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PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

In mathematics, *PDE* is the acronym for *Partial Differential Equation*. In other words, a PDE is an equation that contains partial derivatives. A derivative is simply the rate of change of a *dependent variable* with respect to an *independent variable*.
For example, in the linear equation *y = mx + c*, y is the dependent variable and x is the independent variable. The derivative of y with respect to x,is generally represented as *dy/dx*. The process of determining a derivative is called *differentiation*, an elaborate branch of calculus.

A differential equation (an equation that contains derivatives) is an *ordinary differential equation* (ODE) if the unknown function (the dependent variable) depends on only one variable. A differential equation is a *partial differential equation* if the unkown function depends on more than one variable.

Suppose that u is a function of x, and t. Then we can state the following:

u(x,t) is the function u with its independent variablea x and t.
Some partial derivatives from u(x,t) are:
δu/δt (first partial derivative with respect to t)
δu/δx (first partial derivative with respect to x)
δ^{2}u/δx^{2} (second partial derivative with respect to x).

*PDEs* are very important because most natural problems can only be expressed as multivariate functions whose derivatives represent natural phenomena such as *velocity, acceleration, force, friction, current, flux*, etc. The process of *formulating the PDEs associated with a natural problem is called mathematical modeling*.

There are various classes of partial differential equations. In general, the method used to solve a PDE is often dependent on the class to which the PDE belong. One of the basic PDE classifications is as follows:
*Order of the PDE*: the order of a PDE is the order of the *highest partial derivative* in the equation, for example, the partial differential equation:

δu/δt = δ^{2}u/δx^{2} has order 2

Partial differential equations are broadly classified into two groups: *linear* and *nonlinear*.

*Linear PDEs*: in linear partial differential equations, the dependent variable and the partial derivatives are linearly expressed. In other words, the following terms are absent in the equation: powers of the dependent variable ≥2, powers of the partial derivatives, products of the dependent variable and the partial derivatives.

Suppose u(x,y) is a function of the independent variables x and y. Then a second order linear PDE of u(x,y) is of the form:

*Aδ ^{2}u/δx^{2} + Bδ^{2}u/δxδxy + δ^{2}u/δy^{2} + Dδu/δx + Eδu/δy + Fu(x,y) = G* ---(1).

where A, B, C, D, E, F and G can be constants or given functions of x and y.

Equation (1) is *homogeneous* if the right-hand side G(x,y) is zero. Otherwise, it is *nonhomogeneous*.

There are three basic types of linear PDEs with the form indicated in equation (1):
*Parabolic PDEs*: describe heat flow and difussion processes. B^{2} - 4AC = 0. e.g. δu/δy = δ^{2}u/δx^{2}
*Hyperbolic PDEs*: describe vibrating systems and wave motion. B^{2} - 4AC > 0. e.g. δ^{2}u/δy^{2} = δ^{2}u/δx^{2}
*Elliptic PDEs*: describe *steady-state* phenomena. B^{2} - 4AC <0. e.g. δ^{2}u/δx^{2}

Find all functions u(x,y) that satisfy the following PDE:

δu(x,y)/δx = 0. Answer = g(y), that is, the function g(y).

Peter Oye Sagay