Partial differential equation

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Heat

Partial differential equations (PDEs) are mathematical models that involve functions of several variables and their partial derivatives. They are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. PDEs are fundamental to the description of physical, biological, and social systems, including the propagation of sound and heat, the dynamics of fluids, the elasticity of materials, the evolution of populations, and the spread of diseases.

Definition

A partial differential equation is an equation that contains unknown multivariable functions and their partial derivatives. A PDE can be written in the form: \[ F(x_1, x_2, ..., x_n, u, \frac{\partial u}{\partial x_1}, \frac{\partial u}{\partial x_2}, ..., \frac{\partial u}{\partial x_n}, \frac{\partial^2 u}{\partial x_1^2}, \frac{\partial^2 u}{\partial x_1 \partial x_2}, ..., \frac{\partial^2 u}{\partial x_n^2}) = 0 \] where \(u\) is the unknown function of the variables \(x_1, x_2, ..., x_n\), and \(F\) is a given function.

Classification

PDEs are classified according to their order, linearity, and homogeneity.

Order

The order of a PDE is the order of the highest derivative of the unknown function that appears in the equation. For example, if the highest derivative is a second derivative, the PDE is of second order.

Linearity

A PDE is linear if it can be written so that the unknown function and its derivatives appear to the first power and are not multiplied together. Otherwise, it is nonlinear.

Homogeneity

A homogeneous PDE is one in which every term contains the unknown function or its derivatives. If there is a term that does not involve the unknown function or its derivatives, the PDE is nonhomogeneous.

Methods of Solution

Solving PDEs can be challenging and depends on the type of PDE. Some common methods include:

- Separation of variables: This method involves assuming that the solution can be written as a product of functions, each of which depends on only one of the variables. - Method of characteristics: Used primarily for first-order PDEs, this method transforms the PDE into a system of ordinary differential equations (ODEs) along certain curves called characteristics. - Finite difference method: A numerical method that approximates solutions by replacing the derivatives in the PDE with finite differences. - Finite element method: Another numerical method that solves PDEs by converting them into an equivalent problem of finding the stationary value of an integral.

Applications

PDEs are used in various fields such as physics, engineering, economics, and biology. Some examples include:

- The Navier-Stokes equations in fluid dynamics - The Heat equation for heat conduction - The Wave equation for sound and light waves - The Black-Scholes equation in financial mathematics

See Also

- Ordinary differential equation - Differential equation - Mathematical model - Numerical analysis