7.1 Classification of Partial Differential Equations (PDEs)
9 min read•august 14, 2024
Partial Differential Equations (PDEs) are equations involving functions of multiple variables and their partial derivatives. They're crucial for modeling complex physical phenomena in science and engineering. This section focuses on classifying PDEs based on their , , and coefficients.
Understanding PDE classification is essential for choosing appropriate solution methods. We'll explore , , and PDEs, which model steady-state, diffusion, and wave propagation problems respectively. This knowledge forms the foundation for solving real-world problems using PDEs.
Classifying PDEs
Order, Linearity, and Coefficients
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The order of a PDE is determined by the highest order partial derivative present in the equation
First-order PDEs contain only first-order partial derivatives (u_x, u_y, u_t)
The classification of a PDE as linear or nonlinear, and the nature of its coefficients, determine the appropriate solution techniques and the complexity of the problem
Linear PDEs with constant coefficients are generally easier to solve than nonlinear PDEs or PDEs with variable coefficients
Techniques such as , Fourier series, and Laplace transforms are more applicable to linear PDEs with constant coefficients
Canonical Forms and Characteristics
The canonical form of a second-order linear PDE is Auxx+2Buxy+Cuyy+Dux+Euy+Fu=G, where A, B, C, D, E, F, and G are functions of x and y
The coefficients A, B, and C determine the classification of the PDE as elliptic, parabolic, or hyperbolic
The discriminant Δ=B2−AC is used to classify the PDE
The characteristics of a PDE are curves or surfaces along which the PDE reduces to an ordinary differential equation (ODE)
Characteristics are important in the study of hyperbolic PDEs and the
The characteristic curves of a first-order PDE a(x,y)ux+b(x,y)uy=c(x,y) are given by dy/dx=b(x,y)/a(x,y)
Elliptic, Parabolic, and Hyperbolic PDEs
Elliptic PDEs
Elliptic PDEs have the general form Auxx+2Buxy+Cuyy+lower order terms=0, where B2−AC<0
They model steady-state or equilibrium problems, such as potential theory and elasticity
Examples include Laplace's equation (uxx+uyy=0) and Poisson's equation (uxx+uyy=f(x,y))
Elliptic PDEs require on a closed domain
Dirichlet boundary conditions specify the value of the dependent variable on the boundary
Neumann boundary conditions specify the normal derivative of the dependent variable on the boundary
Elliptic PDEs do not involve time derivatives and describe phenomena that have reached equilibrium or steady-state
Parabolic PDEs
Parabolic PDEs have the general form Auxx+2Buxy+Cuyy+lower order terms=ut, where B2−AC=0
They model time-dependent diffusion processes, such as heat conduction and mass transfer
The most common example is the (ut−α2uxx=0)
Parabolic PDEs require (specifying the dependent variable at t=0) and boundary conditions (specifying the dependent variable or its normal derivative on the spatial boundary)
Initial conditions describe the state of the system at the beginning of the diffusion process
Boundary conditions describe the interaction of the system with its surroundings
Parabolic PDEs involve first-order time derivatives and second-order spatial derivatives, describing the evolution of diffusive processes over time
Hyperbolic PDEs
Hyperbolic PDEs have the general form Auxx+2Buxy+Cuyy+lower order terms=utt, where B2−AC>0
They model wave propagation and vibration problems, such as acoustics and electromagnetic waves
The most common example is the (utt−c2uxx=0)
Hyperbolic PDEs require initial conditions (specifying the dependent variable and its time derivative at t=0) and boundary conditions (specifying the dependent variable or its normal derivative on the spatial boundary)
Initial conditions describe the initial displacement and velocity of the wave
Boundary conditions describe the interaction of the wave with its surroundings (reflection, absorption, or transmission)
Hyperbolic PDEs involve second-order time derivatives and second-order spatial derivatives, describing the propagation of waves through a medium
Physical Phenomena Modeled by PDEs
Elliptic PDEs: Steady-State and Equilibrium Problems
Electrostatics: Laplace's equation (∇2ϕ=0) and Poisson's equation (∇2ϕ=−ρ/ε0) describe the electric potential ϕ in a charge-free region and a region with charge density ρ, respectively
Laplace's equation models the electric potential in a region without charges
Poisson's equation models the electric potential in a region with a given charge distribution
Magnetostatics: Poisson's equation (∇2A=−μ0J) describes the magnetic vector potential A in a region with current density J
The magnetic field B is related to the vector potential by B=∇×A
Elasticity: Navier's equations (μ∇2u+(λ+μ)∇(∇⋅u)+f=0) describe the displacement field u in an elastic solid under body forces f, where λ and μ are Lamé constants
Navier's equations model the deformation of elastic materials under applied forces
The equations are derived from the balance of linear momentum and Hooke's law
Heat conduction: The heat equation (∂t∂T−α∇2T=Q) describes the temperature distribution T in a medium with thermal diffusivity α and heat source Q
The heat equation models the diffusion of heat in a material over time
The thermal diffusivity α is related to the thermal conductivity k, density ρ, and specific heat capacity cp by α=k/(ρcp)
Mass transfer: The diffusion equation (∂t∂c−D∇2c=R) describes the concentration c of a species in a medium with diffusion coefficient D and reaction rate R
The diffusion equation models the transport of mass due to concentration gradients
The reaction rate R accounts for the generation or consumption of the species due to chemical reactions
Groundwater flow: Darcy's law (q=−K∇h) and the continuity equation (∇⋅q=−S∂t∂h) combine to form the groundwater flow equation (S∂t∂h−∇⋅(K∇h)=0), which describes the hydraulic head h in an aquifer with hydraulic conductivity K and specific storage S
The groundwater flow equation models the movement of water in porous media, such as aquifers
The hydraulic head h is the sum of the pressure head and the elevation head, and it drives the flow of groundwater
Hyperbolic PDEs: Wave Propagation and Vibration Problems
Acoustics: The wave equation (∂t2∂2p−c2∇2p=0) describes the acoustic pressure p in a medium with sound speed c
The wave equation models the propagation of sound waves in fluids and gases
The sound speed c is related to the bulk modulus K and density ρ of the medium by c=K/ρ
Electromagnetic waves: Maxwell's equations (∇×E=−∂t∂B, ∇×H=J+∂t∂D, ∇⋅D=ρ, ∇⋅B=0) describe the electric field E, magnetic field B, electric displacement D, and magnetic field intensity H in a medium with current density J and charge density ρ
Maxwell's equations model the propagation of electromagnetic waves, such as light and radio waves
In a vacuum, the equations reduce to the wave equation for the electric and magnetic fields
Elastic waves: Lamé's equations (ρ∂t2∂2u=(λ+μ)∇(∇⋅u)+μ∇2u+f) describe the displacement field u in an elastic solid with density ρ, Lamé constants λ and μ, and body forces f
Lamé's equations model the propagation of elastic waves, such as seismic waves and ultrasonic waves, in solid materials
The equations account for both longitudinal (P) waves and transverse (S) waves, which propagate at different speeds depending on the elastic properties of the material
Solution Techniques for PDEs
Elliptic PDEs: Steady-State Solution Methods
Separation of variables: Assumes the solution can be written as a product of functions, each depending on only one variable
Applicable to linear PDEs with homogeneous boundary conditions
Leads to a set of ordinary differential equations (ODEs) that can be solved analytically or numerically
Eigenfunction expansions: Expresses the solution as an infinite series of eigenfunctions, which are determined by the boundary conditions
Applicable to linear PDEs with homogeneous boundary conditions
The eigenfunctions form a complete orthonormal basis for the
Green's functions: Represents the solution as an integral of the product of the Green's function and the source term or boundary conditions
Applicable to linear PDEs with inhomogeneous boundary conditions or source terms
The Green's function is the fundamental solution of the PDE and satisfies the homogeneous boundary conditions
Numerical methods: Discretize the domain and approximate the solution using techniques such as finite differences, finite elements, or spectral methods
Applicable to both linear and nonlinear PDEs, with various boundary conditions
The accuracy and efficiency of the solution depend on the choice of discretization scheme and the resolution of the mesh
Parabolic PDEs: Time-Dependent Solution Methods
Separation of variables: Assumes the solution can be written as a product of a spatial function and a temporal function
Applicable to linear PDEs with homogeneous boundary conditions and simple initial conditions
Leads to an eigenvalue problem for the spatial function and an ODE for the temporal function
Fourier series: Expresses the solution as an infinite series of sinusoidal functions, which are determined by the initial and boundary conditions
Applicable to linear PDEs with homogeneous boundary conditions and periodic or semi-infinite domains
The Fourier coefficients are determined by the initial condition using Fourier analysis
Laplace transforms: Converts the PDE into an ordinary differential equation (ODE) in the transform domain, which is then solved and inverted back to the time domain
Applicable to linear PDEs with constant coefficients and simple initial and boundary conditions
The Laplace transform simplifies the time derivative and reduces the PDE to an ODE
Numerical methods: Discretize the domain in both space and time, and approximate the solution using techniques such as finite differences, finite elements, or spectral methods
Applicable to both linear and nonlinear PDEs, with various initial and boundary conditions
The accuracy and stability of the solution depend on the choice of discretization scheme, the resolution of the mesh, and the time step size
Method of characteristics: Transforms the PDE into a system of ODEs along characteristic curves, which represent the paths of wave propagation
Applicable to first-order hyperbolic PDEs and systems of hyperbolic conservation laws
The characteristic curves are determined by the coefficients of the PDE and the initial conditions
D'Alembert's formula: Expresses the solution as a sum of two traveling waves, each depending on a characteristic variable
Applicable to the one-dimensional wave equation with homogeneous boundary conditions and simple initial conditions
The solution is determined by the initial displacement and velocity of the wave
Fourier series: Expresses the solution as an infinite series of sinusoidal functions, which are determined by the initial and boundary conditions
Applicable to linear PDEs with homogeneous boundary conditions and periodic or semi-infinite domains
The Fourier coefficients are determined by the initial conditions using Fourier analysis
Numerical methods: Discretize the domain in both space and time, and approximate the solution using techniques such as finite differences, finite volumes, or discontinuous Galerkin methods
Applicable to both linear and nonlinear PDEs, with various initial and boundary conditions
The accuracy and stability of the solution depend on the choice of discretization scheme, the resolution of the mesh, and the time step size, as well as the proper treatment of shocks and discontinuities