exercise 04

一.摘要

Many problems encountered in physics involve ordinary differential equations.First,we should learn to solve a problem involving a first-order differential equation with initial limits.

二.背景介绍

To solve first-order differential problems,there is the Euler method which is based on Taylor expansion:

$$f(x_0+\Delta{}x)=f(x_0)+\frac{df(x)}{dx}_{x=x_0}\Delta{}x+\frac{1}{2}\frac{d^2f(x)}{dx^2}_{x=x_0}\Delta{}x^2...$$
where $f(x_0)$ is the value of our function at $x=x_0$.If we take $\Delta$x to be small,then it is usually a good approximation to simply ignore the terms that involve second and higher powers of $\Delta$x,leaving us with:
$$f(x_0+\Delta{}x)\approx{}f(x_0)+\frac{df(x)}{dx}_{x=x_0}\Delta{}x$$
For problem 1.5,it dives us two relevant $N_A$ and $N_B$ which are characterised by two first-order differential equations.We can raplace $\frac{dN_A}{dt}$ with former equation,so,
$$N_A(t+\Delta{}t)\approx{}N_A(t)+(\frac{N_B}{\tau}-\frac{N_A}{\tau})\Delta{}t$$
with same principle,we can conclude $N_B$:
$$N_B(t+\Delta{}t)\approx{}N_B(t)+(\frac{N_A}{\tau}-\frac{N_B}{\tau})\Delta{}t$$
This approximation forms the basis for a numerical solution of our radioactive decay problem.$\tau$ is the time constant.Given that we know the value of $N_A$ and $N_B$ at some value of t,we can use this to estimate its value a time $\Delta$t later.
In this pronlem,we are given the initial value of the function,that is,the value at time t=0.We can then employ value at t=$\Delta$t.This result can be used in turn to estimate the value at t=2$\Delta$t,3$\Delta$t,etc.,and thereby lead to an approximate solution $N_A(n\Delta{}t)$ and $N_B(n\Delta{}t)$ at times $n\Delta$t where n is an integer.We delete the error term of order $\Delta{}t^2$,and cannot emphasize too strongly that the numerical"solution"obtained in this way is only an approximation to the"true",or exact,solution.we want to make the difference between the two negligible.

三.正文

1.calculate results for different initial conditions

code

I choose three different initial conditions,the results are as followings:

for $N_A=100,N_B=0,\tau=1$:

for $N_A=80,N_B=20,\tau=1$:

for $N_A=80,N_B=20,\tau=2$:

2.test my program

code

Before the program's calculation,I calculate the differential equation through the pen.I draw the following results:

$$N_A(t)=\frac 12[N-(N-2N_A(0)e^\frac{-2t}{\tau}]$$
$$N_B(t)=\frac 12[N+(N-2N_A(0)e^\frac{-2t}{\tau}]$$
where $N=N_A(t)+N_B(t)=constant$.Compare the figure in terms of above-mentioned equations with the figure that is plotted through Euler method.This is my result:

By comparison,the two figures are similar to each other as a whole.It proves that the Euler method is very reliable within the allowed error range.

3.Always check that my program gives the same answer for different"step sizes."

code

Change the step size and compare their differences:

From the figures,we feel the results are normal and we can accept it.

四.结论

The Euler method can apply first-order differential equations and its results are in acceptable range.In this problem,$N_A$ and $N_B$ are constant finally when the system reachs equilibrium.In such a steady state,the time derivatives $\frac{dN_A}{dt}$ and $\frac{dN_B}{dt}$ should vanish.

This assignment is an original works.