An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation 1. Ot² (1) Equation 1 can be written in the form of Equation 2 using the central finite difference approximation of the second derivative of their respective variables. The subscripts represent locations, while superscripts represent time steps. y¹¹ = By₁₁₁ + C y₁ + Dy₁₁₁+ Ey (2) a) Express B, C, D and E in terms of A t and A x. Note that the coefficient of y' is equal to one. b) The string is discretised into five nodes equally distanced, with A t = 0.2. Write down equations as a result of applying Equation 2 to the domain. Subscripts must be replaced with correct node numbers, while superscripts can be retained. not required to incorporate the boundary and initial conditions yet. c) Assemble the three equations above in the form of matrix {y}+= [A] {y}'+ {y}+{BC}, where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}t+1, {y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively. d) Initially, the string is at rest at the profile {y2 y3 y4} = {0.4 0.7 0.3}. Use the answer in part c) to find {y2 y3 y4}¹. Show all the necessary calculations. e) Thus, further calculations to show one cycle (periodical process) are obtained by tabulating a table.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve for e) only, I have posted a) b) c) d) on the Barterby and received the answer.

Transcribed Image Text:An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation 1. Ot² (1) Equation 1 can be written in the form of Equation 2 using the central finite difference approximation of the second derivative of their respective variables. The subscripts represent locations, while superscripts represent time steps. y¹¹ = By₁₁₁ + C y₁ + Dy₁₁₁+ Ey (2) a) Express B, C, D and E in terms of A t and A x. Note that the coefficient of y' is equal to one. b) The string is discretised into five nodes equally distanced, with A t = 0.2. Write down equations as a result of applying Equation 2 to the domain. Subscripts must be replaced with correct node numbers, while superscripts can be retained. not required to incorporate the boundary and initial conditions yet. c) Assemble the three equations above in the form of matrix {y}+= [A] {y}'+ {y}+{BC}, where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}t+1, {y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively. d) Initially, the string is at rest at the profile {y2 y3 y4} = {0.4 0.7 0.3}. Use the answer in part c) to find {y2 y3 y4}¹. Show all the necessary calculations. e) Thus, further calculations to show one cycle (periodical process) are obtained by tabulating a table.

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