数学物理方程的求解
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数学物理定解问题的求解方法
行波法
分离变量法
幂级数解
格林函数法
积分变换
包角变换
变分法
数值解
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
import scipy.special as ss
import sympy as sy
特殊函数
开始前，一定要仔细检查文献中这些函数的具体定义。文献不同，定义也会小有不同！
Airy Functions
Elliptic Functions
Gamma Functions
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d
import scipy.special as ss
from scipy.special import factorial
import sympy as sy
fig = plt.figure(figsize=(8,24),tight_layout=True)
fig.suptitle("Gamma")
ax1 = plt.subplot(311)
ax1.set_title("Gamma Function: $\Gamma(z)=\int_0^{\infty}e^{-t}t^{z-1}dt$")
X = np.linspace(0,5,101)
Y = ss.gamma(X)
ax1.plot(X,Y,label="Gamma Function in Real Sets")
plt.legend()
ax2 = plt.subplot(312)
ax2.set_title("Beta Function: $ B(P,Q)=\int_0^1 x^{P-1}(1-x)^{Q-1} dx $")
for p in range(2,6):
for q in range(2,6):
Y = ss.beta(p,q)
ax2.scatter(p,q)
ax2.text(p+0.1,q-0.1,str(round(Y,3)))
ax2.set_xlabel("p")
ax2.set_ylabel("q")
plt.legend()
ax3 = plt.subplot(313)
ax3.set_title("Psi Function $\Psi$")
Y = ss.psi(X)
ax3.plot(X,Y,label="$\Psi function$")
ax3.text(3,-2,"$Euler Const:-\psi(1)=0.577 215 ...$")
plt.pause(0.2)
plt.savefig("0.jpg")
Bessel Funcitons
v阶贝塞尔方程
v不为整数
v任意值
汉克尔函数
fig = plt.figure(figsize=(8,8))
fig.suptitle("Bessel")
ax1 = plt.subplot(211)
ax1.set_title("FIRST KIND")
X = np.linspace(0,10,101)
for i in range(5):
Y = ss.jv(i,X)
ax1.plot(X,Y,label=str(i)+"th order of first kind Bessel function")
plt.legend()
ax1.axhline(0)
ax2 = plt.subplot(212)
ax2.set_title("SECOND KIND")
for i in range(5):
Y = ss.yn(i,X[10:])
ax2.plot(X[10:],Y,label=str(i)+"th order of second kind Bessel function")
plt.legend()
plt.pause(0.2)
plt.savefig("0.jpg")
Legendre Functions
还有递推公式：
注意：python scipy.special.lpmv 里的连带勒让德函数乘上了一个 (-1)^m
help(ss.legendre)
ss.legendre(6)
poly1d([ 1.44375000e+01, 0.00000000e+00, -1.96875000e+01, 1.60288449e-15,
6.56250000e+00, 0.00000000e+00, -3.12500000e-01])
help(ss.lpmv)
X = np.linspace(-1,1,102)
Y = ss.lpmv(1,2,X)
plt.plot(X,Y)
f = lambda x:3*x*(1-x**2)**0.5
plt.plot(X,f(X))以俯仰角为变量的勒让德函数
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
import scipy.special as ss
import sympy as sy
fig = plt.figure(figsize=(12,4))
ax1 = plt.subplot(131,projection = "polar")
ax2 = plt.subplot(132,projection = "polar")
ax3 = plt.subplot(133)
def abs_lpmv_polar(m,l,x):
return abs(((-1)**m)*ss.lpmv(m,l,x))
def lpmv_polar(m,l,x):
return ((-1)**m)*ss.lpmv(m,l,x)
def lpmv_rect(m,l,x):
return ((-1)**m)*ss.lpmv(m,l,x)
X = np.linspace(0,np.pi,100)
Y1 = abs_lpmv_polar(1,3,np.cos(X))
Y2 = lpmv_polar(1,3,np.cos(X))
Y3 = lpmv_rect(1,3,np.cos(X))
ax1.plot(X,Y1,label="m1l3")
ax1.set_title("abs_lpmv_polar")
ax1.legend()
ax2.plot(X,Y2,label="m1l3")
ax2.set_title("lpmv_polar")
ax2.legend()
ax3.plot(X,Y3,label="m1l3")
ax3.legend()
plt.pause(0.2)
实球谐函数
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d
import scipy.special as ss
from scipy.special import factorial
import sympy as sy
fig = plt.figure(figsize=(12,4))
ax1 = plt.subplot(131,projection = "3d")
ax2 = plt.subplot(132,projection = "3d")
ax3 = plt.subplot(133,projection = "3d")
def lpmv(m,l,x):
return ((-1)**m)*ss.lpmv(m,l,x)
def Yfun(l,theta,phi):
A = ((2*l+1)/4/np.pi)**0.5
return A * lpmv(0,l,theta)
Theta = np.linspace(0,np.pi,101)
Phi = np.linspace(0,2*np.pi,201)
Theta,Phi = np.meshgrid(Theta,Phi)
X = Yfun(1,Theta,Phi)*np.sin(Theta)*np.cos(Phi)
Y = Yfun(1,Theta,Phi)*np.sin(Theta)*np.sin(Phi)
Z = Yfun(1,Theta,Phi)*np.cos(Theta)
ax1.plot_surface(X,Y,Z)
X = Yfun(2,Theta,Phi)*np.sin(Theta)*np.cos(Phi)
Y = Yfun(2,Theta,Phi)*np.sin(Theta)*np.sin(Phi)
Z = Yfun(2,Theta,Phi)*np.cos(Theta)
ax2.plot_surface(X,Y,Z)
X = Yfun(3,Theta,Phi)*np.sin(Theta)*np.cos(Phi)
Y = Yfun(3,Theta,Phi)*np.sin(Theta)*np.sin(Phi)
Z = Yfun(3,Theta,Phi)*np.cos(Theta)
ax3.plot_surface(X,Y,Z)
plt.pause(0.2)
plt.savefig("0.jpg")
