Parabolic partial differential equation

A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments. To define the simplest kind of parabolic PDE, consider a real-valued function u ( x , y ) {displaystyle u(x,y)} of two independent real variables, x {displaystyle x} and y {displaystyle y} . A second-order, linear, constant-coefficient PDE for u {displaystyle u} takes the form and this PDE is classified as being parabolic if the coefficients satisfy the condition Usually x {displaystyle x} represents one-dimensional position and y {displaystyle y} represents time,and the PDE is solved subject to prescribed initial and boundary conditions. The name 'parabolic' is used because the assumption on the coefficients is the same as the conditionfor the analytic geometry equation A x 2 + 2 B x y + C y 2 + D x + E y + F = 0 {displaystyle Ax^{2}+2Bxy+Cy^{2}+Dx+Ey+F=0} to define a planar parabola. The basic example of a parabolic PDE is the one-dimensional heat equation, where u ( x , t ) {displaystyle u(x,t)} is the temperature at time t {displaystyle t} and at position x {displaystyle x} along a thin rod, and α {displaystyle alpha } is a positive constant (the thermal diffusivity). The symbol u t {displaystyle u_{t}} signifies the partial derivative of u {displaystyle u} with respect to the time variable t {displaystyle t} , and similarly u x x {displaystyle u_{xx}} is the second partial derivative with respect to x {displaystyle x} . For this example, t {displaystyle t} plays the role of y {displaystyle y} in the general second-order linear PDE: A = α {displaystyle A=alpha } , E = − 1 {displaystyle E=-1} , and the other coefficients are zero.