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PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equations (PDEs) is an important branch of applied and pure mathematics. These differential equations describe changes in parameters with respect to more than one independent variable (for example, space and time).
The PDE can be classified as linear if every term in the equation is a constant and/or a function of the unknown variable. If this is not the case, then the PDE is nonlinear. An example of a linear PDE is Laplace’s equation, which is a simplification of the nonlinear Navier-Stokes equations.
The order of the PDE is determined by the highest derivative in the equation. An equation with a second and first derivative of the variable x would be a second-order PDE. A PDE can be classified with different orders depending on the independent variable. A PDE that has a second derivative of x, but a first derivative in time, t, then the PDE is second order in x and first order in t.
This remainder of this topic provides a very brief introduction of classifying a very narrow type of PDEs that are used frequently in aerospace engineering. Linear second-order partial differential equations of a function u in two dimensions (u = u(x, y)) can be written as
(1) A(∂2u)/(∂x2) + 2B(∂2u)/(∂x∂y) + C(∂2u)/(∂y2) + D(∂u)/(∂x) + E(∂u)/(∂y) + F = 0
This equation can be classified by replacing the differentials (∂ ⁄ ∂x and ∂ ⁄ ∂y) by a new variable, X and Y, respectively. Since a higher derivative can be decomposed into a series of first derivatives (∂2 ⁄ ∂x = ∂ ⁄ ∂x(∂ ⁄ ∂x)), eq. 1↑ becomes:
(2) AX2 + 2BXY + CY2 + DX + EY + F = 0
To analyze this, only the second-order terms are analyzed:
(3) AX2 + 2BXY + CY2 = 0
Dividing by Y2 results in the quadratic equation for the variable (X ⁄ Y):
(4) A(X ⁄ Y)2 + 2B(X ⁄ Y) + C = 0
The solution of this equation (u) behaves differently depends of the relation between A, B and C. The X ⁄ Y describes a characteristic line of the PDE, across which the variables are assumed to change. This analysis is known as method of characterics.
  • Elliptic equation
    If B2 − 4AC < 0 then eq. 1↑ is an elliptic equation. Laplace’s and Poisson’s equations are among the very famous elliptic equations. Steady subsonic fluids are governed by elliptic equations. Any change in the flow field is felt everywhere in the flow field because there are no real valued characteristic lines as the solution to the quadratic equation are imaginary numbers.
  • Parabolic equation
    If B2 − 4AC = 0 then eq. 1↑ is an parabolic equation. Heat diffusion equation, governing equations of traffic are among the very famous parabolic equations. This type of equation has one characteristic line.
  • Hyperbolic equation
    If B2 − 4AC > 0 then eq. 1↑ is an hyperbolic equation. Waves are the most important physical phenomena that can be described by hyperbolic equation. This type of equation has two characteristic lines and describes steady supersonic flows.
This classification can be generalized to higher dimensions than two, but the second-order equations are used most frequently in aerospace engineering.
A common and powerful approach to solve many linear partial differential equations is the separation of variables.