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week1 - Week 1 notes Math 716 1 PDE Order Linear Nonlinear...

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Week 1 notes: Math 716 1 PDE: Order, Linear & Nonlinear A differential equation that involves more than one independent variable is called a partial differential equation , abbreviated as PDE. The order of the highest derivative occurring in the PDE is defined as the order of the PDE. For instance, the following 2 u ∂x 2 + 2 u ∂y 2 = 0 (1) is a second order PDE for u ( x,y ), which is usually called the Laplace’s equation in two variables. Another example of a second order linear differential equation is the so-called heat equation for u ( x,t ) in one space variable with a source: ∂u ∂t = κ 2 u ∂x 2 + g ( x,t ) for some constant κ (2) As with ODEs, if the differential operator L is linear, i.e. for constant c 1 and c 2 , L ( c 1 u 1 + c 2 u 2 ) = c 1 L u 1 + c 2 L u 2 (3) then L u = g is called a linear differential differential equation. For instance, equations (1) and (2) above are linear differential equations since in those cases we can check easily the linearity of the corresponding differential operators L = 2 ∂x 2 + 2 ∂y 2 or L = ∂t 2 ∂x 2 , acting on appropriate class of functions 1 Note, that in (1), g = 0. In that case, the PDE is linear and homogeneous. Partial differential equations that are not linear are called nonlinear ; the following semi-linear heat equation is an example: u t u xx = u 2 , (4) where for notational brevity, the partial derivatives are denoted by subscripts. As for ODEs, linear PDEs are usually simpler to analyze/solve than nonlinear PDEs. Non- linear PDEs arise in many applications; but general theory rarely exists. 2 Some physical applications where PDEs arise PDEs abound in physical sciences and engineering. Here we give some examples. 2.1 Dispersion of pollutants in a river stream Consider predicting concentration ρ (measured in some units, say Kg/m 3 ) of some pollutant as a function of position x and time t , in a river. We denote the x domain by Ω R 3 . Let the fluid velocity field in the river (domain Ω) be given by u ( x ,t ) (measured in some units, say m/sec ). We will assume that the pollutant is passive, meaning it’s inertia is neglected. Consider a small but fixed volume V centered around at a point x (See Figure 1), entirely contained in Ω. The 1 Classically, this would be C 2 ( R 2 ) in the first case and C 2 in x and C 1 in time for the second. However, this set can be extended further by introducing the concept of weak solutions. 1
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. x V δ V n F Figure 1: Control Volume V for pollutant rate of change of mass of fluid inside that volume equals d dt integraltext V ρ ( x ,t ) dV . Assuming no pollutant is created or destroyed within V , this must equal the net inward flow of pollutants into V : integraldisplay ∂V F · n dA where F ( x ,t ) is the flux (measured in units like Kg/m 2 /sec ) of pollutants at any point x and n is the outward normal to the boundary of volume V . A reasonable expression for flux is: F = u ρ κ ρ where the first term is due to fluid motion carrying pollutants, usually called advection , while
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