Explanation Given: The given initial boundary value problem is: ∂ u ∂ t = 4 ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , ∂ u ∂ x ( 0 , t ) = 0 , u ( π , t ) = 0 , t > 0 , u ( x , 0 ) = f ( x ) , 0 < x < π Formula used: The general heat flow model is: ∂ u ∂ t ( x , t ) = β ∂ 2 u ∂ x 2 , 0 < x < L ∂ u ∂ x ( 0 , t ) = U 1 , u ( L , t ) = U 2 , t > 0 u ( x , 0 ) = f ( x ) , 0 < x < L Then, the solution of heat flow problem is: u ( x , t ) = U 1 + ( U 2 − U 1 ) x L + ∑ n = 1 ∞ c n e − β ( n π L ) 2 t sin n π x L Calculation: Consider the initial boundary value problem: ∂ u ∂ t = 4 ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , ∂ u ∂ x ( 0 , t ) = 0 , u ( π , t ) = 0 , t > 0 , u ( x , 0 ) = f ( x ) , 0 < x < π Let U ( x , t ) = X ( x ) T ( t ) Substitute in the above equation, we get: X ″ ( x ) − K X ( x ) = 0 and T ′ ( t ) − β K T ( x ) = 0 . K is a constant. and, X ′ ( 0 ) T ( t ) = 0 X ( L ) T ( t ) = 0 , t > 0 Hence either T ( t ) = 0 for all t > 0 which implies that u ( x , t ) = 0 or X ′ ( 0 ) = X ( L ) = 0 . So, ignoring the trivial solution u ( x , t ) = 0 result in the boundary value problem, X ″ ( x ) − K X ( x ) = 0 and X ′ ( 0 ) = X ( L ) = 0 . Let X ( t ) = e r t . The auxiliary equation is: r 2 − K = 0 and only K < 0 gives the nontrivial solution. When K < 0 ; X ( x ) = C 1 cos − K x + C 2 sin − k x ... ( 1 ) Now, X ′ ( 0 ) = 0 gives C 2 = 0 . Equation (1) reduces to X ( x ) = C 1 cos − K x : Setting X ( L ) = 0 gives: C 1 cos − K x = 0 C 1 cos − K x = cos ( 2 n + 1 ) π 2 K = − ( 2 n + 1 2 L ) 2 π 2 = − ( n + 1 2 ) 2 π 2 L 2 Now X n ( x ) = a n cos [ ( n + 1 2 ) π L ] x , here a n ' s are arbitrary nonzero constants

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Fundamentals of Differential Equations and Boundary Value Problems

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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To find:

The formal solution for the given initial boundary value problem.