Partial Differential Equations (PDE)

Peter Oye Sagay

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)
δ2u/δx2 (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 = δ2u/δx2 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:

2u/δx2 + Bδ2u/δxδxy + δ2u/δy2 + 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. B2 - 4AC = 0. e.g. δu/δy = δ2u/δx2
Hyperbolic PDEs: describe vibrating systems and wave motion. B2 - 4AC > 0. e.g. δ2u/δy2 = δ2u/δx2
Elliptic PDEs: describe steady-state phenomena. B2 - 4AC <0. e.g. δ2u/δx2

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) .

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