Multivariate differentiation in Python

Question

I have a function

that I want to simplify and differentiate in Python, defined as**

def u(x, t):
    return math.erf((x + 1) / (2 * (k * t) ** (1 / 2)))

** Please correct me if I am wrong.

I have all the necessary imports as follows:

import math
import scipy
import matplotlib
from sympy import *

As well as defining the symbols

x, k, t = symbols('x k t')

This works perfectly fine:

def f(x):
     return x ** 4

diff(f(x))

Which returns the correct answer,

4x^3

However, this

diff(u(x, t))

or this

diff(u(x, t), t)

returns an error as follows

TypeError Traceback (most recent call last) in () ----> 1 diff(u(x, t))

in u(x, t) 1 def u(x, t): ----> 2 return math.erf((x + 1) / (2 * (k * t) ** (1 / 2)))

C:\Anaconda\lib\site-packages\sympy\core\expr.py in float(self) 223 if result.is_number and result.as_real_imag()1: 224 raise TypeError("can't convert complex to float") --> 225 raise TypeError("can't convert expression to float") 226 227 def complex(self):

TypeError: can't convert expression to float

In Matlab I could easily do it:

syms x;
syms k;
syms t;
u = erf((x + 1)/(2 * sqrt(k * t)));
LHS = simplify(diff(u, t))
RHS = k * simplify(diff(u, x, 2))

My question is, how can I differentiate and/or simplify a mathematical function of more than one variable in Python?

Answer 1

You need to use sympy.erf, not math.erf:

>>> import sympy
>>> x, k, t = sympy.symbols('x k t')
>>> def u(x, t):
...     return sympy.erf((x + 1) / (2 * (k * t) ** (1 / 2)))
>>> sympy.diff(u(x, t), x, t)
(0.25*(k*t)**(-1.5)*(x + 1)**2 - 0.5*(k*t)**(-0.5))*exp(-(k*t)**(-1.0)*(x + 1)**2/4)/(sqrt(pi)*t)