PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Applied Mathematics ↓


Partial Differential Equations


Introduction

Partial Differential Equations (PDEs) play a vital role in mathematical modeling of various phenomena in applied sciences. These equations are used to describe the dynamics of systems involving functions of multiple variables such as time and space. In simple terms, PDEs are equations that involve rates of change with respect to continuous variables.

Basic concepts

Generally, PDE is expressed as:

F(x_1, x_2, ..., x_n, u, u_{x1}, u_{x2}, ..., u_{xn}, u_{x1x1}, ..., u_{xn...xn}) = 0

Here, u_{xi} denotes the partial derivative of u with respect to x_i. To understand, let us first understand some basic concepts:

The order of the P.D.E.

The order of a PDE is determined by the highest order derivative present in the equation. For example, the equation:

u_{tt} - c^2 u_{xx} = 0

is a second-order PDE because the highest derivatives, u_{tt} and u_{xx}, are second-order derivatives.

Linear and nonlinear PDEs

A PDE is linear if it can be expressed as a linear combination of the unknown function and its derivatives; it is nonlinear if the PDE contains nonlinear terms of the function or its derivatives. For example,

u_t + u u_x = 0

is a nonlinear PDE because it contains a nonlinear term u u_x.

Common types of PDEs

There are several standard PDE types used in modeling physical phenomena. The most common types are:

  • Heat equation
  • Wave equation
  • Laplace equation
  • Poisson's equation

Visual example

Let us give an example of a simple 2D heat distribution modeled by a PDE. Consider a metal plate where the heat distribution u(x, y, t) changes with time due to heat conduction. The corresponding PDE is the 2D heat equation:

u_t = alpha (u_{xx} + u_{yy})

The solution of this equation gives a function u(x, y, t), which represents the temperature at time t and position (x, y).

Metal plate heat distribution u(x, y, t)

Examples of P.D.E.

Heat equation

The heat equation is a parabolic PDE that describes the distribution of heat over a given region over time.

u_t = alpha u_{xx}

This equation represents the rate of change of temperature distribution in a rod. Here, alpha is the thermal diffusion constant.

Example problem

A rod has an initial temperature distribution u(x, 0) = sin(pi x) and boundary conditions u(0, t) = u(1, t) = 0, solve the heat equation for t > 0.

The solution involves variable separation method, Fourier series expansion, and applying initial/boundary conditions to determine the coefficients and find the general solution.

Wave equation

The wave equation is a second-order linear PDE that describes wave motion such as sound or light waves.

u_{tt} = c^2 u_{xx}

Here, c is the speed of the wave. This PDE models the vibrations in a stretched string.

Laplace and Poisson's equations

Laplace's and Poisson's equations are elliptic PDEs that are important in engineering and physics.

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

These equations arise in potential theory (e.g., electric potential) and describe steady state processes.

Solution to P.D.E.

Solving PDEs can be very challenging due to their complexity and the nature of real-world boundary conditions. Various methods exist for solving PDEs:

Analytical methods

These methods provide exact solutions and include the following techniques:

  • Characterization method
  • Separation of variables
  • Transformation methods (e.g., Fourier and Laplace transforms)

Numerical methods

Practical problems in PDEs often require numerical approaches where analytical solutions are impossible. Major numerical methods include:

  • Finite Difference Method
  • Finite element method
  • Finite volume method

Example: Numerical solution of the heat equation

Consider using the finite difference method to numerically solve the heat equation:

Divide time and space into grid points and estimate the derivatives using finite differences. Apply the recurrence relation for temperature at each grid point over time.
// Python-like pseudocode for numerical solution
for n in range(time_steps):
    for i in range(1, space_steps - 1):
        u[i][n+1] = u[i][n] + alpha * (u[i+1][n] - 2*u[i][n] + u[i-1][n])

Application

PDEs are important in a wide variety of fields due to their ability to model complex systems. Some application domains include:

  • Physics: quantum mechanics (Schrödinger equation), electromagnetism (Maxwell's equations)
  • Engineering: structural analysis, thermal management
  • Biology: population dynamics, chemical reaction processes
  • Finance: Black-Scholes Model for Option Pricing

Conclusion

Partial differential equations are an important part of applied mathematics and are indispensable for modeling and solving problems in various scientific fields. Understanding both the theoretical and practical perspectives of PDEs enables mathematicians and engineers to effectively tackle complex real-world challenges.


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