Basic concepts in ODEs

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Applied ODEComplex System modeling

Basic Notions

An ODE of the \(k\) -th order is in the expression of the form

\[\frac{\mathrm{d}^kx}{\mathrm{d}t^k}=f\left(t,x,\frac{\mathrm{d}x}{\mathrm{d}t},\ldots,\frac{\mathrm{d}^{k-1}x}{\mathrm{d}t^{k-1}}\right)\]

A function \(\phi (x)\) is called a solution to above ODE on the interval \(I=(a, b)\) if this function \(k\) times continously differentiable on \(I\) or say that \(\phi\in C^{k}(I,\mathbb{R})\) .

An ODE plus the initial conditions are called an initial value problem (IVP for short) or Cauchy’s problem.

Malthus population model

We will denote \(N (t)\) as the number of individuals in a given populations at time moment \(t\) . Then the population number changes during a short time interval \(h\) we have

\[N(t+h)=N(t)+bh(t)-dhN(t)\]

where \(b\) and \(d\) here stands for birth rate and death rate, and then we will have

\[\frac{N(t+h)-N(t)}{h}=(b-d)N(t)\]

if we postulate that the existence of a derivative, then we will have

\[\frac{\mathrm{d}N}{\mathrm{d}t}=(b-d)N\]

if we denote \(b-d\) as a parameter \(m\) then we will have the population growth model to be an IVP

\[\dot{N}=mN,N(0)=N_0\]

where \(N(t)\) is the population size at time moment \(t\). But we cannot always know the direct form of the model, maybe we can use assume that the law of growth has the general form:

\[\dot{N}=NF(N)\]

Where \(F\) is some function, which has to be negative for sufficiently large values of \(N\), if this function is smooth enough, we have its Taylor formula around \(N=0\) as

\[F(N)=F(0)+\frac{F'(0)}{1!}N+\frac{F''(0)}{2!}N^2+o(N^2)\]

If we only keep up constant terms and them we will have the equation

\[\dot{N}=mN\]

where \(m=N(0)\), and if you keep two terms we will have

\[\dot{N}=NF(N)=N(F(0)+F'(0)N)=mN\left(1-\frac{N}{K}\right)\]

where \(K=-F(0)/F\'(0)\) to be another parameter called \(K\) in above equation.

Well-posed problems

(Salomon Hadamard) A mathematical problem is well posed if

  • its solution exists.
  • its solution is unique.
  • its solution depends continuously on the initial data.

Theorem(local uniqueness)

Consider the IVP and assume that function \(f\) is continous in \(t\) and continously differentiable in \(x\) for \((t,x)\in (a,b)\times(c,d)\) for some constant \(a,b,c,d\), assume that \((t_0,x_0)\in(a,b)\times(c,d)\) then there exists am \(\epsilon>0\) such that the solution \(\phi(x,t)\) to the equation exists and unique for \(t\in(t_0-\epsilon,t_0+\epsilon)\).

However the theorem is local, it only guarantees that the solution exists and is unique on some smaller interval \((t_0-\epsilon,t_0+\epsilon)\subseteq (a,b)\). However the solution of ODE can below up i.e. approach infinity for a finite \(t\).

Examples of the uniqueness theorem. Consider the ODE

\[\dot{x}=1+x^2=f(t,x)\]

the right hand side is a polynomial for any \((t,x)\in\mathbb{R}\), its solution is given by

\[x(t)=\tan{(t+C)}\]

and hence for each fixed \(C\) is defined only on the interval \((-\pi/2-C,\pi/2-C)\).

Consider the ODE

\[\dot{x}=\sqrt{x},x(0)=x_0,x\geq0\]

One solution can be given as

\[x(t)=\frac{(t+2\sqrt{x_0})^2}{4}\]

if \(x_0=0\) then the solution is exactly \(x(t)=0\), therefore the solution is not unique! A strong theorem on IVP is given below

Theorem

Let IVPs satisfy the local conditions, where two IVPs are defined as

\[\begin{gathered} \dot{x}=f(x,t),x(t_0)=x_0\\ \dot{x}=f(x,t),x(t_0)=x_1 \end{gathered}\]

and \(t\mapsto x_0(t)\) and \(t\mapsto x_1(t)\) be the solution to IVP at the same time \(t\) and then we will have

\[|x_1(t)-x_0(t)|\leq |x_1-x_0|\mathrm{e}^{L|t-t_0|}\]

where \(L\) is a constant that depends on \(f\).

Above theorem shows that the solution to a first order ODE depends continously on the initial condition.

Visualizing a ODE system.

Consider the ODE

\[\dot{x}=\sin{x}\]

and we plot out its phase diagram. 9f48a11b019613d1dcbd25486a5f5fa0.png