The Central Idea
A partial differential equation, or PDE, relates a multivariable function to one or more of its partial derivatives.
If
\[ u=u(x,t), \]then
\[ u_x \]describes change with respect to \(x\), while
\[ u_t \]describes change with respect to \(t\).
A PDE combines these directions of change into one mathematical model.
Ordinary versus Partial Differential Equations
An ordinary differential equation uses derivatives with respect to one independent variable.
For example,
\[ y''+4y=0. \]A partial differential equation uses partial derivatives because the unknown function depends on several independent variables.
For example,
\[ u_t=\alpha u_{xx}. \]Here \(u\) depends on both position \(x\) and time \(t\).
Common PDE Notation
If
\[ u=u(x,y,t), \]then common partial-derivative notation includes
\[ u_x=\frac{\partial u}{\partial x}, \] \[ u_y=\frac{\partial u}{\partial y}, \] \[ u_t=\frac{\partial u}{\partial t}, \] \[ u_{xx}=\frac{\partial^2u}{\partial x^2}, \] \[ u_{xy}=\frac{\partial^2u}{\partial y\,\partial x}. \]The subscript notation is especially common in PDE work.
What Is the Unknown?
In a PDE, the unknown is usually a function of several variables.
Examples include:
- \(T(x,t)\): temperature at position \(x\) and time \(t\);
- \(u(x,t)\): displacement of a string;
- \(V(x,y)\): electric potential;
- \(C(x,t)\): concentration of a substance;
- \(p(x,y,z,t)\): pressure in a fluid.
Order of a PDE
The order of a PDE is determined by the highest-order partial derivative that appears.
For example,
\[ u_t+cu_x=0 \]is first order.
The equation
\[ u_t=\alpha u_{xx} \]is second order because \(u_{xx}\) appears.
Linear and Nonlinear PDEs
A PDE is linear when the unknown function and its derivatives appear only to the first power and are not multiplied together.
The equation
\[ u_t-\alpha u_{xx}=0 \]is linear.
The equation
\[ u_t+u\,u_x=0 \]is nonlinear because the unknown \(u\) multiplies its derivative \(u_x\).
Homogeneous and Nonhomogeneous PDEs
A linear PDE is homogeneous when the forcing term is zero.
For example,
\[ u_t-\alpha u_{xx}=0. \]A nonhomogeneous version might be
\[ u_t-\alpha u_{xx}=q(x,t), \]where \(q(x,t)\) represents an external source or input.
Why PDEs Need Conditions
A PDE usually has many possible solutions.
To select a physically meaningful solution, we impose:
- initial conditions;
- boundary conditions;
- sometimes both.
The equation and its conditions together form the complete model.
Initial Conditions for PDEs
If a PDE involves time, an initial condition often specifies the entire spatial profile at \(t=0\).
For example,
\[ \boxed{ u(x,0)=f(x) }. \]This is different from an ODE initial condition such as \(y(0)=y_0\), because the PDE condition specifies a whole function of space.
Boundary Conditions for PDEs
Boundary conditions describe behavior at the edge of the spatial domain.
On an interval
\[ 0\le x\le L, \]examples include
\[ u(0,t)=0, \qquad u(L,t)=0. \]These fix the solution values at both endpoints.
Dirichlet Conditions
A Dirichlet boundary condition specifies the value of the solution.
For example,
\[ \boxed{ u(0,t)=T_0 }. \]In a heat problem, this could represent a boundary held at a fixed temperature.
Neumann Conditions
A Neumann condition specifies a normal derivative or flux-related quantity.
In one dimension,
\[ \boxed{ u_x(0,t)=0 }. \]In a heat-conduction problem, this can represent an insulated boundary with zero heat flux.
Robin Conditions
A Robin condition combines the function and a derivative.
A one-dimensional example is
\[ \boxed{ au(0,t)+bu_x(0,t)=c }. \]This can model exchange between a system and its environment.
The Heat Equation
One of the central PDEs is the one-dimensional heat equation:
\[ \boxed{ u_t=\alpha u_{xx} }. \]Here:
- \(u(x,t)\) is temperature;
- \(x\) is position;
- \(t\) is time;
- \(\alpha>0\) is thermal diffusivity.
Interpreting the Heat Equation
The second spatial derivative
\[ u_{xx} \]measures local curvature of the temperature profile.
Where the profile is curved upward,
\[ u_{xx}>0, \]so
\[ u_t>0. \]Temperature tends to increase there.
Where the profile is curved downward, temperature tends to decrease.
Diffusion Smooths Profiles
The heat equation is a diffusion equation.
Diffusion tends to smooth differences.
High values spread toward lower values, reducing sharp spatial variation over time.
Similar mathematics describes diffusion of concentration and other transported quantities.
Example Heat Problem
A rod of length \(L\) may be modeled by
\[ u_t=\alpha u_{xx}, \qquad 0and initial temperature profile
\[ u(x,0)=f(x). \]The PDE describes evolution, while the conditions specify the actual physical problem.
The Wave Equation
Another fundamental PDE is the one-dimensional wave equation:
\[ \boxed{ u_{tt}=c^2u_{xx} }. \]Here \(u(x,t)\) may represent displacement of a string or another wave-like quantity.
The parameter \(c\) is the wave speed.
Interpreting the Wave Equation
The quantity
\[ u_{tt} \]describes acceleration in time.
The quantity
\[ u_{xx} \]describes curvature in space.
The equation says that spatial curvature creates temporal acceleration.
Initial Data for the Wave Equation
Because the wave equation is second order in time, it typically needs two initial conditions:
\[ \boxed{ u(x,0)=f(x) } \]and
\[ \boxed{ u_t(x,0)=g(x) }. \]These represent initial displacement and initial velocity.
Fixed-End Wave Problem
A string fixed at both ends may satisfy
\[ u(0,t)=0, \qquad u(L,t)=0. \]Together with
\[ u_{tt}=c^2u_{xx}, \]these conditions allow only certain spatial vibration modes.
Laplace's Equation
A fundamental steady-state PDE is Laplace's equation:
\[ \boxed{ u_{xx}+u_{yy}=0 }. \]In higher dimensions,
\[ \boxed{ \nabla^2u=0 }. \]Solutions are called harmonic functions.
Where Laplace's Equation Appears
Laplace's equation appears in models of:
- steady temperature;
- electrostatic potential;
- gravitational potential;
- incompressible potential flow;
- steady diffusion without sources.
Poisson's Equation
A related equation includes a source term:
\[ \boxed{ \nabla^2u=f }. \]This is Poisson's equation.
Laplace's equation is the special case
\[ f=0. \]Three Major PDE Types
Second-order linear PDEs are often classified as:
- elliptic;
- parabolic;
- hyperbolic.
The heat equation, wave equation, and Laplace equation provide standard examples of these categories.
Parabolic Behavior
The heat equation
\[ u_t=\alpha u_{xx} \]is the standard introductory parabolic equation.
Parabolic equations often describe diffusion, smoothing, and irreversible spreading.
Hyperbolic Behavior
The wave equation
\[ u_{tt}=c^2u_{xx} \]is the standard introductory hyperbolic equation.
Hyperbolic equations often describe propagation of signals and waves at finite speeds.
Elliptic Behavior
Laplace's equation
\[ u_{xx}+u_{yy}=0 \]is elliptic.
Elliptic equations often describe equilibrium or steady-state spatial configurations.
Separation of Variables
One of the first systematic methods for solving certain PDEs is separation of variables.
Suppose we seek a solution of the product form
\[ \boxed{ u(x,t)=X(x)T(t) }. \]This assumption converts one PDE into ordinary differential equations for \(X\) and \(T\).
Separate the Heat Equation
Start with
\[ u_t=\alpha u_{xx}. \]Let
\[ u(x,t)=X(x)T(t). \]Then
\[ u_t=X(x)T'(t) \]and
\[ u_{xx}=X''(x)T(t). \]Substitute:
\[ XT'=\alpha X''T. \]The Separation Constant
Divide by
\[ \alpha XT \]where appropriate:
\[ \frac{T'}{\alpha T} = \frac{X''}{X}. \]The left side depends only on \(t\).
The right side depends only on \(x\).
For them to be equal for all \(x\) and \(t\), both must equal a constant.
Write
\[ \boxed{ \frac{X''}{X} = \frac{T'}{\alpha T} = -\lambda }. \]The Resulting ODEs
Separation gives
\[ \boxed{ X''+\lambda X=0 } \]and
\[ \boxed{ T'+\alpha\lambda T=0 }. \]The PDE has been reduced to two ordinary differential equations.
Boundary Conditions Select Spatial Modes
Suppose
\[ X(0)=0, \qquad X(L)=0. \]Then nonzero solutions of
\[ X''+\lambda X=0 \]occur only for
\[ \boxed{ \lambda_n = \left( \frac{n\pi}{L} \right)^2 }. \]The corresponding spatial modes are
\[ \boxed{ X_n(x) = \sin\left( \frac{n\pi x}{L} \right) }. \]Time Dependence of Heat Modes
For each eigenvalue
\[ \lambda_n = \left( \frac{n\pi}{L} \right)^2, \]the time equation is
\[ T_n' + \alpha\lambda_nT_n = 0. \]Therefore,
\[ \boxed{ T_n(t) = e^{-\alpha\lambda_nt} }. \]Higher-frequency spatial modes decay faster.
A Separated Heat Solution
A single separated solution has the form
\[ \boxed{ u_n(x,t) = \sin\left( \frac{n\pi x}{L} \right) e^{-\alpha(n\pi/L)^2t} }. \]Because the heat equation is linear, sums of such modes are also solutions.
Superposition in Linear PDEs
If \(u_1\) and \(u_2\) solve a homogeneous linear PDE, then
\[ C_1u_1+C_2u_2 \]is also a solution.
This superposition principle allows complicated initial data to be built from simpler modes.
Fourier-Series Connection
For a fixed-end heat problem, a general solution may be represented as
\[ \boxed{ u(x,t) = \sum_{n=1}^{\infty} b_n \sin\left( \frac{n\pi x}{L} \right) e^{-\alpha(n\pi/L)^2t} }. \]The coefficients \(b_n\) are chosen so that the initial profile is matched.
This creates a direct connection between PDEs and Fourier series.
Matching the Initial Condition
At
\[ t=0, \]the heat-series solution becomes
\[ u(x,0) = \sum_{n=1}^{\infty} b_n \sin\left( \frac{n\pi x}{L} \right). \]Therefore, if
\[ u(x,0)=f(x), \]then the coefficients \(b_n\) are the sine-series coefficients of \(f\).
Separation for the Wave Equation
For
\[ u_{tt}=c^2u_{xx}, \]let
\[ u=X(x)T(t). \]Substitution gives
\[ XT''=c^2X''T. \]Divide by
\[ c^2XT: \] \[ \frac{T''}{c^2T} = \frac{X''}{X}. \]Set both sides equal to
\[ -\lambda. \]Wave Separation Equations
We obtain
\[ \boxed{ X''+\lambda X=0 } \]and
\[ \boxed{ T''+c^2\lambda T=0 }. \]The spatial boundary problem again selects eigenvalues, while the time equation produces oscillation.
Standing-Wave Modes
With fixed endpoints, the spatial modes are
\[ \sin\left( \frac{n\pi x}{L} \right). \]The corresponding angular frequencies are
\[ \boxed{ \omega_n = \frac{n\pi c}{L} }. \]These are the natural frequencies of the ideal string.
Normal Modes
A separated wave solution has the form
\[ u_n(x,t) = \sin\left( \frac{n\pi x}{L} \right) \left[ A_n\cos(\omega_nt) + B_n\sin(\omega_nt) \right]. \]Each value of \(n\) represents a normal mode of vibration.
Why Eigenfunctions Matter
In ordinary linear algebra, eigenvectors provide special directions that are preserved by a linear transformation.
In PDE boundary problems, eigenfunctions play an analogous role.
They provide special spatial patterns that evolve independently in simple ways.
The Laplacian
In two dimensions, the Laplacian is
\[ \boxed{ \nabla^2u = u_{xx}+u_{yy} }. \]In three dimensions,
\[ \boxed{ \nabla^2u = u_{xx}+u_{yy}+u_{zz} }. \]The Laplacian measures a form of local spatial curvature.
Steady State from a Time-Dependent PDE
A steady state does not change with time.
Therefore,
\[ u_t=0. \]For the heat equation
\[ u_t=\alpha\nabla^2u, \]steady state satisfies
\[ \boxed{ \nabla^2u=0 }. \]Thus Laplace's equation appears naturally as the steady-state heat equation.
Conservation and PDEs
Many PDEs arise from conservation principles.
A typical conservation law has the form
\[ \boxed{ \text{rate of accumulation} = \text{inflow} - \text{outflow} + \text{sources} }. \]When this balance is applied locally in space, partial derivatives naturally appear.
The Transport Equation
A simple first-order PDE is
\[ \boxed{ u_t+cu_x=0 }. \]This equation describes a profile moving with speed \(c\) without changing shape.
A solution has the form
\[ \boxed{ u(x,t)=f(x-ct) }. \]Verify the Transport Solution
Let
\[ u(x,t)=f(x-ct). \]Then
\[ u_t=-cf'(x-ct) \]and
\[ u_x=f'(x-ct). \]Therefore,
\[ u_t+cu_x = -cf'(x-ct)+cf'(x-ct) = 0. \]The solution is verified.
Characteristics: Introductory Idea
In first-order PDEs, information often travels along special curves called characteristics.
For
\[ u_t+cu_x=0, \]the characteristic curves satisfy
\[ x-ct=\text{constant}. \]The solution value remains constant along each such line.
Numerical PDE Methods
Many PDEs are solved numerically rather than analytically.
Common numerical approaches include:
- finite-difference methods;
- finite-volume methods;
- finite-element methods;
- spectral methods.
Each replaces the continuous spatial problem with a finite computational representation.
Finite Differences for Space
On a grid
\[ x_i=ih, \]the second derivative can be approximated by
\[ \boxed{ u_{xx}(x_i,t) \approx \frac{ u_{i+1}(t)-2u_i(t)+u_{i-1}(t) }{h^2} }. \]This converts a PDE into a large system of ordinary differential equations in time.
Method of Lines
The strategy of discretizing space while keeping time continuous is called the method of lines.
After spatial discretization, standard ODE solvers can advance the resulting system through time.
A Simple Explicit Heat Scheme
Using a time step \(\Delta t\) and spatial step \(\Delta x\), a simple finite-difference heat update is
\[ \boxed{ u_i^{n+1} = u_i^n + r \left( u_{i+1}^n - 2u_i^n + u_{i-1}^n \right) } \]where
\[ \boxed{ r = \frac{\alpha\Delta t}{(\Delta x)^2} }. \]Numerical Stability for the Heat Equation
The simple explicit heat scheme is stable only when the time step is sufficiently small relative to the spatial grid.
In the standard one-dimensional case, a common condition is
\[ \boxed{ r\le\frac12 }. \]Thus,
\[ \Delta t \]and
\[ \Delta x \]cannot be selected independently.
Geometry and PDE Domains
PDEs may be posed on:
- intervals;
- rectangles;
- disks;
- surfaces;
- three-dimensional regions;
- irregular computational domains.
The geometry of the domain affects both the boundary conditions and the solution structure.
Coordinate Systems Matter
Cartesian coordinates are not always the most efficient choice.
Problems with circular or spherical symmetry may be easier in:
- polar coordinates;
- cylindrical coordinates;
- spherical coordinates.
The differential operators change form in these coordinate systems.
Common Error: Treating Partial Derivatives as Ordinary Derivatives
When computing
\[ u_x, \]all other independent variables are treated as constant.
When computing
\[ u_t, \]the spatial variables are held fixed.
Common Error: Forgetting the Domain
A PDE problem is not fully specified by the differential equation alone.
The spatial domain, time interval, boundary conditions, and initial data are all part of the problem.
Common Error: Using Too Few Initial Conditions
The heat equation is first order in time and usually needs one initial time profile.
The wave equation is second order in time and usually needs two:
\[ u(x,0) \]and
\[ u_t(x,0). \]The time order helps determine how much temporal initial data are required.
Common Error: Assuming Separation Always Works
Separation of variables is powerful, but it requires suitable equation structure and compatible geometry and conditions.
Many PDEs cannot be solved by a simple product assumption.
Common Error: Forgetting the Boundary Eigenvalue Condition
After separation, not every value of the separation constant is allowed.
The boundary conditions determine which eigenvalues and eigenfunctions are valid.
Common Error: Confusing Steady State with Zero Solution
A steady state satisfies
\[ u_t=0, \]but it need not satisfy
\[ u=0. \]A nonzero spatial profile can remain constant in time.
Common Error: Ignoring Units
In
\[ u_t=\alpha u_{xx}, \]if \(x\) has units of length and \(t\) has units of time, then
\[ \boxed{ [\alpha] = \frac{\text{length}^2}{\text{time}} }. \]Dimensional consistency is an important model check.
A Reliable Introductory PDE Workflow
- Identify the unknown multivariable function.
- Identify the independent variables and domain.
- Determine the PDE order.
- Determine whether the PDE is linear or nonlinear.
- Identify the physical meaning of each derivative.
- List the initial conditions.
- List the boundary conditions.
- Check units and parameter meanings.
- Look for symmetry or a useful coordinate system.
- Consider separation of variables when the structure allows it.
- Use eigenvalue conditions to satisfy boundaries.
- Use superposition for homogeneous linear problems.
- Use numerical methods when analytical methods are impractical.
- Verify the PDE and every condition.
Example: Verify a Heat-Equation Mode
Consider
\[ u(x,t) = e^{-4\alpha t}\sin2x. \]Compute
\[ u_t = -4\alpha e^{-4\alpha t}\sin2x. \]Also,
\[ u_{xx} = -4e^{-4\alpha t}\sin2x. \]Therefore,
\[ \alpha u_{xx} = -4\alpha e^{-4\alpha t}\sin2x. \]Hence
\[ \boxed{ u_t=\alpha u_{xx} }. \]Example: Verify a Wave-Equation Mode
Let
\[ u(x,t) = \sin(kx)\cos(ckt). \]Then
\[ u_{xx} = -k^2\sin(kx)\cos(ckt) \]and
\[ u_{tt} = -c^2k^2\sin(kx)\cos(ckt). \]Therefore,
\[ \boxed{ u_{tt}=c^2u_{xx} }. \]Practice 1: ODE or PDE?
Classify
\[ y''+3y=0 \]as an ODE or PDE.
Practice 2: ODE or PDE?
Classify
\[ u_t=5u_{xx} \]as an ODE or PDE.
Practice 3: Identify the PDE
Which standard PDE is
\[ u_{tt}=9u_{xx}? \]Practice 4: Wave Speed
For
\[ u_{tt}=25u_{xx}, \]find the wave speed.
Practice 5: Heat Diffusivity Units
In
\[ u_t=\alpha u_{xx}, \]what are the units of \(\alpha\) if \(x\) is measured in meters and \(t\) in seconds?
Practice 6: Initial Data
How many time-initial profiles are typically needed for the one-dimensional wave equation?
Practice 7: Dirichlet or Neumann?
Classify
\[ u_x(0,t)=0. \]Practice 8: Separation
If
\[ u(x,t)=X(x)T(t), \]compute \(u_t\) and \(u_{xx}\).
Practice 9: Laplace Equation
Write Laplace's equation in two Cartesian variables.
Practice 10: Transport Solution
Verify that
\[ u(x,t)=f(x-ct) \]satisfies
\[ u_t+cu_x=0. \]Practice Answers
Practice 1
The unknown depends on one independent variable and uses ordinary derivatives.
Therefore,
\[ \boxed{\text{ODE}}. \]Practice 2
The unknown depends on more than one independent variable and partial derivatives appear.
Therefore,
\[ \boxed{\text{PDE}}. \]Practice 3
The equation has the form
\[ u_{tt}=c^2u_{xx}. \]Therefore, it is the
\[ \boxed{\text{wave equation}}. \]Practice 4
Compare
\[ u_{tt}=25u_{xx} \]with
\[ u_{tt}=c^2u_{xx}. \]Thus,
\[ c^2=25 \]and
\[ \boxed{c=5}. \]Practice 5
Since
\[ u_t \]has units of quantity per second and
\[ u_{xx} \]has units of quantity per square meter,
\(\alpha\) must have units
\[ \boxed{ \frac{\text{m}^2}{\text{s}} }. \]Practice 6
Because the wave equation is second order in time, it typically requires
\[ \boxed{2} \]initial time profiles:
\[ u(x,0) \]and
\[ u_t(x,0). \]Practice 7
The derivative is specified, so this is a
\[ \boxed{\text{Neumann condition}}. \]Practice 8
If
\[ u=XT, \]then
\[ \boxed{ u_t=XT' } \]and
\[ \boxed{ u_{xx}=X''T }. \]Practice 9
In two Cartesian variables,
\[ \boxed{ u_{xx}+u_{yy}=0 }. \]Practice 10
Let
\[ u(x,t)=f(x-ct). \]Then
\[ u_t=-cf'(x-ct) \]and
\[ u_x=f'(x-ct). \]Therefore,
\[ u_t+cu_x = -cf'(x-ct)+cf'(x-ct) = 0. \]Thus,
\[ \boxed{ u(x,t)=f(x-ct) } \]satisfies the transport equation.
What to Remember
A partial differential equation relates a multivariable function to its partial derivatives.
- the heat equation models diffusion and smoothing;
- the wave equation models propagation and vibration;
- Laplace's equation models many steady-state fields;
- initial conditions specify starting profiles;
- boundary conditions constrain the edge of the spatial domain;
- separation of variables can reduce certain PDEs to ODEs;
- boundary conditions select eigenvalues and eigenfunctions;
- superposition builds more general solutions of homogeneous linear PDEs;
- Fourier series naturally arise from modal expansions;
- numerical methods are essential for many realistic PDE problems.
The central structural idea is
\[ \boxed{ \text{multivariable state} + \text{directional rates of change} \longrightarrow \text{distributed evolution} }. \]Where We Go Next
This lesson completes the introductory Differential Equations sequence. First-order equations, higher-order equations, systems, qualitative analysis, numerical methods, and PDEs are all different expressions of one common idea: a mathematical law specifies how a state changes, and solutions reveal the resulting evolution.
From here, these tools connect naturally to advanced analysis, dynamical systems, numerical analysis, mathematical physics, engineering models, optimization, probability, and scientific computing.