$ and $u_2$ coincides with $u_1$ on the small interval $J_1$. By a maximal solution, we mean a solution $(J,u)$ that can not be extended to another solution of (E). Finally, a global solution of the differential equation (E) is a solution $(J,u)$ such that $J=I$.


Existence of solutions to nonlinear differential equation: Pean theorem


As we work in spaces of finite dimension, the maximal solution exists under the condition that the vectors field $f$ is continuous on $I\times\Omega$. This result is due to Giuseppe Peano; an Italian mathematician. This result has been published in the year 1886. In this case, for any $(t_0,x_0)\in I\times \Omega,$ there exists an interval $J$ of $I$ centred in $t_0$ and at least a function $u:J\to\Omega$ such that $u(t_0)=x_0$ and satisfies the nonlinear differential equation (E). Moreover, it satisfies the following integral equation


\begin{align*}\tag{IE}u(t)=x_0+\int^t_0 f(s,u(s))ds,\quad t\in J.\end{align*}


We mention that the Peano theorem is not true if we work on infinite-dimensional state spaces. There are examples of that.


Existence and uniqueness: Cauchy-Lipischtz theorem


In practice (engineering sciences), we are looking for only one solution and we focus on it. This is why we need the solution to be unique. We can give simple examples in which the continuity of $ f $ is not sufficient to obtain the uniqueness of the solution. An additional condition on $ f $ is necessary for nonlinear ordinary differential equations.


We say that the function $f$ is locally Lipschitz on $I\times \Omega$ if for all $(t_0,x_0)\in I\times \Omega$ then exists an open set $V_{ (t_0,x_0) }$ of $I\times \Omega$ centred in $(t_0,x_0) $, and a constant $C>0$ such that for any $(t,x)$ and $(t,y)$ in $V_{ (t_0,x_0) }$,


\begin{align*}\|f(t,x)-f(t,y)\|\le C\|x-y\|.\end{align*}


Now if $f$ is continuous and locally Lipschitz on $I\times \Omega$ then there exists a unique maximal solution of the nonlinear differential equation (E). This result is known as the Cauchy-Lipshitz theorem; it holds also even if we work in infinite-dimensional spaces.


The proof of this result is based on a Banach fixed point theorem. If fact, we define the map


\begin{align*}\Phi(u)(t)=x_0+\int^t_{t_0}f(s,u(s))ds. \end{align*}


We work locally in a compact set of $I\times \Omega$ in order to prove that there exists $n\in \mathbb{N}$ such that the iterated $\Phi^n$ of $\Phi$ is a contraction. Then the fixed point $u$ exists for $\Phi^n,$ then for $\Phi$. This fixe point is unique and it is the solution of (E).

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