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：
egin{equation}
frac{dN(t)}{dt}=-frac{N(t)}{ au}
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。( au)为时间常数。
方程eqref{eq1}有解析解：
[N(t)=N(0)exp(-t/ au)
]
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}approxfrac{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}{ au}
]
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：
