Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 9 - Linear, First-Order Pa

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 9 - Linear, First-Order Pa

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CHAPTER 9 Linear, First-Order Partial Diferential Equations I n this chapter, we will discuss the frst oF several special classes oF pde s that can be solve via analytical techniques. In particular, we will investigate linear, frst-order pde s a ( x , t ) u t + b ( x , t ) u x + f ( x , t ) u = g ( x , t ) ,( 9 . 1 ) where u = u ( x , t ) is our dependent variable, and the Functions a , b , f , and g are given. Our goal is to develop a systematic method For determining all Functions u that satisFy the pde . A little geometric intuition will help use devise a rather useFul technique For solving such equations, and For that reason we will review a Few notions From multivariate calculus. Suppose that f ( x , y ) is a diFFerentiable Function oF two variables, and let v =( v 1 , v 2 ) denote a unit vector in R 2 . Defnition 9 . 0 . 2 . The directional derivative oF f in the direction oF the vector v is given by lim h 0 f ( x + v 1 h , y + v 2 h ) - f ( x , y ) h , provided that this limit exists. It is straightForward to prove (see your multivariate calculus text) that this limit is equal to the dot product f ( x , y ) v , thereby providing us with an equivalent (and computationally convenient) deF- inition oF the directional derivative. The directional derivative measures the 236
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linear , first - order partial differential equations 237 instantaneous rate of change of f in the direction of the vector v . If we choose the unit vector v =( 1, 0 ) , which points in the direction of the positive x -axis, then the directional derivative is simply f ( x , y ) v = ± f x , f y ² ( 1, 0 )= f x , the partial derivative of f with respect to x . Similarly, the directional derivative in the direction of v 0, 1 ) is given by f / y . We now use the notion of directional derivatives to give a quick, clean solution of a special case of Equation ( 9 . 1 ). A linear, homogeneous, constant-coefFcient equation. Consider the pde α u t + β u x = 0, ( 9 . 2 ) where α and β are non-zero constants. Our goal is to determine all functions u ( x , t ) that satisfy this equation. Observe that since the gradient of u is given by u u x , u t ) , the pde ( 9 . 2 ) is equivalent to the equation u ( x , t ) ( β , α 0. In other words, the solutions of the pde are precisely those functions u ( x , t ) whose directional derivative in the direction of the vector ( β , α ) is 0 . Geometrically, this implies that u ( x , t ) must remain constant as we move along any line parallel to the vector ( β , α ) . These lines have slope α / β in the xt plane, and the general equation for such lines is t α / β ) x + constant. Equivalently, u ( x , t ) must remain constant along any line of the form α x - β t = C , where C is an arbitrary constant. Since u remains constant along each such line, u depends only on the value of the constant C . In other words, u depends only upon the quantity α x - β t , but otherwise has no restrictions at all. It follows that the general solution of Equation ( 9 . 2 ) is u ( x , t f ( α x - β t ) ,( 9 . 3 ) where f is any (differentiable) function of a single variable.
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238 Defnition 9 . 0 . 3 . In the above example, the “geometric” technique used to solve the pde is called the
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This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 9 - Linear, First-Order Pa

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