How to put higher order boundary conditions while solving 4th order time dependent pde (1d)

[divyanshjnv1]

Dear all, I am trying to solve Cahn hilliard eqn (a 4th order time dependent pde) but im not able to figure out syntax to put 2nd and 3rd order boundary conditions . 1- Do i need to put all 4 BCs for 4th order eqn or less BCs may work . 2- And please look , is my pde is defined correctly ? (specially 'f_prime' term)

reference eqn is uploaded , code what i have written is also added, please help.

import deepxde as dde
import numpy as np

from deepxde.backend import tf
import tensorflow as tf
import matplotlib.pyplot as plt

def dy_xx(x, y):
return dde.grad.hessian(y, x, i=0, j=0)

def dy_xxx(x, y):
return dde.grad.jacobian(dy_xx(x, y), x, i=0, j=0)

def boundary_l(x, on_boundary):
return on_boundary and dde.utils.isclose(x[0], 0)

def boundary_r(x, on_boundary):
return on_boundary and dde.utils.isclose(x[0], 1)

def pde(x, y):
M, kappa = 1.0, 1.0
f_prime = 2y - 3y**3 # First derivative of f w.r.t. c

dy_t = dde.grad.jacobian(y, x, i=0, j=1)        # 1st derivative of c
dy_xx = dde.grad.hessian(y, x, i=0, j=0)        # 2nd derivative of c
dy_xxx = dde.grad.jacobian(dy_xx, x, i=0, j=0)  # 3rd derivative of c
dy_xxxx = dde.grad.hessian(dy_xx, x, i=0, j=0)  # 4th derivative of c

d2f_prime_dx2 = dde.grad.hessian(f_prime, x, i=0, j=0)  # 2nd derivative of f_prime

return dy_t - M * (d2f_prime_dx2 - kappa * dy_xxxx)
def func(x):
return 0.5*(1 + np.tanh(0.5* x [:,0:1]))

geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

bc = dde.icbc.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary)
bc3 = dde.icbc.OperatorBC(geom, lambda x, y, _: dy_xx , boundary_r)
bc4 = dde.icbc.OperatorBC(geom, lambda x, y, _: dy_xxx, boundary_r)

ic = dde.icbc.IC(geomtime, func, lambda _, on_initial: on_initial)
data = dde.data.TimePDE(
geomtime,
pde,
[bc,bc3,bc4, ic],
num_domain=40,
num_boundary=20,
num_initial=10,
solution=func,
num_test=10000,
)

layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.nn.FNN(layer_size, activation, initializer)

model = dde.Model(data, net)

model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(iterations=10000)

dde.saveplot(losshistory, train_state, issave=True, isplot=True)```

Thanks.
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[praksharma]

Can you please use markdown code block to write the code? Here is some help.