我正在创建一个函数,以使用欧拉方法创建微分方程的近似解。我可以让代码正常工作,但在尝试将其转换为函数时遇到了麻烦。也就是说,我很难让我的函数正确调用公式。
这个原始代码运行良好:
#define your inital values
y = 1
x = 0
#h represents step size (incremental shift over the directional field)
h = 0.1
solutions = []
x_values = []
#generates a list of 10 solutions, increase the number in the range to get more solutions
for s in range (0, 10):
#enter your differential equation
diff_eq = x+y
#Euler's method to approximate the solutions
s = y + h*(diff_eq)
solutions.append(s)
#replace initial values with the new values
y = s
x = x + h
#create a list of the x values
x_values.append(x)
#creating a dataframe from the solutions
Euler_values = pd.DataFrame(list(zip(x_values, solutions)), columns = ['xₙ', 'yₙ'])
#create an index for the dataframe starting at one and increasing by one
Euler_values.index = Euler_values.index + 1
Euler_values.rename_axis('n', inplace = True)
Euler_values
这就是将上面的代码变成函数的开始:
#eulers method to approximate the solutions function
def eulers_method(x, y, diff_eq, h, n):
#empty solution and x_value lists to be used in the euler function
solutions = []
x_values = []
for s in range (0, n):
# eq = diff_eq <-- this does not work
# if I call the equation here it does work, but I want it to be entered into the function
eq = x + y
s = y + h*(eq)
#replace initial values with the new values and adds them to the solution and x_value lists
y = s
solutions.append(s)
x = x + h
x_values.append(x)
#creates a dataframe from the solutions
Euler_values = pd.DataFrame(list(zip(x_values, solutions)), columns = ['xₙ', 'yₙ'])
#creates an index for the dataframe starting at one and increasing by one
Euler_values.index = Euler_values.index + 1
Euler_values.rename_axis('n', inplace = True)
return Euler_values
#enter an initial x value, initial y value, the differential equation, the step size, and the number of solutions to generate
# the diff_eq entry is giving me trouble
eulers_method(0,1, x+y, 0.1, 10)
有几种方法可以解决您的问题。第一个(IMO 更佳)解决方案是将一个函数传递给您的
eulers_method
函数,即
def eulers_method(x, y, diff_eq, h, n):
#empty solution and x_value lists to be used in the euler function
solutions = []
x_values = []
for s in range (0, n):
# compute the function
eq = diff_eq(x, y)
s = y + h*(eq)
#replace initial values with the new values and adds them to the solution and x_value lists
y = s
solutions.append(s)
x = x + h
x_values.append(x)
#creates a dataframe from the solutions
Euler_values = pd.DataFrame(list(zip(x_values, solutions)), columns = ['xₙ', 'yₙ'])
#creates an index for the dataframe starting at one and increasing by one
Euler_values.index = Euler_values.index + 1
Euler_values.rename_axis('n', inplace = True)
return Euler_values
#enter an initial x value, initial y value, the differential equation, the step size, and the number of solutions to generate
eulers_method(0, 1, lambda x, y:x+y, 0.1, 10)
输出:
xₙ yₙ
n
1 0.1 1.100000
2 0.2 1.220000
3 0.3 1.362000
4 0.4 1.528200
5 0.5 1.721020
6 0.6 1.943122
7 0.7 2.197434
8 0.8 2.487178
9 0.9 2.815895
10 1.0 3.187485
第二种并不是真正理想的方法是传递函数的字符串版本,并将其放在
eval
函数中:eulers_method
输出是一样的。