Explanation Given: The boundary value problem is ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + cot θ r 2 ∂ u ∂ θ = 0 , r > c , 0 < θ < π and the boundary conditions are u ( c , θ ) = 0 and lim r → ∞ u ( r , θ ) = − E z = − E r cos θ . Calculation: The given boundary value problem is, ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + cot θ r 2 ∂ u ∂ θ = 0 Consider the equation, u ( r , θ ) = R ( r ) Θ ( θ ) (1) Take separation of the variables of the above equation and separation constant is λ , r 2 R ″ + r R ′ R = λ and − Θ ″ + cot θ Θ ′ Θ = λ The equation of variable R is, r 2 R ″ + r R ′ − λ R = 0 (2) The equation of variable θ is, sin θ Θ ″ + cos θ Θ ′ + λ sin θ Θ = 0 (3) Substitute x = cos θ in the above equation, ( 1 − x 2 ) d 2 Θ d x 2 − 2 x d Θ d x + λ Θ = 0 , − 1 ≤ x ≤ 1 The above equation is the form of Legendre’s equation which has continuous derivatives on the closed interval [ − 1 , 1 ] is the Legendre polynomials P n ( x ) corresponding to the λ = n ( n + 1 ) where n is nonnegative integer. The general solution of the equation (3) is, Θ ( θ ) = P n ( cos θ ) (4) Equation (2) is the form of Cauchy Euler equation and substitute λ = n ( n + 1 ) then the general solution of the equation (2) is expressed as, R ( r ) = c 1 r n + c 2 r − ( n + 1 ) (5) Substitute the value of Θ ( θ ) from equation (4) and the value of R ( r ) from equation (5) in equation (1), u ( r , θ ) = ( c 1 r n + c 2 r − ( n + 1 ) ) P n ( cos θ ) Rewrite the term c 1 = A n and c 2 = B n in the above equation and apply superposition principle in the above equation, u ( r , θ ) = ∑ n = 0 ∞ ( A n r n + B n r − ( n + 1 ) ) P n ( cos θ ) (6) At given boundary condition u ( c , θ ) = 0 &

6th Edition

ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

See similar textbooks

Videos

Question

Chapter 14.3, Problem 12E

To determine

To show: The given solution of the boundary value problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.3, Problem 11E

Chapter 14.3, Problem 13E

Students have asked these similar questions

Problems Determine I, and Iy for the cross-sections 6.11 See the steel tables in the Appendix Table A3. |x= = " -2x8 STEEL PLATE CENTERED -W14x82

Problems Construct shear and moment diagrams using the equilib- rium method. Indicate the magnitudes of Vmax and Mmax. 7.4 8=2K/FT. W=2K/FT. 10' 10'

Problems Determine I, and Iy for the cross-sections 6.8 89 2" 1011 2" ส 10 |x= ly=

Chapter 14 Solutions

EBK ADVANCED ENGINEERING MATHEMATICS

Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

The given solution of the boundary value problem.

Solution Summary: The author explains the given solution of the boundary value problem.

Videos

To show: The given solution of the boundary value problem.

Want to see the full answer?

Chapter 14 Solutions