Mathwords logoReference LibraryMathwords

Partial Differential Equation

Partial Differential Equation

A differential equation that contains at least one partial derivative.

 

Example equation: the second partial derivative of z with respect to x squared plus the second partial derivative of z with...

 

 

See also

Ordinary differential equation

Key Formula

F ⁣(x1,x2,…,xn,  u,  ∂u∂x1,∂u∂x2,…,∂2u∂x12,…)=0F\!\left(x_1, x_2, \ldots, x_n,\; u,\; \frac{\partial u}{\partial x_1}, \frac{\partial u}{\partial x_2}, \ldots, \frac{\partial^2 u}{\partial x_1^2}, \ldots\right) = 0
Where:
  • x1,x2,…,xnx_1, x_2, \ldots, x_n = Independent variables (e.g., position, time)
  • uu = Unknown function of the independent variables
  • ∂u∂xi\frac{\partial u}{\partial x_i} = Partial derivatives of u with respect to each independent variable
  • FF = A given relationship that the function and its partial derivatives must satisfy

Worked Example

Problem: Verify that the function u(x, t) = sin(x) · e^(−t) satisfies the partial differential equation ∂u/∂t = ∂²u/∂x².
Step 1: Write down the given function.
u(x,t)=sin⁡(x) e−tu(x, t) = \sin(x)\, e^{-t}
Step 2: Compute the partial derivative of u with respect to t, treating x as a constant.
∂u∂t=sin⁡(x)⋅(−e−t)=−sin⁡(x) e−t\frac{\partial u}{\partial t} = \sin(x) \cdot (-e^{-t}) = -\sin(x)\, e^{-t}
Step 3: Compute the first partial derivative of u with respect to x, treating t as a constant.
∂u∂x=cos⁡(x) e−t\frac{\partial u}{\partial x} = \cos(x)\, e^{-t}
Step 4: Compute the second partial derivative of u with respect to x.
∂2u∂x2=−sin⁡(x) e−t\frac{\partial^2 u}{\partial x^2} = -\sin(x)\, e^{-t}
Step 5: Compare the two results. Both sides of the PDE are equal.
∂u∂t=−sin⁡(x) e−t=∂2u∂x2\frac{\partial u}{\partial t} = -\sin(x)\, e^{-t} = \frac{\partial^2 u}{\partial x^2}
Answer: Since ∂u/∂t equals ∂²u/∂x², the function u(x, t) = sin(x) · e^(−t) is indeed a solution to the heat equation ∂u/∂t = ∂²u/∂x².

Frequently Asked Questions

What is the difference between a partial differential equation and an ordinary differential equation?
An ordinary differential equation (ODE) involves a function of one independent variable and its ordinary derivatives. A partial differential equation (PDE) involves a function of two or more independent variables and its partial derivatives. For example, dy/dx = 2x is an ODE (one variable x), while ∂u/∂t = ∂²u/∂x² is a PDE (two variables x and t).
What are the most common types of partial differential equations?
The three classical types are the heat equation (∂u/∂t = k ∂²u/∂x², modeling diffusion), the wave equation (∂²u/∂t² = c² ∂²u/∂x², modeling vibrations and waves), and Laplace's equation (∂²u/∂x² + ∂²u/∂y² = 0, modeling steady-state systems). These represent the three fundamental categories of second-order linear PDEs: parabolic, hyperbolic, and elliptic.

Partial Differential Equation (PDE) vs. Ordinary Differential Equation (ODE)

An ODE contains derivatives with respect to a single independent variable, such as dy/dx + y = 0. A PDE contains partial derivatives with respect to two or more independent variables, such as ∂u/∂t + ∂u/∂x = 0. ODEs typically have solutions that are curves (functions of one variable), while PDE solutions are surfaces or higher-dimensional objects (functions of multiple variables). PDEs generally require boundary conditions in addition to initial conditions, making them significantly harder to solve.

Why It Matters

Partial differential equations are the mathematical language of physics and engineering. They describe heat conduction, fluid flow, electromagnetic fields, quantum mechanics, and the propagation of sound and light. Nearly every physical system that varies across both space and time is modeled by a PDE, making them one of the most widely applied areas of mathematics.

Common Mistakes

Mistake: Confusing partial derivatives with ordinary derivatives when a function has multiple variables.
Correction: When differentiating u(x, t) with respect to t, you must hold x constant and use the partial derivative symbol ∂, not d. Writing du/dt when you mean ∂u/∂t changes the mathematical meaning entirely.
Mistake: Thinking a PDE has a single solution like many ODEs do.
Correction: PDEs generally have infinitely many solutions. To determine a unique solution, you need boundary conditions and/or initial conditions. The PDE alone only describes the relationship; the conditions pin down which specific solution applies to your problem.

Related Terms