An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation 1. = a² y_a² y at² Ox² (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. y=By+Cy+ Dy₁₁ + Ey i+1 (2) +1 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}+1 = [A] {y}'+ {y}+{BC}, where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}+1, {y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively.
An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation 1. = a² y_a² y at² Ox² (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. y=By+Cy+ Dy₁₁ + Ey i+1 (2) +1 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}+1 = [A] {y}'+ {y}+{BC}, where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}+1, {y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively.
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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![An equation representing a vibrating string of one unit long, fixed at both ends, is given in
Equation 1.
=
a² y_a² y
at² Ox²
(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.
y=By+Cy+ Dy₁₁ + Ey
i+1
(2)
+1
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}+1 = [A] {y}'+ {y}+{BC},
where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}+1,
{y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4000e28a-5cf8-47ab-bb9c-d9b564c038d2%2Fea63b7a0-01ed-4a4a-bbea-712858369fe0%2Fllxp097_processed.jpeg&w=3840&q=75)
Transcribed Image Text:An equation representing a vibrating string of one unit long, fixed at both ends, is given in
Equation 1.
=
a² y_a² y
at² Ox²
(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.
y=By+Cy+ Dy₁₁ + Ey
i+1
(2)
+1
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}+1 = [A] {y}'+ {y}+{BC},
where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}+1,
{y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively.
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