analytical_solution.sagews

%typeset_mode True

var('x, y, t')

# E_z =  sin(2 * pi * (x + t)) + cos(2 * pi * (y - t))
# B_x =  sin(2 * pi * (x + t)) + cos(2 * pi * (y - t))
# B_y = -cos(2 * pi * (x + t)) + sin(2 * pi * (y - t))

# E_z =  sin(2 * pi * (x + t)) + cos(2 * pi * (y - t))
# B_x =  cos(2 * pi * (y - t))
# B_y =  sin(2 * pi * (x + t))

E_z = sin(2 * pi * (x + t)) + cos(2 * pi * (y - t))
B_x = cos(2 * pi * (x + t)) + sin(2 * pi * (y - t))
B_y = 2 * pi * sin(2 * pi * (x + t)) * (y - t)

E_z
B_x
B_y

($\displaystyle x$, $\displaystyle y$, $\displaystyle t$)

$\displaystyle \cos\left(-2 \, \pi t + 2 \, \pi y\right) + \sin\left(2 \, \pi t + 2 \, \pi x\right)$

$\displaystyle \cos\left(2 \, \pi t + 2 \, \pi x\right) + \sin\left(-2 \, \pi t + 2 \, \pi y\right)$

$\displaystyle -2 \, \pi {\left(t - y\right)} \sin\left(2 \, \pi t + 2 \, \pi x\right)$

eq_1 = diff(E_z, t) - diff(B_y, x) + diff(B_x, y)
eq_1

eq_2 = diff(B_x, t) + diff(E_z, y)
eq_2

eq_3 = diff(B_y, t) - diff(E_z, x)
eq_3

$\displaystyle 2 \, {\left(\pi + 2 \, \pi^{2} t\right)} \cos\left(2 \, \pi t\right) \cos\left(2 \, \pi x\right) - 2 \, {\left(\pi + 2 \, \pi^{2} t\right)} \sin\left(2 \, \pi t\right) \sin\left(2 \, \pi x\right) - 4 \, {\left(\pi^{2} \cos\left(2 \, \pi t\right) \cos\left(2 \, \pi x\right) - \pi^{2} \sin\left(2 \, \pi t\right) \sin\left(2 \, \pi x\right)\right)} y + 2 \, {\left(\pi \cos\left(2 \, \pi t\right) - \pi \sin\left(2 \, \pi t\right)\right)} \cos\left(2 \, \pi y\right) + 2 \, {\left(\pi \cos\left(2 \, \pi t\right) + \pi \sin\left(2 \, \pi t\right)\right)} \sin\left(2 \, \pi y\right)$

$\displaystyle -2 \, \pi \cos\left(2 \, \pi x\right) \sin\left(2 \, \pi t\right) - 2 \, \pi \cos\left(2 \, \pi t\right) \sin\left(2 \, \pi x\right) - 2 \, {\left(\pi \cos\left(2 \, \pi t\right) - \pi \sin\left(2 \, \pi t\right)\right)} \cos\left(2 \, \pi y\right) - 2 \, {\left(\pi \cos\left(2 \, \pi t\right) + \pi \sin\left(2 \, \pi t\right)\right)} \sin\left(2 \, \pi y\right)$

$\displaystyle 4 \, {\left(\pi^{2} \cos\left(2 \, \pi t\right) \cos\left(2 \, \pi x\right) - \pi^{2} \sin\left(2 \, \pi t\right) \sin\left(2 \, \pi x\right)\right)} y - 2 \, {\left({\left(\pi + 2 \, \pi^{2} t\right)} \cos\left(2 \, \pi t\right) + \pi \sin\left(2 \, \pi t\right)\right)} \cos\left(2 \, \pi x\right) - 2 \, {\left(\pi \cos\left(2 \, \pi t\right) - {\left(\pi + 2 \, \pi^{2} t\right)} \sin\left(2 \, \pi t\right)\right)} \sin\left(2 \, \pi x\right)$