Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Differential Equations → Partial Differential Equations ↓


Classification of PDEs


Partial differential equations (PDEs) are important in describing a variety of phenomena such as heat, sound, fluid dynamics, elasticity, and quantum mechanics. Understanding PDEs helps us model and solve complex problems where changes depend on many variables.

This topic aims to provide in-depth insight into classifying PDEs based on certain characteristics and conditions. It is essential for graduate students and researchers who deal with mathematical models and computational simulations.

What is a partial differential equation?

A partial differential equation (PDE) is an equation that contains partial derivatives of a function of several independent variables. Unlike ordinary differential equations, which involve derivatives with respect to one variable, PDEs involve partial derivatives with respect to several variables.

    Example: Equations 
    ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0 
    is a PDE that describes a harmonic function in two dimensions x and y.

Types of PDE

PDEs can be classified according to their order, linearity, and the number of independent variables included.

1. Order of the PDE

The order of a PDE is determined by the order of the highest derivative present in the equation.

    First Order : ∂u/∂x + ∂u/∂y = 0
    Second Order : ∂^2u/∂x^2 + ∂^2u/∂y^2 + c = 0

2. Linearity in PDEs

PDEs are classified as linear or nonlinear:

  • Linear PDEs: The dependent variable and its derivatives appear linearly.
  • Nonlinear PDEs: the dependent variable and/or its derivatives appear nonlinearly (e.g., squares, products of functions).
    Linear example: 
    a(x, y) ∂u/∂x + b(x, y) ∂u/∂y + c(x, y) u = f(x, y)

    Nonlinear example: 
    
    u ∂u/∂x + ∂u/∂y = 0

Main categories of second-order PDEs

Second-order PDEs are widely studied as they appear in a variety of physical problems. They are mainly classified into three types - elliptic, parabolic, and hyperbolic. The classification depends on the discriminant of the equation.

Elliptic PDEs

In elliptic PDEs, the equations look like the Laplace equation. They usually describe steady-state problems (e.g., steady temperature distributions).

    general form:
    A ∂^2u/∂x^2 + B ∂^2u/∂x∂y + C ∂^2u/∂y^2 = F(x, y)
    
    Differential Position:
    
    B^2 - 4AC < 0

    Example: Laplace equation
    
    ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0

Parabolic PDE

Parabolic PDEs are associated with diffusion-type problems, such as heat conduction. The simplest form of a parabolic PDE looks like the heat equation.

    general form: 
    A ∂^2u/∂x^2 + B ∂^2u/∂x∂y + C ∂^2u/∂y^2 = F(x, y)
    
    Differential Position:
    
    B^2 - 4AC = 0

    Example: the heat equation
    
    ∂u/∂t = ∂^2u/∂x^2

Hyperbolic PDE

Hyperbolic PDEs generally describe wave phenomena. The classic wave equation is a fundamental example of this category.

    general form: 
    A ∂^2u/∂x^2 + B ∂^2u/∂x∂y + C ∂^2u/∂y^2 = F(x, y)
    
    Differential Position:
    
    B^2 - 4AC > 0

    Example: wave equation
    
    ∂^2u/∂t^2 = c^2 ∂^2u/∂x^2

Viewing the PDE classification

To better understand how these PDEs are classified, let's look at some of the equations in a systematic way. The illustrations may help to see the grid of classification in terms of their normal forms found above.

Classification of PDE elliptical parabolic hyperbolic

This SVG representation provides a simplified visual layout classifying PDEs into elliptic, parabolic, and hyperbolic types. It helps to remember their primary applications and distinctive features.

Boundary and initial conditions

In addition to their classification, solving PDEs requires the specification of boundary and initial conditions. These conditions help determine a specific solution to the PDE that models a particular physical situation.

1. Boundary conditions

  • Dirichlet boundary condition: Specifies the value of a function on a surface.
  • Neumann boundary condition: Specifies the value of the normal derivative for the surface.
  • Robin boundary condition: a combination of the Dirichlet and Neumann conditions.

2. Initial conditions

These are often needed for parabolic and hyperbolic PDEs, which specify the state of the system at the beginning of the observation.

As an example, consider the heat equation:

    Heat equation: ∂u/∂t = α ∂^2u/∂x^2

    Initial condition: u(x, 0) = g(x), since x is a spatial variable
    Boundary condition: u(0, t) = A, u(L, t) = B, where A, B are constant temperatures

Applications of PDE

Classifying PDEs into elliptic, parabolic, and hyperbolic helps to identify appropriate numerical methods for solving them. These equations are used in many areas:

  • Elliptic PDEs: used in steady state phenomena such as electric and gravitational potentials.
  • Parabolic PDE: Used for diffusion and heat conduction problems.
  • Hyperbolic PDE: Used for dynamic systems such as wave propagation and acoustics.

Conclusion

The classification of PDEs is a fundamental concept in the analysis and application of differential equations. Understanding the nature of PDEs helps researchers select appropriate analytical and numerical techniques for structural, thermodynamic, electromagnetic, and fluid mechanical analyses. Accurate knowledge of PDEs facilitates progress in scientific research and technological innovation.

From theoretical insights to practical implementations, the study of PDEs and their classification remains a cornerstone of advanced mathematics.


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Classification of PDEs

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