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2024-07-04
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2024-07-04
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import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
from scipy.special import gamma, beta
from sympy import symbols, integrate, diff, Function, pi, exp, sin, cos, tan, sqrt, I, Eq, dsolve, simplify
import sympy as sp
# Define symbols
x, t, v, theta, r = sp.symbols('x t v theta r')
phi = sp.Function('phi')(x, t)
n = sp.Function('n')(x, t)
v_x = sp.Function('v')(x, t)
f = sp.Function('f')(x, v, t)
C_f = sp.Function('C')(f)
psi = sp.Function('psi')(x, t)
V = sp.Function('V')(x, t)
g, m, hbar, nu = sp.symbols('g m hbar nu')
u = sp.Function('u')(x, t)
J = sp.Function('J')(x, t)
E = sp.Function('E')(x, t)
B = sp.Function('B')(x, t)
rho = sp.Function('rho')(x, t)
epsilon_0, mu_0 = symbols('epsilon_0 mu_0')
# Define equations
# Reynolds' Transport Theorem
RTT = integrate(diff(phi, t), x) - integrate(n * phi * v_x, x)
print("Reynolds' Transport Theorem:")
sp.pprint(RTT)
print("\n")
"""
Reynolds' Transport Theorem:
⌠
⌠ ⎮ ∂
- ⎮ n(x, t)⋅φ(x, t)⋅v(x, t) dx + ⎮ ──(φ(x, t)) dx
⌡ ⎮ ∂t
⌡
"""
# Boltzmann Transport Equation
BTE = sp.Function('F')(x, t) * diff(f, v) + v_x * diff(f, x) + diff(f, t) - C_f
print("Boltzmann Transport Equation:")
sp.pprint(BTE)
print("\n")
"""
∂ ∂ ∂
-C(f(x, v, t)) + F(x, t)⋅──(f(x, v, t)) + v(x, t)⋅──(f(x, v, t)) + ──(f(x, v,t))
∂v ∂x ∂t
"""
# Gross-Pitaevskii Equation
GPE = I * hbar * diff(psi, t) - V * psi - g * psi * abs(psi)**2 + (hbar**2 / (2 * m)) * diff(psi, x, x)
print("Gross-Pitaevskii Equation:")
sp.pprint(GPE)
print("\n")
"""
Gross-Pitaevskii Equation:
>>> sp.pprint(GPE)
2
2 ∂
h̅ ⋅───(ψ(x, t))
2
2 ∂x ∂
- g⋅ψ(x, t)⋅│ψ(x, t)│ + ─────────────── + ⅈ⋅h̅⋅──(ψ(x, t)) - V(x, t)⋅ψ(x, t)
2⋅m ∂t
"""
# Transform of Burgers' Equation
Burgers = -nu * diff(u, x, x) + u * diff(u, x) + diff(u, t)
print("Transform of Burgers' Equation:")
sp.pprint(Burgers)
print("\n")
"""
2
∂ ∂ ∂
- ν⋅───(u(x, t)) + u(x, t)⋅──(u(x, t)) + ──(u(x, t))
2 ∂x ∂t
∂x
"""
# Schrödinger Equation
Schrodinger = I * hbar * diff(psi, t) - V * psi
print("Schrödinger Equation:")
sp.pprint(Schrodinger)
print("\n")
"""
∂
ⅈ⋅h̅⋅──(ψ(x, t)) - V(x, t)⋅ψ(x, t)
∂t
"""
# Heat Equation
Heat = diff(u, t) - diff(u, x, x)
print("Heat Equation:")
sp.pprint(Heat)
print("\n")
"""
2
∂ ∂
──(u(x, t)) - ───(u(x, t))
∂t 2
∂x
"""
# Laplace's Equation
u, x, y, z = sp.symbols('u x y z')
Laplace = diff(u, x, x) + diff(u, y, y) + diff(u, z, z)
print("Laplace's Equation:")
sp.pprint(Laplace)
print("\n")
"""
0
"""
class rho:
def __init__(self, x, y, z):
self.x = x
self.y = y
self.z = z
def __call__(self, x, y, z):
return self.x + self.y + self.z
# Navier-Stokes Equations (Simplified form)
Navier_Stokes_x = sp.Function('u')(x, t) * sp.diff(sp.Function('u')(x, t), x) + sp.diff(sp.Function('u')(x, t), t)
print("Navier-Stokes Equation (x-component):")
sp.pprint(Navier_Stokes_x)
print("\n")
"""
∂ ∂
u(x, t)⋅──(u(x, t)) + ──(u(x, t))
∂x ∂t
"""