微分方程的同情系统：未实现

I am trying to solve a complicated system of differential equations using sympy. I use sympy to calculate time derivatives quickly, after which I have two equations of derivatives that contain derivatives themselves. The equations are not linear and do not fit the type of equations sympy recognizes, so it throws a Not Implemented Error. Is there an easier way to solve these equations (even numerically), and get their respective law of motion (values across time)? This is probably very inefficient, so if someone know of a more efficient process, I am all ears, I primarily started using sympy as calculating derivatives is fairly quick (otherwise I spend lots of time wasting paper).

我试图用sympy解决一个复杂的微分方程组。我使用sympy来快速计算时间导数，之后我有两个包含导数本身的导数方程。方程式不是线性的，并且不适合同意识别的方程式，因此它会抛出未实现的错误。是否有更简单的方法来解决这些方程（甚至数字），并获得各自的运动定律（时间上的值）？这可能非常低效，所以如果有人知道一个更有效的过程，我很满意，我主要是开始使用sympy，因为计算衍生物相当快（否则我会花很多时间浪费纸张）。

import sympy as sym

a,b = sym.S(['a','b'])
S1,S2, alpha, r, c1 ,c2 = sym.symbols('S1, S2, alpha, r, c1, c2',  negative=False)
t = sym.var('t')
x1 = sym.Function("x1")(t)
x2 = sym.Function("x2")(t)
lam = sym.Function('lam')(t)
gam = sym.Function('gam')(t)

p = (1/3)*a*(alpha*x1 + (1-alpha)*x2)**3 + (1/2)*b*(alpha*x1 + (1-alpha)*x2)
lagrangian = p - c1/2*alpha*x1 - c2/2*(1-alpha)*x2 + lam*(S1 - 0.5*x1**2) + gam*(S2 - 0.5*x2**2)

FOC_x1 = sym.diff(lagrangian,x1) 
FOC_x2 = sym.diff(lagrangian,x2)

x1_lam = sym.solve(FOC_x1,x1)[0]
lam_x1 = sym.solve(FOC_x1,lam)[0]

x2_gam = sym.solve(FOC_x2,x2)[0]
gam_x2 = sym.solve(FOC_x2,gam)[0]

dx1_dt = sym.diff(x1_lam,t).subs(lam,lam_x1)
dx2_dt = sym.diff(x2_gam,t).subs(gam,gam_x2)

x1dot = sym.Derivative(x1,t)
x2dot = sym.Derivative(x2,t)

eq = [sym.Eq(x1dot,dx1_dt),sym.Eq(x2dot,dx2_dt)]

sym.dsolve(eq)

To clarify: My state variables are S1 and S2, respectively, my control variables are x1 and x2, respectively, and finally my costate variables are lam and gam, respectively.

为了澄清：我的状态变量分别是S1和S2，我的控制变量分别是x1和x2，最后我的costate变量分别是lam和gam。

1 个解决方案

#1

I'm not quite able to identify what your dependent variables are, x1 and x2? For numerical integration of a (system) of first order ordinary differential equations expressed as SymPy expressions you might be interested in using SymbolicSys (example found here) from pyodesys.

我不太能确定你的因变量是什么，x1和x2？对于表示为SymPy表达式的一阶常微分方程（系统）的数值积分，您可能有兴趣使用来自pyodesys的SymbolicSys（此处的示例）。