Partial Differential Equations (Formula Guide)
A partial differential equation (PDE) describes an unknown function of several variables and its partial derivatives. To use a formula well, identify the equation’s order and type, specify its domain and initial or boundary data, then choose a method that fits. This guide summarizes core formulas, assumptions, solution steps, checks, and common models.
The first time you meet a long PDE formula, it can feel like a wall of symbols. Think of the equations in a science-fiction story: a model may describe how heat spreads through a ship, or how a wave moves across a surface. The interesting part is not just the equation. It is the information about the setting and starting conditions that lets you solve it.
I recommend reading any PDE in a fixed order: identify its variables, find its highest derivatives, note the domain, and inspect the supplied data. These steps help you choose a method instead of trying formulas at random.
PDE Overview, Classification & Course Preparation
A partial differential equation links an unknown function to its partial derivatives. Its solutions depend on the equation and on data given over a domain or its boundary. Classification helps narrow the possible methods, but it does not provide a solution by itself.
What do order and linearity mean?
The order of a PDE is the highest derivative that appears. For example, (u_t=\kappa u_{xx}) is second order because it includes a second derivative, (u_{xx}).
A PDE is linear if the unknown function and its derivatives appear only to the first power, are not multiplied by each other, and have coefficients that do not depend on the unknown function. For example, (u_t-\kappa u_{xx}=0) is linear. An equation containing (u\,u_x) is nonlinear.
A solution is a function that satisfies the PDE and the required conditions. Those conditions matter: one equation can have many solutions until enough data are given.
How do you classify a second-order PDE?
For a two-variable equation whose highest-order terms are
[ A u_{xx}+2B u_{xy}+C u_{yy}, ]
calculate the discriminant
[ \Delta=B^2-AC. ]
With the mixed term written as (2B u_{xy}), the equation is elliptic if (\Delta<0), parabolic if (\Delta=0), and hyperbolic if (\Delta>0). This convention matters. If the mixed term is instead written (D u_{xy}), use (B=D/2) in the formula.
| Type | Discriminant | Typical model | Common behavior |
|---|---|---|---|
| Elliptic | (\Delta<0) | (u_{xx}+u_{yy}=0) | Boundary data often describe the solution across a region |
| Parabolic | (\Delta=0) | (u_t=\kappa u_{xx}) | Describes diffusion or smoothing over time |
| Hyperbolic | (\Delta>0) | (u_{tt}=c^2u_{xx}) | Describes wave-like changes that can travel through a domain |
These descriptions are useful guides, not full solution rules. Lower-order terms, the shape of the domain, and the given conditions also affect the problem.
What should you review before studying PDEs?
A typical upper-division PDE course draws on multivariable calculus, ordinary differential equations (ODEs), and introductory linear algebra. Review partial derivatives, gradients, divergence, the Laplacian, eigenvalues and eigenvectors, and common ODE methods.
Before enrolling, compare your background with the specific course’s stated prerequisites. Requirements differ by institution. If you need the course for a degree, also ask the department or adviser whether it fits your program and whether transfer credit will apply. Accreditation alone does not guarantee that a course will transfer or count toward a particular requirement.
Core Equations, Boundary Conditions & Solution Methods
A PDE formula is useful only when paired with its variables, domain, and conditions. Initial conditions describe a solution at a starting time; boundary conditions describe it at the edge of a domain. Together, these details help determine which solutions are allowed.
What are the main model equations?
Here, (u) is the unknown function, (t) is time, (x) is position, (\kappa>0) is a diffusion coefficient, and (c>0) is a wave speed. In several spatial dimensions, the Laplacian is written (\Delta u); in two dimensions, (\Delta u=u_{xx}+u_{yy}).
| Model | Formula | Common data |
|---|---|---|
| Heat or diffusion equation | (u_t=\kappa\Delta u) | Initial temperature or concentration, plus boundary conditions |
| Wave equation | (u_{tt}=c^2\Delta u) | Initial displacement and velocity, plus boundary conditions |
| Laplace equation | (\Delta u=0) | Boundary values on the edge of a region |
For example, a one-dimensional heat problem on (0<x<L) may require (u(x,0)=f(x)) and fixed-end values (u(0,t)=u(L,t)=0). The equation alone does not specify this complete problem.
How do you solve a linear first-order PDE?
For
[ a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u), ]
the characteristic equations are
[ \frac{dx}{ds}=a,\qquad \frac{dy}{ds}=b,\qquad \frac{du}{ds}=c. ]
They turn the PDE into a system of ODEs along curves called characteristics. A practical procedure is:
- Write the characteristic ODEs.
- Solve them, or find quantities that stay constant along each curve.
- Use the data on the starting curve to set the solution.
- Check that the starting curve is non-characteristic: it must not run along the characteristic direction at the point where data are given.
For instance, (u_x+u_y=0) gives (dx/ds=1), (dy/ds=1), and (du/ds=0). Since (y-x) and (u) stay constant along each characteristic, the general solution is (u=F(y-x)), where (F) is an arbitrary function. Data along a suitable curve determine (F).
How does separation of variables work?
Separation of variables assumes a product form, such as (u(x,t)=X(x)T(t)). Substitute it into the PDE, divide by the product when allowed, and set the expression in (x) equal to a constant and the expression in (t) to its opposite. This creates separate ODEs.
For the heat equation (u_t=\kappa u_{xx}), the product assumption gives
[ \frac{T’}{\kappa T}=\frac{X”}{X}=-\lambda. ]
Thus (X”+\lambda X=0) and (T’+\kappa\lambda T=0). With (X(0)=X(L)=0), the allowed values are (\lambda_n=(n\pi/L)^2), and (X_n(x)=\sin(n\pi x/L)). The time factors are (T_n(t)=e^{-\kappa(n\pi/L)^2t}).
A solution built from these modes is
[ u(x,t)=\sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right) e^{-\kappa(n\pi/L)^2t}. ]
The coefficients (b_n) are determined by the initial function (f(x)), using its Fourier sine series under suitable conditions. The domain and boundary conditions determine the eigenvalues and eigenfunctions; they are not set by the PDE alone.
Which method should you try first?
| Problem feature | Method to consider | What you must check |
|---|---|---|
| First-order equation with a clear transport direction | Characteristics | Starting data must be compatible and non-characteristic |
| Simple domain and boundary conditions | Separation of variables | The resulting eigenfunctions fit the domain and boundary conditions |
| Periodic or infinite spatial domain | Fourier series or Fourier transform | The data and solution meet the method’s assumptions |
| Complicated geometry or coefficients | Numerical method | The numerical setup represents the domain and conditions correctly |
These are starting points, not guarantees. A method that works for one boundary condition may fail for another.
Study Strategy, Common Errors & Mathematical Verification
A reliable PDE workflow moves from the problem statement to a checked solution. Classify first, choose a method second, and verify last. This order prevents a common mistake: applying a familiar formula before confirming that the problem meets its assumptions.
What steps should you follow?
- Prepare: Review multivariable derivatives, vector calculus, ODEs, and eigenvalue problems. Mark any gaps before starting a new chapter.
- Model: Write the PDE in standard form. Identify its order, type, variables, domain, coefficients, and data.
- Check compatibility: Confirm that the initial and boundary conditions do not contradict each other. For example, if fixed endpoint values are prescribed, the initial function must agree with them at the endpoints when the solution is required to satisfy both there.
- Choose a method: Use characteristics, separation of variables, Fourier or transform methods, or a suitable numerical method.
- Verify: Substitute the proposed solution into the PDE, then test every initial and boundary condition.
What errors should you watch for?
- Mixing discriminant conventions: The formula (\Delta=B^2-AC) assumes the mixed term is (2B u_{xy}).
- Ignoring the domain: Eigenvalues and eigenfunctions depend on the domain and boundary conditions.
- Dropping a condition: A function can solve the PDE but fail the initial or boundary data.
- Treating a formal series as automatically valid: A series solution also requires conditions that allow the series to represent the data and satisfy the equation in the intended sense.
- Forgetting units: In the heat equation, (\kappa) has units of length squared per time when (x) is measured in length and (t) in time. In the wave equation, (c) has units of length per time.
I use a short verification checklist: Does the solution have the right variables? Does substitution recover the equation? Does it meet the data? Are its units consistent? A “no” often points to a missing assumption or algebra error.
Applications, Further Study & Career Pathways
PDEs model changes across space and, often, time. The same equation can describe different settings when its variables and coefficients are interpreted in different ways. These examples show how a model, its data, and its solution method fit together.
How do PDEs appear in applied problems?
- Heat flow: Temperature (u(x,t)) in a rod may follow (u_t=\kappa u_{xx}). The domain represents the rod, (\kappa) describes diffusion, and endpoint conditions state how the ends are treated.
- Vibrating string: Displacement (u(x,t)) may satisfy (u_{tt}=c^2u_{xx}). The starting shape and speed provide initial data; the support at each end provides boundary conditions.
- Steady-state field: A quantity that has reached equilibrium may satisfy (\Delta u=0). Boundary values on the edge of a region help determine the field inside.
These are standard model forms, not complete descriptions of every physical system. Real models may need additional terms or more detailed assumptions.
When does PDE study become more advanced?
At the undergraduate level, many courses focus on classical solutions: functions smooth enough for the derivatives in the equation to exist in the usual sense. Graduate study often adds functional analysis, distributions, and weak solutions. A weak solution meets the equation through an integral formulation, which can work even when ordinary derivatives are not well behaved.
PDE knowledge can support further study in mathematics, physics, and engineering. A course or formula sheet does not by itself establish admission eligibility, degree credit, professional licensure, or employment. Those depend on the institution, program, and role.
What is the best way to keep a PDE formula sheet?
For each method, record the equation, notation, assumptions, domain, required conditions, and a short verification step. Keep one example beside each formula, and label changes in convention, such as whether the mixed derivative is written (2B u_{xy}) or (D u_{xy}).
The key habit is to treat a PDE and its data as one problem. When you can state the equation, domain, conditions, and method in plain language, you are ready to work through the symbols.
Frequently Asked Questions
These short answers review the main choices involved in classifying and solving PDEs. They are quick references, not substitutes for checking a problem’s domain, notation, and conditions.
What is a partial differential equation?
A PDE is an equation involving an unknown function of two or more variables and one or more of its partial derivatives.
How do I find the order of a PDE?
Find the highest derivative that appears. If the highest derivative is second order, the PDE is second order.
How do I classify a second-order PDE in two variables?
For (A u_{xx}+2B u_{xy}+C u_{yy}), calculate (B^2-AC). A negative result is elliptic, zero is parabolic, and a positive result is hyperbolic.
Why does the mixed-derivative convention matter?
The discriminant formula assumes the coefficient of (u_{xy}) is written (2B). If it is written (D), set (B=D/2) before calculating.
What are characteristics in a first-order PDE?
Characteristics are curves along which a first-order PDE becomes a system of ordinary differential equations.
What does separation of variables assume?
It assumes a product form, such as (u(x,t)=X(x)T(t)). The boundary conditions and domain determine which separated modes are allowed.
What conditions does the heat equation need?
A typical heat problem needs an initial temperature and boundary conditions. The exact conditions depend on the domain and how its boundaries are modeled.
How can I check a proposed solution?
Substitute it into the PDE, test every initial and boundary condition, and check that its variables and units match the problem.
(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)
