def ide(x, y, int_mat):
"""int_0^x y(t)dt
"""
lhs1 = tf.matmul(int_mat, y)
lhs2 = tf.gradients(y, x)[0]
rhs = 2 * np.pi * tf.cos(2 * np.pi * x) + tf.sin(np.pi * x) ** 2 / np.pi
return lhs1 + (lhs2 - rhs)[: tf.size(lhs1)]
def fpde(x, y, int_mat):
"""(D_{0+}^alpha + D_{1-}^alpha) u(x)"""
if isinstance(int_mat, (list, tuple)) and len(int_mat) == 3:
int_mat = tf.SparseTensor(*int_mat)
lhs = tf.sparse_tensor_dense_matmul(int_mat, y)
else:
lhs = tf.matmul(int_mat, y)
lhs /= 2 * tf.cos(alpha * np.pi / 2)
rhs = gamma(alpha0 + 2) * x
return lhs - rhs[:tf.size(lhs)]
def fpde(x, y, int_mat):
"""\int_theta D_theta^alpha u(x)"""
if isinstance(int_mat, (list, tuple)) and len(int_mat) == 3:
int_mat = tf.SparseTensor(*int_mat)
lhs = tf.sparse_tensor_dense_matmul(int_mat, y)
else:
lhs = tf.matmul(int_mat, y)
lhs = lhs[:, 0]
lhs *= gamma((1 - alpha) / 2) * gamma((3 + alpha) / 2) / (2 * np.pi**2)
x = x[:tf.size(lhs)]
rhs = (2**alpha * gamma(2 + alpha / 2) * gamma((3 + alpha) / 2) /
gamma(3 / 2) * (1 - (1 + alpha / 3) * tf.reduce_sum(x**2, axis=1)))
return lhs - rhs
def fpde(x, y, int_mat):
"""du/dt + (D_{0+}^alpha + D_{1-}^alpha) u(x) = f(x)"""
if isinstance(int_mat, (list, tuple)) and len(int_mat) == 3:
int_mat = tf.SparseTensor(*int_mat)
lhs = -tf.sparse_tensor_dense_matmul(int_mat, y)
else:
lhs = -tf.matmul(int_mat, y)
dy_t = tf.gradients(y, x)[0][:, 1:2]
x, t = x[:, :-1], x[:, -1:]
rhs = -dy_t - tf.exp(-t) * (
x ** 3 * (1 - x) ** 3
+ gamma(4) / gamma(4 - alpha) * (x ** (3 - alpha) + (1 - x) ** (3 - alpha))
- 3 * gamma(5) / gamma(5 - alpha) * (x ** (4 - alpha) + (1 - x) ** (4 - alpha))
+ 3 * gamma(6) / gamma(6 - alpha) * (x ** (5 - alpha) + (1 - x) ** (5 - alpha))
- gamma(7) / gamma(7 - alpha) * (x ** (6 - alpha) + (1 - x) ** (6 - alpha))
)
return lhs - rhs[: tf.size(lhs)]
def fpde(x, y, int_mat):
"""(D_{0+}^alpha + D_{1-}^alpha) u(x) = f(x)"""
if isinstance(int_mat, (list, tuple)) and len(int_mat) == 3:
int_mat = tf.SparseTensor(*int_mat)
lhs = tf.sparse_tensor_dense_matmul(int_mat, y)
else:
lhs = tf.matmul(int_mat, y)
rhs = (gamma(4) / gamma(4 - alpha) * (x**(3 - alpha) +
(1 - x)**(3 - alpha)) -
3 * gamma(5) / gamma(5 - alpha) * (x**(4 - alpha) +
(1 - x)**(4 - alpha)) +
3 * gamma(6) / gamma(6 - alpha) * (x**(5 - alpha) +
(1 - x)**(5 - alpha)) -
gamma(7) / gamma(7 - alpha) * (x**(6 - alpha) +
(1 - x)**(6 - alpha)))
# lhs /= 2 * np.cos(alpha * np.pi / 2)
# rhs = gamma(alpha + 2) * x
return lhs - rhs[:tf.size(lhs)]
def ide(x, y, int_mat):
rhs = tf.matmul(int_mat, y)
lhs1 = tf.gradients(y, x)[0]
return (lhs1 + y)[:tf.size(rhs)] - rhs
