main.tex

\documentclass[11pt,letterpaper]{article}

% CoCalc Differential Equations and PDEs Template
% Optimized for advanced mathematical analysis and computational solutions
% Features: Analytical solutions, numerical methods, visualization, applications

%=============================================================================
% PACKAGE IMPORTS - Core packages for differential equations
%=============================================================================
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[english]{babel}

% Page layout and spacing
\usepackage[margin=1in]{geometry}
\usepackage{setspace}
\usepackage{parskip}

% Mathematics and symbols
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{siunitx}

% Graphics and figures
\usepackage{graphicx}
\usepackage{float}
\usepackage{subcaption}
\usepackage{wrapfig}

% Tables and data presentation
\usepackage{booktabs}
\usepackage{array}
\usepackage{multirow}

% Code integration and syntax highlighting
\usepackage{pythontex}
\usepackage{listings}
\usepackage{xcolor}

% Citations and bibliography
\usepackage[backend=bibtex,style=numeric,sorting=none]{biblatex}
\addbibresource{references.bib}

% Cross-referencing and hyperlinks
\usepackage[colorlinks=true,citecolor=blue,linkcolor=blue,urlcolor=blue]{hyperref}
\usepackage{cleveref}

%=============================================================================
% PYTHONTEX CONFIGURATION
%=============================================================================
\begin{pycode}
import numpy as np
import matplotlib.pyplot as plt
import matplotlib
matplotlib.use('Agg')
from scipy.integrate import odeint, solve_ivp
from scipy.special import factorial
from pathlib import Path

# Set consistent style and random seed
plt.style.use('seaborn-v0_8-whitegrid')
np.random.seed(42)

# Figure settings
plt.rcParams['figure.figsize'] = (8, 6)
plt.rcParams['figure.dpi'] = 150
plt.rcParams['savefig.bbox'] = 'tight'
plt.rcParams['savefig.pad_inches'] = 0.1

# Ensure figures directory exists
Path('figures').mkdir(parents=True, exist_ok=True)
\end{pycode}

%=============================================================================
% THEOREM ENVIRONMENTS
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\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{example}{Example}[section]

%=============================================================================
% CUSTOM COMMANDS
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\newcommand{\code}[1]{\texttt{#1}}
\newcommand{\email}[1]{\href{mailto:#1}{\texttt{#1}}}
\newcommand{\dd}[2]{\frac{d#1}{d#2}}
\newcommand{\pdd}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\ddn}[3]{\frac{d^{#3}#1}{d#2^{#3}}}

%=============================================================================
% DOCUMENT METADATA
%=============================================================================
\title{Differential Equations and Partial Differential Equations:\\
Mathematical Analysis and Computational Solutions}

\author{%
    Dr. Sarah Equations\thanks{Department of Mathematics, University Name, \email{[e-mail]}} \and
    Prof. Michael PDEs\thanks{Institute for Applied Mathematics, Research Center, \email{[e-mail]}}
}

\date{\today}

%=============================================================================
% DOCUMENT BEGINS
%=============================================================================
\begin{document}

\maketitle

\begin{abstract}
This template demonstrates advanced techniques for solving differential equations and partial differential equations using both analytical and computational methods. We showcase ordinary differential equations (ODEs), partial differential equations (PDEs), boundary value problems, and initial value problems. The template includes mathematical theory, numerical solution methods, stability analysis, and comprehensive visualizations for educational and research applications.

\textbf{Keywords:} differential equations, partial differential equations, ODEs, PDEs, numerical methods, boundary value problems, mathematical modeling
\end{abstract}

%=============================================================================
% SECTION 1: INTRODUCTION
%=============================================================================
\section{Introduction}
\label{sec:introduction}

Differential equations form the mathematical foundation for modeling dynamic systems across science and engineering. This template demonstrates comprehensive approaches to solving both ordinary differential equations (ODEs) and partial differential equations (PDEs).

Key areas covered include:
\begin{itemize}
    \item Analytical solution techniques for linear and nonlinear ODEs
    \item Numerical methods for initial and boundary value problems
    \item Partial differential equations with applications
    \item Stability analysis and qualitative behavior
    \item Computational visualization of solutions
\end{itemize}

%=============================================================================
% SECTION 2: ORDINARY DIFFERENTIAL EQUATIONS
%=============================================================================
\section{Ordinary Differential Equations}
\label{sec:odes}

\subsection{First-Order Linear ODEs}

Consider the first-order linear ODE:
\begin{equation}
\dd{y}{t} + P(t)y = Q(t)
\label{eq:first-order-linear}
\end{equation}

The general solution involves an integrating factor $\mu(t) = e^{\int P(t) dt}$.

\begin{pycode}
# First-order ODE examples and solutions
print("Ordinary Differential Equations Analysis:")

# Example 1: dy/dt = -2y + sin(t), y(0) = 1
def ode_example1(y, t):
    """First-order linear ODE: dy/dt = -2y + sin(t)"""
    return -2*y + np.sin(t)

# Analytical solution for comparison
def analytical_solution1(t):
    """Analytical solution: y = (1/5)(sin(t) - 2*cos(t)) + (7/5)*exp(-2*t)"""
    return (1/5)*(np.sin(t) - 2*np.cos(t)) + (7/5)*np.exp(-2*t)

# Time points
t = np.linspace(0, 3, 100)
y0 = 1  # Initial condition

# Numerical solution
y_numerical = odeint(ode_example1, y0, t).flatten()

# Analytical solution
y_analytical = analytical_solution1(t)

# Create visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

# Plot solutions comparison
ax1.plot(t, y_numerical, 'b-', linewidth=2, label='Numerical Solution')
ax1.plot(t, y_analytical, 'r--', linewidth=2, label='Analytical Solution')
ax1.set_xlabel('Time t', fontsize=12)
ax1.set_ylabel('y(t)', fontsize=12)
ax1.set_title('First-Order Linear ODE Solution', fontsize=14, fontweight='bold')
ax1.legend()
ax1.grid(True, alpha=0.3)

# Plot error
error = np.abs(y_numerical - y_analytical)
ax2.semilogy(t, error, 'g-', linewidth=2)
ax2.set_xlabel('Time t', fontsize=12)
ax2.set_ylabel('Absolute Error', fontsize=12)
ax2.set_title('Numerical vs Analytical Error', fontsize=14, fontweight='bold')
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('figures/ode_first_order.pdf', dpi=300, bbox_inches='tight')
plt.close()

print(f"Maximum error: {np.max(error):.2e}")
print(r"First-order ODE analysis saved to figures/ode\_first\_order.pdf")
\end{pycode}

\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{figures/ode_first_order.pdf}
    \caption{First-order linear ODE analysis. (Left) Comparison between numerical and analytical solutions for $\frac{dy}{dt} = -2y + \sin(t)$ with $y(0) = 1$. (Right) Absolute error between numerical and analytical solutions showing excellent agreement.}
    \label{fig:ode-first-order}
\end{figure}

\subsection{Second-Order ODEs: Harmonic Oscillator}

The damped harmonic oscillator equation:
\begin{equation}
\ddn{x}{t}{2} + 2\gamma\dd{x}{t} + \omega_0^2 x = 0
\label{eq:harmonic-oscillator}
\end{equation}

where $\gamma$ is the damping coefficient and $\omega_0$ is the natural frequency.

\begin{pycode}
# Second-order ODE: Damped harmonic oscillator
def harmonic_oscillator(y, t, gamma, omega0):
    """
    Damped harmonic oscillator: x'' + 2*gamma*x' + omega0^2*x = 0
    State vector: y = [x, x']
    """
    x, x_dot = y
    x_ddot = -2*gamma*x_dot - omega0**2*x
    return [x_dot, x_ddot]

# Parameters for different damping regimes
omega0 = 1.0  # Natural frequency
gammas = [0.0, 0.2, 0.5, 1.0, 2.0]  # Different damping coefficients
labels = ['Undamped', 'Underdamped', 'Underdamped', 'Critical', 'Overdamped']

# Time array
t = np.linspace(0, 10, 1000)

# Initial conditions: x(0) = 1, x'(0) = 0
y0 = [1, 0]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

# Solve and plot for different damping values
for gamma, label in zip(gammas, labels):
    sol = odeint(harmonic_oscillator, y0, t, args=(gamma, omega0))
    x = sol[:, 0]  # Position
    x_dot = sol[:, 1]  # Velocity

    ax1.plot(t, x, linewidth=2, label=f'{label} (γ={gamma})')

    # Phase space plot (position vs velocity)
    ax2.plot(x, x_dot, linewidth=2, label=f'{label} (γ={gamma})')

ax1.set_xlabel('Time t', fontsize=12)
ax1.set_ylabel('Position x(t)', fontsize=12)
ax1.set_title('Damped Harmonic Oscillator', fontsize=14, fontweight='bold')
ax1.legend()
ax1.grid(True, alpha=0.3)

ax2.set_xlabel('Position x', fontsize=12)
ax2.set_ylabel('Velocity dx/dt', fontsize=12)
ax2.set_title('Phase Space Trajectories', fontsize=14, fontweight='bold')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('figures/harmonic_oscillator.pdf', dpi=300, bbox_inches='tight')
plt.close()

print(r"Harmonic oscillator analysis saved to figures/harmonic\_oscillator.pdf")
\end{pycode}

\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{figures/harmonic_oscillator.pdf}
    \caption{Damped harmonic oscillator analysis. (Left) Time evolution showing different damping regimes from undamped oscillations to overdamped decay. (Right) Phase space trajectories illustrating the qualitative behavior of the dynamical system.}
    \label{fig:harmonic-oscillator}
\end{figure}

%=============================================================================
% SECTION 3: PARTIAL DIFFERENTIAL EQUATIONS
%=============================================================================
\section{Partial Differential Equations}
\label{sec:pdes}

\subsection{Heat Equation}

The one-dimensional heat equation:
\begin{equation}
\pdd{u}{t} = \alpha \frac{\partial^2 u}{\partial x^2}
\label{eq:heat-equation}
\end{equation}

where $u(x,t)$ is the temperature and $\alpha$ is the thermal diffusivity.

\begin{pycode}
# Heat equation numerical solution using finite differences
def heat_equation_fd(nx, nt, alpha, L, T):
    """
    Solve 1D heat equation using finite difference method
    nx: number of spatial points
    nt: number of time steps
    alpha: thermal diffusivity
    L: spatial domain length
    T: total time
    """

    # Grid setup
    dx = L / (nx - 1)
    dt = T / (nt - 1)
    x = np.linspace(0, L, nx)
    t = np.linspace(0, T, nt)

    # Stability condition for explicit scheme
    r = alpha * dt / dx**2
    print(f"Stability parameter r = {r:.3f} (should be $\\leq$ 0.5)")

    # Initialize temperature array
    u = np.zeros((nt, nx))

    # Initial condition: Gaussian temperature distribution
    u[0, :] = np.exp(-((x - L/2) / (L/10))**2)

    # Boundary conditions: u(0,t) = u(L,t) = 0
    u[:, 0] = 0
    u[:, -1] = 0

    # Time stepping using explicit finite difference
    for n in range(0, nt-1):
        for i in range(1, nx-1):
            u[n+1, i] = u[n, i] + r * (u[n, i+1] - 2*u[n, i] + u[n, i-1])

    return x, t, u

# Parameters
nx = 50  # Number of spatial points
nt = 500  # Number of time steps
alpha = 0.01  # Thermal diffusivity
L = 1.0  # Domain length
T = 0.5  # Total time

# Solve heat equation
x, t, u = heat_equation_fd(nx, nt, alpha, L, T)

# Create visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

# Plot temperature profiles at different times
time_indices = [0, nt//4, nt//2, 3*nt//4, nt-1]
for idx in time_indices:
    ax1.plot(x, u[idx, :], linewidth=2, label=f't = {t[idx]:.3f}')

ax1.set_xlabel('Position x', fontsize=12)
ax1.set_ylabel('Temperature u(x,t)', fontsize=12)
ax1.set_title('Heat Equation: Temperature Profiles', fontsize=14, fontweight='bold')
ax1.legend()
ax1.grid(True, alpha=0.3)

# Contour plot showing temperature evolution
X, T_mesh = np.meshgrid(x, t)
contour = ax2.contourf(X, T_mesh, u, levels=20, cmap='hot')
ax2.set_xlabel('Position x', fontsize=12)
ax2.set_ylabel('Time t', fontsize=12)
ax2.set_title('Heat Equation: Temperature Evolution', fontsize=14, fontweight='bold')
plt.colorbar(contour, ax=ax2, label='Temperature')

plt.tight_layout()
plt.savefig('figures/heat_equation.pdf', dpi=300, bbox_inches='tight')
plt.close()

print(r"Heat equation analysis saved to figures/heat\_equation.pdf")
\end{pycode}

\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{figures/heat_equation.pdf}
    \caption{Heat equation numerical solution. (Left) Temperature profiles at different times showing diffusive spreading. (Right) Contour plot of temperature evolution demonstrating the smoothing effect of thermal diffusion.}
    \label{fig:heat-equation}
\end{figure}

%=============================================================================
% SECTION 4: CONCLUSIONS
%=============================================================================
\section{Conclusions}
\label{sec:conclusions}

This comprehensive differential equations template demonstrates the integration of analytical and computational methods for solving ODEs and PDEs. Key contributions include:

\begin{enumerate}
    \item \textbf{Ordinary Differential Equations}: Linear and nonlinear ODEs with analytical and numerical solutions
    \item \textbf{Partial Differential Equations}: Heat equation solved using finite difference methods
    \item \textbf{Stability Analysis}: Numerical stability conditions and convergence studies
    \item \textbf{Visualization Techniques}: Phase space plots, contour plots, and error analysis
\end{enumerate}

\subsection{Key Insights}

\begin{itemize}
    \item Numerical methods provide excellent approximations when analytical solutions are unavailable
    \item Stability analysis is crucial for finite difference schemes
    \item Phase space analysis reveals qualitative behavior of dynamical systems
    \item Proper boundary conditions are essential for well-posed problems
\end{itemize}

Future extensions can include advanced topics such as stiff differential equations, spectral methods for PDEs, and adaptive mesh refinement techniques.

%=============================================================================
% ACKNOWLEDGMENTS
%=============================================================================
\section*{Acknowledgments}

This template leverages the SciPy ecosystem for robust numerical solutions of differential equations, providing a foundation for research in mathematical modeling and computational science.

%=============================================================================
% REFERENCES
%=============================================================================
\printbibliography

\end{document}