Problems

P15.1

Show that δ = Δ 1 + Δ − 1 / 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/equ24_1.jpg"/>

P15.2

Show that δ = 2 sinh h D 2 , μ = cosh h D 2 , μ = 1 + δ 2 4 1 / 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/equ24_2.jpg"/>

P15.3

Show that h 2 D 2 = Δ 2 − Δ 3 + 11 24 Δ 4 − 5 6 Δ 5 ⋯ = δ 2 − 1 12 δ 4 + 1 90 δ 6 ⋯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/equ24_3.jpg"/>

Hint: sinh⁻¹ x = x − x ³/3 + (3/40)x ⁵ − ⋯

P15.4

Show that h ² D ² y _n = [1/12] [−y _n+2 + 16y _n+1 − 30y _n + 16y _n−1 − y _n−2 ⋯].

P15.5

Given d 2 V d x 2 + V = 2 x , 0 < x < 1 and V 0 = 0 , d V d x x = 1 = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/equ24_4.jpg"/>

Determine V(1/2) by the method of weighted residuals. Use point matching at x = 1/2. Use trial function V = C ₁ sin Kx.

P15.6

Given d 2 V d x 2 + V 2 + 4 9 = 0 , 0 < x < 1 and V 1 = 0 , d V d x x = 0 = 0.2 V 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/equ24_5.jpg"/> Determine V(1/2) by the method of weighted residuals. Use point matching at x = 1/2.

Hint: try the trial function V = C ₁(1 − x)(1 − Kx) and determine K so that the boundary condition at x = 0 is satisfied.

346P15.7

Solve P15.6 by finite difference method.

P15.8

Given d 2 V d x 2 + V = 2 x , 0 < x < 1 , V 0 = 0 , d V d x at x = 1 is zero . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/equ24_6.jpg"/> Determine V(1/2) by the method of weighted residuals. Use point matching at x = 1/2.

P15.9

Solve the P15.8 by the finite-difference method. Use h = 1/2.

P15.10

Solve −d² V/dx ² = 1, 0 < x < 1, V(0) = V(1) = 0 by using finite-element method. Use three finite elements. Determine V(1/4).