Explanation Given information: The equation P ( x , t ) = P 0 + P 1 e − k x sin ( A t − k x ) models the ground penetration near a toxic waste dump, where the P denotes the amount of pollution at a depth x and time t measured in feet and years respectively.The remaining variables P 0 , P 1 , A and K are constants. The diffusion equation, ∂ P ∂ t = c 2 ∂ 2 P ∂ x 2 . Formula used: The derivative with respect to a variable for the function of two or more than two variables is partially differentiating with respect to that variable and keeping all other variables as constants. The second order partial derivatives for a function of two variables z = g ( x , y ) are as follows: With respect to x , ∂ 2 f ∂ x 2 g ( x , y ) = ∂ ∂ x ( ∂ g ∂ x ) = ( g x ) x = g x x With respect to y , ∂ 2 f ∂ y 2 g ( x , y ) = ∂ ∂ y ( ∂ g ∂ y ) = ( g y ) y = g y y The derivative of the function e a x is, d d x e a x = a e a x . Trigonometric derivatives are as follows: d d x sin a x = a cos a x and d d x cos a x = − a sin a x Proof: Consider the function, P ( x , t ) = P 0 + P 1 e − k x sin ( A t − k x ) which models the ground penetration near a toxic waste dump, where the P denotes the amount of pollution at a depth x and time t measured in feet and years respectively. The remaining variables P 0 , P 1 , A and K are constants. We have to show the function satisfies, diffusion equation, ∂ P ∂ t = c 2 ∂ 2 P ∂ x 2 , where c is a constant. Since, we know he derivative with respect to a variable for the function of two or more than two variables is partially differentiating with respect to that variable and keeping all other variables as constants. Left hand side of the equation is, ∂ ∂ t ( P ( x , t ) ) = ∂ ∂ t ( P 0 + P 1 e − k x sin ( A t − k x ) ) = A P 1 e − k x cos ( A t − k x ) Since, P 0 , P 1 , A , c and K are constants, so multiplication of all constants will be a constant

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.3, Problem 55PS

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To show: The function P(x,t)=P0+P1e−kxsin(At−kx) satisfies the diffusion equation.

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Chapter 11.3, Problem 54PS

Chapter 11.3, Problem 56PS

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Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

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The function P ( x , t ) = P 0 + P 1 e − k x sin ( A t − k x ) satisfies the diffusion equation.

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To show: The function P(x,t)=P0+P1e−kxsin(At−kx) satisfies the diffusion equation.

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