PDEs

Plot a surface

import matplotlib.pyplot as plt
import numpy as np


def f1(x, y):
    return x ** 2 - y ** 2


if __name__ == '__main__':
    x = np.linspace(-np.pi, np.pi, 100)
    y = np.linspace(-np.pi, np.pi, 100)
    X, Y = np.meshgrid(x, y)
    Z = f1(X, Y)
    fig = plt.figure()
    ax = fig.add_subplot(projection='3d')
    ax.plot_surface(X, Y, Z, cmap='viridis', edgecolor='none', linewidth=0.5)
    plt.xlabel('x')
    plt.ylabel('y')
    plt.title('z = f1(x, y)')
    plt.show()

Partial derivatives of first order

Given a function \(f: \mathbb{R}^n \rightarrow \mathbb{R}\), the partial derivative of \(f\) with respect to \( x_i\) is defined as

\[ \frac{\partial f}{\partial x_i} = \lim_{\Delta y \rightarrow 0} \frac{f(x_1, \ldots, x_i + \Delta y, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{\Delta y}. \]

The gradient of the function \(f\) is defined as

\[ \mathrm{grad}(f) = \nabla f = \left( \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right)^T \]

Partial derivatives of higher order

Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) be a function. If the mixed partial derivatives exist and are continuous at a point \(x_0\in\mathbb{R}^n\), then they are equal at \(x_0\) regardless of the order in which they are taken.

Laplacian

Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) be a function. The Laplacian of \(f\) is defined as

\[ \Delta f = \nabla^2 f = \sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2} = \left( \frac{\partial^2 f}{\partial x_1^2}+\ldots+\frac{\partial^2 f}{\partial x_n^2} \right) \]

Fourier series

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a periodic continuous function with angular frequency \(\omega_0\) and period \(T = 2\pi\div\omega_0\). The Fourier series of \(f\) is defined as

\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos(n\omega_0 x) + b_n \sin(n\omega_0 x) \right) \]

where

\[ \begin{matrix} a_0 = \frac{2}{T} \int_{(T)} f(x)\ dx \\ a_n = \frac{2}{T} \int_{(T)} f(x) \cos(n\omega_0 x)\ dx \\ b_n = \frac{2}{T} \int_{(T)} f(x) \sin(n\omega_0 x)\ dx \end{matrix} \]

(Integration over period T)

Partial Differential Equation

Linear, homogenous PDE of first order

If the coefficients \(a = a(x, y) \) and \(b = b(x, y) \) are continuously differentiable functions \(a: D \subset\mathbb{R}^2\rightarrow\mathbb{R} \) and \(b: D \subset\mathbb{R}^2\rightarrow\mathbb{R} \), then the PDE for \(u = u(x, y) \)

\[ a(x,y)\cdot u_x+b(x,y)\cdot u_y=0 \]

is a linear homogenous PDE of first order. Homogenous means that the right side of the PDE vanishes (is zero).

Superposition principle

If \(u_1 = u_1(x, y) \) and \(u_2 = u_2(x, y) \) are solutions of a linear, homogenous PDE, then \(u_1 + u_2 \) is also a solution.

Linear PDE of second order

notation: \( f_{xy} = \frac{\partial^2 f}{\partial x \partial y} \) is a mixed partial derivative

Consider the continuously differentiable functions \(a,b,c,d,e,f,g: D \subset\mathbb{R}^2\rightarrow\mathbb{R} \). The PDE for \(u = u(x, y) \)

\[ a(x,y)\cdot u_{xx}+2b(x,y)\cdot u_{xy}+c(x,y)\cdot u_{yy}=d(x,y)\cdot u_x+e(x,y)\cdot u_y+f(x,y)\cdot u+g(x,y) \]

is a linear PDE of second order (assuming \(u_{xy}=u_{yx} \).

SCC