def test_pdsolve_variable_coeff(): f, F = map(Function, ['f', 'F']) u = f(x, y) eq = x * (u.diff(x)) - y * (u.diff(y)) + y**2 * u - y**2 sol = pdsolve(eq, hint="1st_linear_variable_coeff") assert sol == Eq(u, F(x * y) * exp(y**2 / 2) + 1) assert checkpdesol(eq, sol)[0] eq = x**2 * u + x * u.diff(x) + x * y * u.diff(y) sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, F(y * exp(-x)) * exp(-x**2 / 2)) assert checkpdesol(eq, sol)[0] eq = y * x**2 * u + y * u.diff(x) + u.diff(y) sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, F(-2 * x + y**2) * exp(-x**3 / 3)) assert checkpdesol(eq, sol)[0] eq = exp(x)**2 * (u.diff(x)) + y sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, y * exp(-2 * x) / 2 + F(y)) assert checkpdesol(eq, sol)[0] eq = exp(2 * x) * (u.diff(y)) + y * u - u sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, F(x) * exp(-y * (y - 2) * exp(-2 * x) / 2))
def test_pde_1st_linear_constant_coeff(): f, F = map(Function, ["f", "F"]) u = f(x, y) eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y) sol = pdsolve(eq) assert sol == Eq(f(x, y), (F(4 * x + 2 * y) + exp(x / S(2) + 4 * y) / S(15)) * exp(x / S(2) - y)) assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral") assert checkpdesol(eq, sol)[0] eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u) sol = pdsolve(eq) assert sol == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / S(3)) assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral") assert checkpdesol(eq, sol)[0] eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x) sol = pdsolve(eq) assert sol == Eq(f(x, y), F(3 * x + y) * exp(x / S(5) - 3 * y / S(5)) - 2 * sin(x) / S(5) - cos(x) / S(5)) assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral") assert checkpdesol(eq, sol)[0] eq = u + u.diff(x) + u.diff(y) + x * y sol = pdsolve(eq) assert sol == Eq(f(x, y), -x * y + x + y + F(x - y) * exp(-x / S(2) - y / S(2)) - 2) assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral") assert checkpdesol(eq, sol)[0] eq = u + u.diff(x) + u.diff(y) + log(x) assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral")
def test_pdsolve_variable_coeff(): f, F = map(Function, ['f', 'F']) u = f(x, y) eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2 sol = pdsolve(eq, hint="1st_linear_variable_coeff") assert sol == Eq(u, F(x*y)*exp(y**2/2) + 1) assert checkpdesol(eq, sol)[0] eq = x**2*u + x*u.diff(x) + x*y*u.diff(y) sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, F(y*exp(-x))*exp(-x**2/2)) assert checkpdesol(eq, sol)[0] eq = y*x**2*u + y*u.diff(x) + u.diff(y) sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, F(-2*x + y**2)*exp(-x**3/3)) assert checkpdesol(eq, sol)[0] eq = exp(x)**2*(u.diff(x)) + y sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, y*exp(-2*x)/2 + F(y)) assert checkpdesol(eq, sol)[0] eq = exp(2*x)*(u.diff(y)) + y*u - u sol = pdsolve(eq, hint='1st_linear_variable_coeff') assert sol == Eq(u, exp((-y**2 + 2*y + 2*F(x))*exp(-2*x)/2))
def test_pde_1st_linear_constant_coeff_homogeneous(): f, F = map(Function, ['f', 'F']) u = f(x, y) eq = 2 * u + u.diff(x) + u.diff(y) assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', ) sol = pdsolve(eq) assert sol == Eq(u, F(x - y) * exp(-x - y)) assert checkpdesol(eq, sol)[0] eq = 4 + (3 * u.diff(x) / u) + (2 * u.diff(y) / u) assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', ) sol = pdsolve(eq) assert sol == Eq(u, F(2 * x - 3 * y) * exp(-S(12) * x / 13 - S(8) * y / 13)) assert checkpdesol(eq, sol)[0] eq = u + (6 * u.diff(x)) + (7 * u.diff(y)) assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', ) sol = pdsolve(eq) assert sol == Eq(u, F(7 * x - 6 * y) * exp(-6 * x / S(85) - 7 * y / S(85))) assert checkpdesol(eq, sol)[0] eq = a * u + b * u.diff(x) + c * u.diff(y) sol = pdsolve(eq) assert checkpdesol(eq, sol)[0]
def test_solvefun(): f, F, G, H = map(Function, ['f', 'F', 'G', 'H']) eq1 = f(x, y) + f(x, y).diff(x) + f(x, y).diff(y) assert pdsolve(eq1) == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2)) assert pdsolve(eq1, solvefun=G) == Eq(f(x, y), G(x - y) * exp(-x / 2 - y / 2)) assert pdsolve(eq1, solvefun=H) == Eq(f(x, y), H(x - y) * exp(-x / 2 - y / 2))
def test_pde_1st_linear_constant_coeff(): f, F = map(Function, ["f", "F"]) u = f(x, y) eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y) sol = pdsolve(eq) assert sol == Eq(f(x, y), (F(4 * x + 2 * y) + exp(x / S(2) + 4 * y) / S(15)) * exp(x / S(2) - y)) assert classify_pde(eq) == ( "1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", ) assert checkpdesol(eq, sol)[0] eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u) sol = pdsolve(eq) assert sol == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / S(3)) assert classify_pde(eq) == ( "1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", ) assert checkpdesol(eq, sol)[0] eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x) sol = pdsolve(eq) assert sol == Eq( f(x, y), F(3 * x + y) * exp(x / S(5) - 3 * y / S(5)) - 2 * sin(x) / S(5) - cos(x) / S(5), ) assert classify_pde(eq) == ( "1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", ) assert checkpdesol(eq, sol)[0] eq = u + u.diff(x) + u.diff(y) + x * y sol = pdsolve(eq) assert sol == Eq(f(x, y), -x * y + x + y + F(x - y) * exp(-x / S(2) - y / S(2)) - 2) assert classify_pde(eq) == ( "1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", ) assert checkpdesol(eq, sol)[0] eq = u + u.diff(x) + u.diff(y) + log(x) assert classify_pde(eq) == ( "1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", )
def test_checkpdesol(): f, F = map(Function, ['f', 'F']) eq1 = a*f(x,y) + b*f(x,y).diff(x) + c*f(x,y).diff(y) eq2 = 3*f(x,y) + 2*f(x,y).diff(x) + f(x,y).diff(y) eq3 = a*f(x,y) + b*f(x,y).diff(x) + 2*f(x,y).diff(y) for eq in [eq1, eq2, eq3]: assert checkpdesol(eq, pdsolve(eq))[0] eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y) eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y) eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y) assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [ (False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))), (False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))]
def test_checkpdesol(): f, F = map(Function, ['f', 'F']) eq1 = a * f(x, y) + b * f(x, y).diff(x) + c * f(x, y).diff(y) eq2 = 3 * f(x, y) + 2 * f(x, y).diff(x) + f(x, y).diff(y) eq3 = a * f(x, y) + b * f(x, y).diff(x) + 2 * f(x, y).diff(y) for eq in [eq1, eq2, eq3]: assert checkpdesol(eq, pdsolve(eq))[0] eq4 = x * f(x, y) + f(x, y).diff(x) + 3 * f(x, y).diff(y) eq5 = 2 * f(x, y) + 1 * f(x, y).diff(x) + 3 * f(x, y).diff(y) eq6 = f(x, y) + 1 * f(x, y).diff(x) + 3 * f(x, y).diff(y) assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [ (False, (x - 2) * F(3 * x - y) * exp(-x / S(5) - 3 * y / S(5))), (False, (x - 1) * F(3 * x - y) * exp(-x / S(10) - 3 * y / S(10))) ]
def test_checkpdesol(): f, F = map(Function, ['f', 'F']) eq1 = a*f(x,y) + b*f(x,y).diff(x) + c*f(x,y).diff(y) eq2 = 3*f(x,y) + 2*f(x,y).diff(x) + f(x,y).diff(y) eq3 = a*f(x,y) + b*f(x,y).diff(x) + 2*f(x,y).diff(y) for eq in [eq1, eq2, eq3]: assert checkpdesol(eq, pdsolve(eq))[0] eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y) eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y) eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y) assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [ (False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))), (False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))] for eq in [eq4, eq5, eq6]: assert checkpdesol(eq, pdsolve(eq))[0] sol = pdsolve(eq4) sol4 = Eq(sol.lhs - sol.rhs, 0) raises(NotImplementedError, lambda: checkpdesol(eq4, sol4, solve_for_func=False))
def ForwardKolmogorovEquation(drift, diffusion): p = Function('p') u = p(x, t) ut = u.diff(t) tdrift = u * drift tdiffusion = u * (1/2*diffusion**2) tdt = tdrift.diff(x) tdifft = tdiffusion.diff(x) tdifftt = tdifft.diff(x) eq = Eq(ut + tdt - tdifftt) return pdsolve(eq)
def ForwardKolmogorovEquation(drift, diffusion): p = Function('p') u = p(x, t) ut = u.diff(t) tdrift = u * drift tdiffusion = u * (1 / 2 * diffusion**2) tdt = tdrift.diff(x) tdifft = tdiffusion.diff(x) tdifftt = tdifft.diff(x) eq = Eq(ut + tdt - tdifftt) return pdsolve(eq)
def test_pdsolve_all(): f, F = map(Function, ["f", "F"]) u = f(x, y) eq = u + u.diff(x) + u.diff(y) + x ** 2 * y sol = pdsolve(eq, hint="all") keys = ["1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", "default", "order"] assert sorted(sol.keys()) == keys assert sol["order"] == 1 assert sol["default"] == "1st_linear_constant_coeff" assert sol["1st_linear_constant_coeff"] == Eq( f(x, y), -x ** 2 * y + x ** 2 + 2 * x * y - 4 * x - 2 * y + F(x - y) * exp(-x / S(2) - y / S(2)) + 6 )
def test_pdsolve_all(): f, F = map(Function, ['f', 'F']) u = f(x,y) eq = u + u.diff(x) + u.diff(y) + x**2*y sol = pdsolve(eq, hint = 'all') keys = ['1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral', 'default', 'order'] assert sorted(sol.keys()) == keys assert sol['order'] == 1 assert sol['default'] == '1st_linear_constant_coeff' assert sol['1st_linear_constant_coeff'] == Eq(f(x, y), -x**2*y + x**2 + 2*x*y - 4*x - 2*y + F(x - y)*exp(-x/S(2) - y/S(2)) + 6)
def test_pde_1st_linear_constant_coeff(): f, F = map(Function, ['f', 'F']) u = f(x, y) eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y) sol = pdsolve(eq) assert sol == Eq(f( x, y), (F(4 * x + 2 * y) * exp(x / 2) + exp(x + 4 * y) / 15) * exp(-y)) assert classify_pde(eq) == ('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral') assert checkpdesol(eq, sol)[0] eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u) sol = pdsolve(eq) assert sol == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / 3) assert classify_pde(eq) == ('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral') assert checkpdesol(eq, sol)[0] eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x) sol = pdsolve(eq) assert sol == Eq( f(x, y), F(3 * x + y) * exp(x / 5 - 3 * y / 5) - 2 * sin(x) / 5 - cos(x) / 5) assert classify_pde(eq) == ('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral') assert checkpdesol(eq, sol)[0] eq = u + u.diff(x) + u.diff(y) + x * y sol = pdsolve(eq) assert sol.expand() == Eq( f(x, y), x + y + (x - y)**2 / 4 - (x + y)**2 / 4 + F(x - y) * exp(-x / 2 - y / 2) - 2).expand() assert classify_pde(eq) == ('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral') assert checkpdesol(eq, sol)[0] eq = u + u.diff(x) + u.diff(y) + log(x) assert classify_pde(eq) == ('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral')
def test_pde_1st_linear_constant_coeff_homogeneous(): f, F = map(Function, ["f", "F"]) u = f(x, y) eq = 2 * u + u.diff(x) + u.diff(y) assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",) sol = pdsolve(eq) assert sol == Eq(u, F(x - y) * exp(-x - y)) assert checkpdesol(eq, sol)[0] eq = 4 + (3 * u.diff(x) / u) + (2 * u.diff(y) / u) assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",) sol = pdsolve(eq) assert sol == Eq(u, F(2 * x - 3 * y) * exp(-S(12) * x / 13 - S(8) * y / 13)) assert checkpdesol(eq, sol)[0] eq = u + (6 * u.diff(x)) + (7 * u.diff(y)) assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",) sol = pdsolve(eq) assert sol == Eq(u, F(7 * x - 6 * y) * exp(-6 * x / S(85) - 7 * y / S(85))) assert checkpdesol(eq, sol)[0] eq = a * u + b * u.diff(x) + c * u.diff(y) sol = pdsolve(eq) assert checkpdesol(eq, sol)[0]
def test_pdsolve_all(): f, F = map(Function, ["f", "F"]) u = f(x, y) eq = u + u.diff(x) + u.diff(y) + x**2 * y sol = pdsolve(eq, hint="all") keys = [ "1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral", "default", "order", ] assert sorted(sol.keys()) == keys assert sol["order"] == 1 assert sol["default"] == "1st_linear_constant_coeff" assert sol["1st_linear_constant_coeff"] == Eq( f(x, y), -(x**2) * y + x**2 + 2 * x * y - 4 * x - 2 * y + F(x - y) * exp(-x / S(2) - y / S(2)) + 6, )
def test_PredictorCorrector(input): Xs = input[0] Ys = input[1] exp = input[2] technique = input[3] xs = input[4] approx_error = input[5] exp = exp.replace('m', '*') exp = exp.replace('p', '**') exp = exp[1:len(exp)] try: f = sympy.sympify(exp) except: print('sympify error') pass # print(f) # print(exp) try: ys, errors, finalerror = PredictorCorrector(Xs, Ys, str(f), technique, xs, approx_error) except: pass x = sympy.symbols('x') y = sympy.symbols('y') # try: # # fx = lambdify([x,y] , f) # # print(fx) # except: # print('lambdify error') # pass ux = sympy.diff(f, x) uy = sympy.diff(f, y) eq = Eq(1 + (2 * (ux / f)) + (3 * (uy / f)), 0) try: sol = pdsolve(eq) print(sol) except: pass # ux = sympy.sympify() pass
from sympy.solvers.pde import pdsolve from sympy import Function, diff, Eq from sympy.abc import x, y f = Function('f') u = f(x, y) uxx = u.diff(x).diff(x) uyy = u.diff(y).diff(y) eq = Eq(1 - (2*(uxx)) - (3*(uyy))) print(pdsolve(eq))
import numpy as np import sdeint import matplotlib.pyplot as plt from sympy.solvers.pde import pdsolve from sympy import Function, diff, Eq, cos, sin from sympy.abc import x, t # Weiner Process Simulation f = Function('f') u = f(x, t) ux = u.diff(x) ut = u.diff(t) uxx = ux.diff(x) eq = Eq(ut + (1 - cos(2 * x) * u + (x + 2 - sin(2 * x)) * ux - 0.5 * (sin(x)**2) * uxx)) pdsolve(eq) # Time steps n = 1000 T = 1 ############################################################################### def Wiener(mu, sig, n, T): Delta = T / n # t = np.arange(1, step=Delta) S = np.zeros(n, np.dtype(float)) x = range(1, len(S)) for i in x: dW = np.random.normal(mu, sig) dt = np.sqrt(Delta)
def test_solvefun(): f, F, G, H = map(Function, ["f", "F", "G", "H"]) eq1 = f(x, y) + f(x, y).diff(x) + f(x, y).diff(y) assert pdsolve(eq1) == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2)) assert pdsolve(eq1, solvefun=G) == Eq(f(x, y), G(x - y) * exp(-x / 2 - y / 2)) assert pdsolve(eq1, solvefun=H) == Eq(f(x, y), H(x - y) * exp(-x / 2 - y / 2))
import numpy as np import sdeint import matplotlib.pyplot as plt from sympy.solvers.pde import pdsolve from sympy import Function, diff, Eq, cos, sin from sympy.abc import x, t # Weiner Process Simulation f = Function('f') u = f(x, t) ux = u.diff(x) ut = u.diff(t) uxx = ux.diff(x) eq = Eq(ut + (1 - cos(2*x)*u + (x + 2 - sin(2*x))*ux - 0.5*(sin(x)**2)*uxx)) pdsolve(eq) # Time steps n = 1000 T = 1 ############################################################################### def Wiener(mu, sig, n, T): Delta = T/n # t = np.arange(1, step=Delta) S = np.zeros(n, np.dtype(float)) x = range(1, len(S)) for i in x: dW = np.random.normal(mu, sig) dt = np.sqrt(Delta)