Definition

1. Noun:
   • A partial differential equation is a type of mathematical equation that involves an unknown function of multiple independent variables and its partial derivatives with respect to those variables. It describes a relationship between the function and the rates at which it changes in different spatial or temporal directions.

Usage

• Partial differential equations are fundamental for modeling continuous systems and phenomena in physics, engineering, finance, and other sciences.
• They are used to describe how quantities like heat, sound, fluid pressure, or electromagnetic fields change over space and time.
• Solving a partial differential equation typically means finding the specific function that satisfies the given equation and any associated boundary or initial conditions.

Examples

• Noun:
  • The partial differential equation governing heat diffusion is known as the heat equation.
  • Finding a general solution to this nonlinear partial differential equation is extremely challenging.
  • In fluid dynamics, the Navier-Stokes equations are a set of partial differential equations.

Advanced Usage

• "To solve a partial differential equation": To find the function or set of functions that satisfy the equation.
  • Numerical methods are often required to solve complex partial differential equations.
• "A system of partial differential equations": Multiple interrelated partial differential equations that must be solved simultaneously.
  • Maxwell's equations form a system of partial differential equations that describe electromagnetism.

Variants and Related Words

• PDE (n): A common abbreviation for "partial differential equation."
  • The researcher specialized in the analysis of PDEs.
• Ordinary Differential Equation (ODE) (n): A differential equation involving a function of a single variable and its derivatives. This is a contrasting term.
  • An ODE involves ordinary derivatives, while a PDE involves partial derivatives.

Synonyms

• Differential equation (n): A broader category that includes both ordinary and partial differential equations.

Related Phrases

• Boundary value problem: A problem involving a partial differential equation where the solution is determined by conditions specified at the boundaries of the domain.
• Initial value problem: A problem involving a partial differential equation where the solution is determined by conditions specified at an initial time.

Noun

1. a differential equation involving a functions of more than one variable