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pde1 - PDE EXAMPLES Math 21a BURGER EQUATION ft f fx = fxx...

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PDE EXAMPLES Math 21a An equation which involves partial derivatives for an unknown function f ( x,y ) is called a partial differential equation or shortly a PDE . The topic of PDE’s would fill a course by itself. Finding and understanding solutions of PDE’s can be difficult. The topic is introduced here in the context of partial differentiation . You should be able to verify that a given function f ( t,x ) satisfies a specific PDE and know some examples. It is useful as well to understand how these equations are derived. LAPLACE EQUATION. f xx + f yy = 0 . A stationary tempera- ture distribution on a plate satisfies this equation. 1) Verify that f ( x,y ) = x 3 3 xy 2 satisfies the Laplace equation. ADVECTION EQUATION. f t = f x . Models transport in a one- dimensional medium. It is also called a transport equation . 2) Verify that f ( t,x ) = e - ( x + t ) 2 satisfy the advection equation f t ( t,x ) = f x ( t,x ). WAVE EQUATION. f tt = f xx . For fixed time t , the function x mapsto→ f ( t,x ) describes a string at that time.
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