Partial Differential Equations (PDEs): Mathematical Foundations of Multivariable Dynamics

Table of Contents

1. Introduction
2. What Are Partial Differential Equations?
3. Order, Linearity, and Classification
4. First-Order PDEs
5. Method of Characteristics
6. Second-Order Linear PDEs
7. Classification: Elliptic, Parabolic, Hyperbolic
8. Canonical Forms and Transformation of Variables
9. The Heat Equation
10. The Wave Equation
11. Laplace’s and Poisson’s Equations
12. Boundary and Initial Conditions
13. Separation of Variables
14. Fourier Series Solutions
15. Green’s Functions and Integral Methods
16. Numerical Methods for PDEs
17. Applications in Physics and Engineering
18. Conclusion

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1. Introduction

Partial Differential Equations (PDEs) involve multivariable functions and their partial derivatives. They are the natural language of physics — describing phenomena from heat and sound propagation to electromagnetism and quantum mechanics.

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2. What Are Partial Differential Equations?

A PDE relates a function \( u(x_1, x_2, …, x_n) \) to its partial derivatives:

\[
F\left(x_i, u, \frac{\partial u}{\partial x_i}, \frac{\partial^2 u}{\partial x_i \partial x_j}, \dots \right) = 0
\]

They generalize ODEs to functions of several variables.

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3. Order, Linearity, and Classification

• Order: Highest derivative present
• Linear: Linear in \( u \) and all derivatives
• Quasilinear: Linear in highest derivatives only
• Nonlinear: Nonlinear in derivatives or the function

Example:

• Linear: \( u_{xx} + u_{yy} = 0 \)
• Nonlinear: \( u_t + u u_x = 0 \)

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4. First-Order PDEs

General form:

\[
a(x, y, u) \frac{\partial u}{\partial x} + b(x, y, u) \frac{\partial u}{\partial y} = c(x, y, u)
\]

Methods:

• Method of characteristics
• Change of variables

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5. Method of Characteristics

Converts PDE into a system of ODEs:

\[
\frac{dx}{a} = \frac{dy}{b} = \frac{du}{c}
\]

Solves for characteristic curves along which the PDE reduces to an ODE.

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6. Second-Order Linear PDEs

General form in two variables:

\[
A u_{xx} + B u_{xy} + C u_{yy} + D u_x + E u_y + Fu = G
\]

The type of PDE depends on the discriminant \( B^2 – 4AC \):

• Elliptic: \( B^2 – 4AC < 0 \)
• Parabolic: \( B^2 – 4AC = 0 \)
• Hyperbolic: \( B^2 – 4AC > 0 \)

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7. Classification: Elliptic, Parabolic, Hyperbolic

• Elliptic: steady-state (e.g., Laplace’s equation)
• Parabolic: diffusion-type (e.g., heat equation)
• Hyperbolic: wave-like (e.g., wave equation)

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8. Canonical Forms and Transformation of Variables

PDEs can often be simplified by change of variables:

• Elliptic → \( u_{\xi\xi} + u_{\eta\eta} = 0 \)
• Hyperbolic → \( u_{\xi\eta} = 0 \)

This reduces complexity and aids analytical solutions.

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9. The Heat Equation

Describes temperature evolution:

\[
\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}
\]

Parabolic; models diffusion of heat, particles, etc.

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10. The Wave Equation

Describes wave propagation:

\[
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}
\]

Hyperbolic; solution involves traveling waves and d’Alembert’s formula.

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11. Laplace’s and Poisson’s Equations

• Laplace: \( \nabla^2 u = 0 \)
• Poisson: \( \nabla^2 u = f(x, y) \)

Elliptic; describe electrostatics, fluid flow, and potential theory.

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12. Boundary and Initial Conditions

Well-posed problems require:

• Initial condition for time evolution
• Boundary conditions (Dirichlet, Neumann, or Robin) on spatial domain

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13. Separation of Variables

Assumes:

\[
u(x, t) = X(x)T(t)
\]

Reduces PDE to ODEs in separate variables, solvable via eigenfunction expansions.

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14. Fourier Series Solutions

Periodic boundary conditions allow expansion in sine and cosine:

\[
u(x, t) = \sum_{n=1}^\infty A_n(t) \sin\left(\frac{n\pi x}{L}\right)
\]

Used to solve heat and wave equations on bounded domains.

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15. Green’s Functions and Integral Methods

A Green’s function \( G(x, x’) \) solves:

\[
L G(x, x’) = \delta(x – x’)
\]

Allows us to write the solution as an integral:

\[
u(x) = \int G(x, x’) f(x’) dx’
\]

Powerful for inhomogeneous boundary value problems.

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16. Numerical Methods for PDEs

When analytical solutions are not possible:

• Finite difference methods
• Finite element methods
• Spectral methods

Used in simulations across physics and engineering.

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17. Applications in Physics and Engineering

• Electromagnetism: Maxwell’s equations
• Quantum mechanics: Schrödinger equation
• Fluid dynamics: Navier–Stokes equations
• Acoustics: Helmholtz equation
• Heat transfer: heat equation

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18. Conclusion

Partial Differential Equations are central to modeling and solving multidimensional, dynamic physical problems. Understanding their classification, solutions, and applications is critical for modern physics, engineering, and applied mathematics.

From electrostatics to quantum fields, PDEs provide the mathematical lens through which we study and simulate the world.

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