Differential equation (nonfiction): Difference between revisions
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[[File:Elmer pump heat equation.png|thumb|Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.]]A '''differential equation''' is a [[Equation (nonfiction)|mathematical equation]] that relates some [[Function (nonfiction)|function]] with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. | [[File:Elmer pump heat equation.png|thumb|Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.]]A '''differential equation''' is a [[Equation (nonfiction)|mathematical equation]] that relates some [[Function (nonfiction)|function]] with its [[Derivative (nonfiction)|derivatives]]. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. | ||
Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. | Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. | ||
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. | In [[Pure mathematics (nonfictcion)|pure mathematics]], differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. | ||
If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. | If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. | ||
The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. | The theory of [[Dynamical system (nonfiction)|dynamical systems]] puts emphasis on qualitative analysis of systems described by differential equations, while many [[Numerical analysis (nonfiction)|numerical methods]] have been developed to determine solutions with a given degree of accuracy. | ||
== In the News == | == In the News == | ||
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== Fiction cross-reference == | == Fiction cross-reference == | ||
* [[Crimes against mathematical constants]] | |||
* [[Gnomon algorithm]] | * [[Gnomon algorithm]] | ||
* [[Gnomon Chronicles]] | |||
* [[Mathematician]] | |||
* [[Mathematics]] | * [[Mathematics]] | ||
== Nonfiction cross-reference == | == Nonfiction cross-reference == | ||
* [[Boundary value problem (nonfiction)]] - a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. | |||
* [[Equation (nonfiction)]] | * [[Equation (nonfiction)]] | ||
* [[Function (nonfiction)]] | * [[Function (nonfiction)]] | ||
* [[Linear differential equation (nonfiction)]] | * [[Linear differential equation (nonfiction)]] | ||
* [[Mathematics (nonfiction)]] | * [[Mathematics (nonfiction)]] | ||
* [[Ordinary differential equation (nonfiction)]] - a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. | |||
* [[Partial differential equation (nonfiction)]] - | |||
* [[Physics (nonfiction)]] | * [[Physics (nonfiction)]] | ||
* [[Separatrix (mathematics) (nonfiction)]] - the boundary separating two modes of behavior in a differential equation. | |||
External links | == External links == | ||
* [https://en.wikipedia.org/wiki/Differential_equation Differential equation] @ Wikipedia | * [https://en.wikipedia.org/wiki/Differential_equation Differential equation] @ Wikipedia | ||
Latest revision as of 19:03, 14 April 2020
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.
Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers.
The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
In the News
Fiction cross-reference
Nonfiction cross-reference
- Boundary value problem (nonfiction) - a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
- Equation (nonfiction)
- Function (nonfiction)
- Linear differential equation (nonfiction)
- Mathematics (nonfiction)
- Ordinary differential equation (nonfiction) - a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
- Partial differential equation (nonfiction) -
- Physics (nonfiction)
- Separatrix (mathematics) (nonfiction) - the boundary separating two modes of behavior in a differential equation.