Matlab数值计算最简单的一个例子——指数衰减

放射性衰变是指数衰减的典型例子。另外还有化学反应某反应物的减少，RC电路电流的减小，大气压随海拔高度的减小等。

指数衰减的方程：

\begin{equation}

\frac{dN(t)}{dt}=-\frac{N(t)}{\tau}

\label{eq1}

\end{equation}

其中，\(N(t)\)为\(t\)时刻的物理量\(N\)，对于放射性衰变，\(N\)就是未衰变的原子核数目。\(\tau\)为时间常数。

方程\eqref{eq1}有解析解：

\[N(t)=N(0)\exp(-t/\tau)
\]

这个解可以通过Matlab符号计算求得：

  dsolve('DN=-N/tau')
  ans =
  C3*exp(-t/tau)

数值求解方程\eqref{eq1}，可用欧拉格式将方程离散化。

\[t_i=(i-1) \Delta t,\quad i=1,2,\dots,\mathrm{npoints}
\]

\[\frac{dN(t)}{dt}\approx\frac{N(t)-N(t-\Delta t)}{\Delta t}
\]

将以上两式带入方程\eqref{eq1}，得离散之后的方程：

\[N(t_{i+1})=N(t_i)-N(t_i)\frac{\Delta t}{\tau}
\]

代入初始条件，即可得解。

下面写个Matlab 脚本文件，重复出Computational Physics_Giordano 2nd Edition的图1.1，pp11

%
% Exponent decay
% 'Computational Physics' book by N Giordano and H Nakanishi
% Section 1.2 p2
% Solve the Equation dN/dt = -N/tau
% by Joyful Physics Blog
% ------------------------------------------------------------

N_nuclei_initial = 100; %initial number of nuclei
npoints = 101; % Discretize time into 100 intervals
dt = 0.05; % set time step
tau=1; % set time constant
N_nuclei = zeros(npoints,1); % initializes N_nuclei, a vector of dimension npoints X 1,to being all zeros
time = zeros(npoints,1); % this initializes the vector time to being all zeros
N_nuclei(1) = N_nuclei_initial; % the initial condition, first entry in the vector N_nuclei is N_nuclei_initial
time(1) = 0; %Initialise time
for step=1:npoints-1 % loop over the timesteps and calculate the numerical solution
N_nuclei(step+1) = N_nuclei(step) - (N_nuclei(step)/tau)*dt;
time(step+1) = time(step) + dt;
end
% calculate analytical solution below
t=0:0.05:5;
N_analytical=N_nuclei_initial*exp(-t/tau);
% Plot both numerical and analytical solution
plot(time,N_nuclei,'ro',t,N_analytical,'b'); %plots the numerical solution in red and the analytical solution in blue
xlabel('Time (s)')
ylabel('Number of nuclei')
text(2,80,'Time constant = 1s')
text(2,70,'Time step = 0.05s')

运行程序，得到：