Partial Differential Equations

1. Partial Differential Equations#

This chapter will introduce the concept of partial differential equations, the different types and the general methodology used to solve partial differential equations using numerical methods.

1.1. Definitions#

from helper import cleaner
from sympy import Derivative as D
from sympy import Eq, Function, exp, pi, sin, symbols

x, y = symbols("x, y")
U = symbols("U", cls=Function)(x, y)

D(U, x)

\frac{\partial}{\partial x} U (x, y)

1.1.1. Partial derivatives#

_U = x**2 + x * y
U_x = D(_U, x).simplify()

Eq(D(U, x), U_x)

\frac{\partial}{\partial x} U (x, y) = 2 x + y

U_y = D(_U, y).simplify()

Eq(D(U, y), U_y)

\frac{\partial}{\partial y} U (x, y) = x

1.1.2. Partial differential equations#

a = symbols("a", cls=Function)(x, y)
b = symbols("b", cls=Function)(x, y)
c = symbols("c", cls=Function)(x, y)
d = symbols("d", cls=Function)(x, y)
e = symbols("e", cls=Function)(x, y)
f = symbols("f", cls=Function)(x, y)

Eq(a * D(U, x) + b * D(U, y) + c * D(U, x, 2) + d * D(U, y, 2) + e * D(D(U, y), x), f)

a (x, y) \frac{\partial}{\partial x} U (x, y) + b (x, y) \frac{\partial}{\partial y} U (x, y) + c (x, y) \frac{\partial^{2}}{\partial x^{2}} U (x, y) + d (x, y) \frac{\partial^{2}}{\partial y^{2}} U (x, y) + e (x, y) \frac{\partial^{2}}{\partial x \partial y} U (x, y) = f (x, y)

1.1.3. Order of a PDE#

1.1.4. Linear PDEs#

1.1.5. Domain#

1.1.6. Initial conditions#

1.1.7. Boundary conditions#

1.1.8. Solution of a PDE#

cleaner(["x", "y", "U", "U_x", "U_y", "a", "b", "c", "d", "e", "f"])

Symbolic variables already cleared

1.1.9. Example $1$ #

Show that the diffusion equation with Homogeneous Dirichlet Boundary conditions

\begin{array}{r} {\begin{cases} U_{t} = k U_{x x} & for 0 < x < L, t > 0. \\ U (0, x) = 2 \sin (\frac{π x}{L}) & for 0 < x < L . \\ U (t, 0) = U (t, L) = 0 & for t > 0. \end{cases} \end{array}

has the solution

U (t, x) = 2 \sin (\frac{π x}{L}) \exp (- \frac{k π^{2} t}{L^{2}}) .

t, x, k, L = symbols("t, x, k, L")
U = symbols("U", cls=Function)(t, x)

heat_equation = Eq(D(U, t), k * D(U, x, 2))

heat_equation

\frac{\partial}{\partial t} U (t, x) = k \frac{\partial^{2}}{\partial x^{2}} U (t, x)

1.1.9.1. Analytical solution#

heat_solution = 2 * sin(pi * x / L) * exp(-k * pi**2 * t / L**2)

Eq(U, heat_solution)

U (t, x) = 2 e^{- \frac{π^{2} k t}{L^{2}}} \sin (\frac{π x}{L})

1.1.9.2. Initial condition#

U_0 = heat_solution.subs({t: 0})

Eq(U.subs({t: 0}), U_0)

U (0, x) = 2 \sin (\frac{π x}{L})

1.1.9.3. Dirichlet Boundary condition#

U_left, U_right = heat_solution.subs({x: 0}), heat_solution.subs({x: L})

Eq(U.subs({x: 0}), U_left)

U (t, 0) = 0

Eq(U.subs({x: L}), U_right)

U (t, L) = 0

1.1.9.4. Checking solution#

U_t = heat_equation.lhs.subs({U: heat_solution}).simplify()

Eq(D(U, t), U_t)

\frac{\partial}{\partial t} U (t, x) = - \frac{2 π^{2} k e^{- \frac{π^{2} k t}{L^{2}}} \sin (\frac{π x}{L})}{L^{2}}

U_xx = heat_equation.rhs.subs({U: heat_solution}).simplify() / k

Eq(D(U, x, 2), U_xx)

\frac{\partial^{2}}{\partial x^{2}} U (t, x) = - \frac{2 π^{2} e^{- \frac{π^{2} k t}{L^{2}}} \sin (\frac{π x}{L})}{L^{2}}

U_t.equals(k * U_xx)

True

cleaner(["t", "x", "k", "L", "U", "U_t", "U_xx"])

Symbolic variables already cleared

1.2. Classifying PDEs#

def classification(a: float, b: float, c: float):
    discriminant = b**2 - 4 * a * c
    if discriminant < 0:
        print("Elliptic PDE")
    elif discriminant == 0:
        print("Parabolic PDE")
    else:
        print("Hyperbolic PDE")

1.2.1. Example $2$ #

For each of the PDEs given below, classify them as either an elliptic, parabolic or hyperbolic PDE.

x, y = symbols("x, y")
U = symbols("U", cls=Function)(x, y)

laplace_equation = Eq(D(U, x, 2) + D(U, y, 2), 0)

laplace_equation

\frac{\partial^{2}}{\partial x^{2}} U (x, y) + \frac{\partial^{2}}{\partial y^{2}} U (x, y) = 0

classification(a=1, b=0, c=1)

Elliptic PDE

t, k = symbols("t, k")
U = symbols("U", cls=Function)(t, x)

heat_equation = Eq(D(U, t), k * D(U, x, 2))

heat_equation

\frac{\partial}{\partial t} U (t, x) = k \frac{\partial^{2}}{\partial x^{2}} U (t, x)

classification(a=-k, b=0, c=0)

Parabolic PDE

c = symbols("c", positive=True)
U = symbols("U", cls=Function)(t, x)

wave_equation = Eq(D(U, t, 2), c**2 * D(U, x, 2))

wave_equation

\frac{\partial^{2}}{\partial t^{2}} U (t, x) = c^{2} \frac{\partial^{2}}{\partial x^{2}} U (t, x)

classification(a=-(c**2), b=0, c=1)

Hyperbolic PDE

cleaner(["t", "x", "y", "k", "c", "U"])

Symbolic variables already cleared

1.2.2. Elliptic PDEs#

x, y = symbols("x, y")
f = symbols("f", cls=Function)(x, y)
U = symbols("U", cls=Function)(x, y)

Eq(D(U, x, 2) + D(U, y, 2), f)

\frac{\partial^{2}}{\partial x^{2}} U (x, y) + \frac{\partial^{2}}{\partial y^{2}} U (x, y) = f (x, y)

1.2.3. Parabolic PDEs#

t, alpha = symbols("t, alpha")

Eq(D(U, t), alpha * D(U, x, 2))

\frac{\partial}{\partial t} U (x, y) = α \frac{\partial^{2}}{\partial x^{2}} U (x, y)

1.2.4. Hyperbolic PDEs#

c = symbols("c")

Eq(D(U, t, 2), c**2 * D(U, x, 2))

\frac{\partial^{2}}{\partial t^{2}} U (x, y) = c^{2} \frac{\partial^{2}}{\partial x^{2}} U (x, y)

v = symbols("v", cls=Function)(x, y)
U = symbols("U", cls=Function)(t, x)

Eq(D(U, t) + v * D(U, x), 0)

v (x, y) \frac{\partial}{\partial x} U (t, x) + \frac{\partial}{\partial t} U (t, x) = 0

Partial Differential Equations

Contents

1. Partial Differential Equations#

1.1. Definitions#

1.1.1. Partial derivatives#

1.1.2. Partial differential equations#

1.1.3. Order of a PDE#

1.1.4. Linear PDEs#

1.1.5. Domain#

1.1.6. Initial conditions#

1.1.7. Boundary conditions#

1.1.8. Solution of a PDE#

1.1.9. Example $1$ #

1.1.9.1. Analytical solution#

1.1.9.2. Initial condition#

1.1.9.3. Dirichlet Boundary condition#

1.1.9.4. Checking solution#

1.2. Classifying PDEs#

1.2.1. Example $2$ #

1.2.2. Elliptic PDEs#

1.2.3. Parabolic PDEs#

1.2.4. Hyperbolic PDEs#

1.3. Domain of dependence#

1.4. Using PDEs to model real world phenomena#

1.4.1. Eulerian and Lagrangian methods#

1.4.2. Exact solution versus numerical approximation#

Partial Differential Equations

Contents

1. Partial Differential Equations#

1.1. Definitions#

1.1.1. Partial derivatives#

1.1.2. Partial differential equations#

1.1.3. Order of a PDE#

1.1.4. Linear PDEs#

1.1.5. Domain#

1.1.6. Initial conditions#

1.1.7. Boundary conditions#

1.1.8. Solution of a PDE#

1.1.9. Example 1#

1.1.9.1. Analytical solution#

1.1.9.2. Initial condition#

1.1.9.3. Dirichlet Boundary condition#

1.1.9.4. Checking solution#

1.2. Classifying PDEs#

1.2.1. Example 2#

1.2.2. Elliptic PDEs#

1.2.3. Parabolic PDEs#

1.2.4. Hyperbolic PDEs#

1.3. Domain of dependence#

1.4. Using PDEs to model real world phenomena#

1.4.1. Eulerian and Lagrangian methods#

1.4.2. Exact solution versus numerical approximation#

1.1.9. Example $1$ #

1.2.1. Example $2$ #