Partial Differential Equation (PDE)
What is Partial Differential Equation (PDE)?
A differential equation containing unknown functions of multiple independent variables and their partial derivatives.
Key formula / rule: One-dimensional Heat Equation
Key points
- Classify PDEs based on order, linearity, and type.
- Understand the physical significance of common PDEs (Heat, Wave, Laplace).
- Apply the method of separation of variables to solve simple PDEs.
- Understand the concept of initial and boundary conditions.
Common exam trap
Confusing partial derivatives with total derivatives.
Definitions
- Term
Partial Differential Equation (PDE)
- Meaning
A differential equation containing unknown functions of multiple independent variables and their partial derivatives.
- Term
Order of a PDE
- Meaning
The order of the highest partial derivative appearing in the equation.
- Term
Linear PDE
- Meaning
A PDE where the unknown function and its derivatives appear only in the first ° and are not multiplied together.
- Term
Homogeneous PDE
- Meaning
A linear PDE where every term contains the unknown function or one of its derivatives.
- Term
Initial Conditions
- Meaning
Conditions that specify the state of the system at an initial time.
- Term
Boundary Conditions
- Meaning
Conditions that specify the behavior of the solution at the boundaries of the domain.
- Term
Elliptic PDE
- Meaning
A type of second-order PDE typically associated with steady-state or equilibrium problems (e.g., Laplace's equation).
- Term
Parabolic PDE
- Meaning
A type of second-order PDE typically associated with diffusion or time-dependent problems (e.g., Heat equation).
- Term
Hyperbolic PDE
- Meaning
A type of second-order PDE typically associated with wave propagation or vibration problems (e.g., Wave equation).
- Term
Separation of Variables
- Meaning
A method for solving certain PDEs by assuming the solution can be expressed as a product of functions, each depending on a single independent variable.
- Term
Method of Characteristics
- Meaning
A technique for solving first-order PDEs by finding curves (characteristics) along which the PDE reduces to an ODE.
Learning objectives
Classify PDEs based on order, linearity, and type.
Understand the physical significance of common PDEs (Heat, Wave, Laplace).
Apply the method of separation of variables to solve simple PDEs.
Understand the concept of initial and boundary conditions.
Solve first-order PDEs using the method of characteristics.
Formulae
- Name
One-dimensional Heat Equation
- Note
Describes heat conduction or diffusion; $u$ is temperature, $t$ is time, $x$ is position, $c2$ is thermal diffusivity.
- Expression
\frac{\partial u}{\partial t} = c2 \frac{\partial2 u}{\partial x2}
- Name
One-dimensional Wave Equation
- Note
Describes wave propagation; $u$ is displacement, $t$ is time, $x$ is position, $c$ is wave speed.
- Expression
\frac{\partial2 u}{\partial t2} = c2 \frac{\partial2 u}{\partial x2}
- Name
Two-dimensional Laplace's Equation
- Note
Describes steady-state phenomena (e.g., electrostatic potential in a charge-free region, steady-state temperature distribution); $u$ is the potential/temperature.
- Expression
\frac{\partial2 u}{\partial x2} + \frac{\partial2 u}{\partial y2} = 0
- Name
Discriminant for Second-Order PDE Classification
- Note
For $A uxx + B uxy + C uyy + D ux + E uy + F u = G$. If $\Δ < 0$, Elliptic; $\Δ = 0$, Parabolic; $\Δ > 0$, Hyperbolic.
- Expression
\Δ = B2 - 4AC
- Name
Characteristic Equations for $A \frac{\partial u}{\partial x} + B \frac{\partial u}{\partial y} = C$
- Note
Used to transform the PDE into a system of ODEs along characteristic curves.
- Expression
\frac{dx}{A} = \frac{dy}{B} = \frac{du}{C}
Prerequisites
Differential Equations (Ordinary Differential Equations)
Multivariable Calculus (Partial Derivatives, Gradients, Divergence, Curl)
Linear Algebra (Eigenvalues, Eigenvectors, Matrix operations)
Fourier Series and Transforms
Laplace Transforms
Common mistakes
Confusing partial derivatives with total derivatives.
Incorrectly applying boundary or initial conditions.
Not checking for linearity or homogeneity before applying specific solution methods.
Errors in separation of variables (e.g., incorrect constant choice).
Forgetting the arbitrary functions/constants when integrating.
Keywords
Partial derivative
differential equation
order
linearity
homogeneity
heat equation
wave equation
Laplace's equation
separation of variables
method of characteristics
initial conditions
boundary conditions
elliptic
parabolic
hyperbolic
Fourier series
Laplace transform
Practice preview
Which of the following partial differential equations is linear?…
easy
A partial differential equation is said to be homogeneous if:…
easy
Using the method of characteristics, the general solution of the PDE x * partial u / partial x + y * partial u / partial y = u is:…
medium