Métodos analíticos de resolución de la Ecuación de Onda (1D, 2D y 3D) 🌊

Sesión 1 (05.12.2022)

Se resuelve el problema de valor inicial de la ecuación de la onda 1D

$$\begin{array}{rl}{u}_{tt}& =c^2{u}_{xx},-\infty <x<\infty ,t>0,\\ u\left(x,0\right)& =\phi \left(x\right),-\infty <x<\infty ,\\ u_t\left(x,0\right)& =\psi \left(x\right),-\infty <x<\infty .\end{array}$$

y se deduce la fórmula de D'Alembert

$$u\left(x,t\right)=\frac{1}{2}\left[\phi \left(x+ct\right)+\phi \left(x-ct\right)\right]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}\psi (s)\mathrm{d}s.$$

Grabación

• Diapositiva

Los programas los puede encontrar en modo de snippets expandiendo aquí.

MATLAB / Octave

#!/usr/bin/env -S octave -qf

% mifi.m
function y = mifi(x)

    if (-1 <= x) && (x <= 1)
        y = 2 -2 * abs(x);
    else
        y = 0;
    end

end

#!/usr/bin/env -S octave -qf

% SolDAlambert.m
clc, close all

x = -5.0:0.01:5;
u = zeros(1, 1001);
conta = 1;
figure(1)

for t = 0:0.5:3
    x1 = x + t;
    x2 = x - t;

    for i = 1:1001
        u(i) = 0.5 * (mifi(x1(i)) + mifi(x2(i)));
    end

    subplot(7, 1, conta)
    plot(x, u), grid on
    axis([-5, 5, -1, 3])
    conta = conta + 1;
end

%% animación
clear x t u
x = -5.0:0.01:5; u = zeros(1, length(x));
figure(2)

for t = 0:0.1:3
    clf
    x1 = x + t;
    x2 = x - t;

    for i = 1:1001
        u(i) = 0.5 * (mifi(x1(i)) + mifi(x2(i)));
    end

    plot(x, u), hold on
    plot(x, zeros(1, length(x)), 'k')% añade un eje central a cada ploteo
    axis([-5, 5, -1, 3]), hold off
    pause(0.4)
end

Python

Tomado de cpp-review-dune/python.

#!/usr/bin/env python
# -*- coding: utf-8 -*-

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation


class Wave1D:
    def __init__(self) -> None:
        self.X = np.linspace(start=-5, stop=5, num=1001)
        self.T = np.linspace(start=0, stop=3, num=7)
        varphi = lambda X: np.piecewise(
            x=X,
            condlist=[abs(X) <= 1, abs(X) >= 1],
            funclist=[lambda t: 2 - 2 * abs(t), 0],
        )
        self.u = lambda X, t: 0.5 * (varphi(X + t) + varphi(X - t))
        self.fig, self.ax = plt.subplots()
        self.yy = [self.u(t, self.X) for t in self.T]

    def make_plot(self):
        fig, axs = plt.subplots(
            nrows=self.T.size,
            ncols=1,
            clear=True,
            constrained_layout=True,
        )
        for ax, t in zip(axs, self.T):
            ax.plot(self.X, self.u(self.X, t), lw=1)

        plt.savefig("wave.png", dpi=300)
        plt.close()

    def update(self, t):
        self.ax.clear()
        self.ax.plot(self.X, self.yy[t])
        self.ax.set_xlim((self.X[0], self.X[-1]))
        self.ax.set_ylim((np.min(self.yy), np.max(self.yy)))
        self.ax.set_title(f"t = {self.T[t]:.2f}")
        self.ax.grid(True)

    def make_animation(self):
        anim = FuncAnimation(
            fig=self.fig, func=self.update, frames=self.T.size, interval=4000
        )
        anim.save(filename="wave1d.mp4", writer="ffmpeg", fps=60, dpi=300)


if __name__ == "__main__":
    Wave1D().make_plot()
    Wave1D().make_animation()

Sesión 2 (12.12.2022)

Se define el dominio de dependencia e intervalo de dependencia.

Se exhibe que la energía total de un sistema se conserva, es decir, $\forall t\ge 0:K\left(t\right)=-P\left(t\right)$, donde

$$K\left(t\right)=\frac{1}{2}{\int }_{-\infty }^{\infty }\rho u_t^2\mathrm{d}x,P\left(t\right)=\frac{T}{2}{\int }_{-\infty }^{\infty }{u}_x^2\mathrm{d}x.$$

El problema de valor inicial de la ecuación de la onda en el caso no homogéneo es

$$\{\begin{array}{ll}{u}_{tt}-c^2{u}_{xx}=f\left(x,t\right),& x\in \mathbb{R},t>0\\ u\left(x,0\right)=\phi \left(x\right),& x\in \mathbb{R},\\ u_t\left(x,0\right)=\psi \left(x\right),& x\in \mathbb{R}.\end{array}$$

Donde $c^2=\frac{T}{\rho }$ y su solución es

$$u\left(x,t\right)=\frac{1}{2}\left[\phi \left(x+ct\right)+\phi \left(x-ct\right)\right]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}\psi (s)\mathrm{d}s+\frac{1}{2c}{\iint }_{D}f\left(y,\tau \right)\mathrm{d}y\mathrm{d}\tau .$$

Se estudia el principio de Duhamel que establece

$$u\left(x,t\right)={\int }_0^{t}w\left(x,t;\tau \right)\mathrm{d}\tau =\frac{1}{2c}{\int }_0^t{\int }_{x-c\left(t-\tau \right)}^{x+c\left(t-\tau \right)}f\left(y,\tau \right)\mathrm{d}y\mathrm{d}\tau .$$

Grabación

• Diapositiva

Sesión 3 (16.12.2022)

Grabación

• Diapositiva 1
• Diapositiva 2
• Diapositiva 3

Referencias

• Partial Differential Equations: Second Edition by Lawrence C. Evans
• Advanced Modern Engineering Mathematics by Glyn James
• Advanced Engineering Mathematics by Dennis Zill
• Advanced Engineering Mathematics by Peter V. O'Neil
• Advanced Engineering Mathematics by Erwin Kreyszig
• Partial Differential Equations: A First Course by Rustum Choksi
• Partial Differential Equations: An Accessible Route through Theory and Applications by András Vasy
• Introduction to Partial Differential Equations with MATLAB
• An Introduction to Partial Differential Equations with MATLAB by Matthew P. Coleman. Páginas 54-58, 62-65, 138-146, 155, 157, 159-160, 195-201, 250, 378-380, 390-397, 414-421, 439-455, 556, 588, 593, 596-597.
• Lecture notes based on L.C.Evans’s textbook, by Baisheng Yan. Páginas 65-82.
• Math 124A/215A by Viktor Grigoryan. Páginas 1-7, 1-4, 1-4, 1-4, 1-4, 1-4, 1-4, 1-4.
• Vector Analysis and an introduction to tensor analysis by Murray R. Spiegel
• A first course in partial differential equations by Russell L. Herman
• Applied Partial Differential Equations by J. David Logan
• Fundamentals of Differential Equations by R. Kent Nagle
• Partial Differential Equations in Action by Sandro Salsa
• Partial Differential Equations: An Introduction by Walter A. Strauss
• Numerical Models for Differential Problems by Alfio Quarteroni
• Fourier Analysis: An introduction by Elias M. Stein and Rami Shakarchi
• A first course in Partial Differential Equations with complex variables and transform methods by H. F. Weinberger

Vídeos acerca de la interpretación física de las leyes de conservación de la energía y del movimiento ondulatorio.