lecture4 - Types of Solutions to Ficks Second Law Error...

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Types of Solutions to Fick’s Second Law Error function solutions – short-time or infinite systems Trigonometric function solutions – long-time or finite systems ( ) Dt l >> size system ( ) Dt l ~ size system Methods of solution: (1) Fourier transforms; (2) separation of variables, typically employed to obtain long-time solutions; (3) Laplace transforms, the most general and powerful method. Definition of Error Function Gaussian function: () = +∞ −∞ = 0 at 1 as 0 as 0 exp 2 x x x x y () π = + dx x 2 exp ( ) 2 exp 0 2 = + dx x Error function: () ( ) ( ) = 0 2 0 2 exp exp du u du u x erf x () ( ) = x du u x erf 0 2 exp 2
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Properties of Error Function 0 ) 0 ( = erf 1 ) ( = erf ( ) x erf x erf = ) ( ( ) x erf x erfc = 1 ) ( Complimentary error function: ( ) () 2 exp 2 x dx x derf = π () x x x erf small at 2 Method of Fourier Transform and Superposition Fourier transform gets rid of the space-derivative, and thus transforms the
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lecture4 - Types of Solutions to Ficks Second Law Error...

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