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Consider the following boundary value problem: d?u du +--u = 0, dx du(0) = 0, u(4) = 1 (Eq 1) dx2 dx In this question, you will need to discretise this equation across 4 equal segments (i.e. with five points xg, X1, ... X4 where xo=0 and x4=4). (a) Write the centered finite difference equation that approximates this equation at the general node "j". Assume a uniform mesh spacing of h. (b) Write the difference equation at j = 0, 1, 2 and 3, making sure to use centered finite difference to approximate the derivative BC at x = 0 and replacing any boundary values with the given BCs. (c) Using the relaxation method, write the system matrix that describes the boundary value problem above. REPLACE h with an appropriate numerical value. (d) What noticeable feature does this matrix system have? How would you solve it for the unknown values y? (Note that you do not have to solve the system matrix).

Consider the following boundary value problem: d?u du +--u = 0, dx du(0) = 0, u(4) = 1 (Eq 1) dx2 dx In this question, you will need to discretise this equation across 4 equal segments (i.e. with five points xg, X1, ... X4 where xo=0 and x4=4). (a) Write the centered finite difference equation that approximates this equation at the general node "j". Assume a uniform mesh spacing of h. (b) Write the difference equation at j = 0, 1, 2 and 3, making sure to use centered finite difference to approximate the derivative BC at x = 0 and replacing any boundary values with the given BCs. (c) Using the relaxation method, write the system matrix that describes the boundary value problem above. REPLACE h with an appropriate numerical value. (d) What noticeable feature does this matrix system have? How would you solve it for the unknown values y? (Note that you do not have to solve the system matrix).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Consider the following boundary value problem: d?u du du(0) 3D 0, и (4) 3D 1 dx (Eq 1) u = 0, dx - dx2 In this question, you will need to discretise this equation across 4 equal segments (i.e. with five points x0, X1, ... X4 where xo=0 and x4=4). (a) Write the centered finite difference equation that approximates this equation at the general node "j". Assume a uniform mesh spacing of h. (b) Write the difference equation at j = 0, 1, 2 and 3, making sure to use centered finite difference to approximate the derivative BC at x = 0 and replacing any boundary values with the given BCs. (c) Using the relaxation method, write the system matrix that describes the boundary value problem above. REPLACE h with an appropriate numerical value. (d) What noticeable feature does this matrix system have? How would you solve it for the unknown values y? (Note that you do not have to solve the system matrix). (e) Clearly describe the process you would use to implement the shooting method on this problem to find the solution of the ODE. You must explicitly include responses to the following 2 items (1) The set of 1st order equation you must solve. (ii) Clearly state the boundary conditions, constraint and adjustable parameter
Transcribed Image Text:Consider the following boundary value problem: d?u du du(0) 3D 0, и (4) 3D 1 dx (Eq 1) u = 0, dx - dx2 In this question, you will need to discretise this equation across 4 equal segments (i.e. with five points x0, X1, ... X4 where xo=0 and x4=4). (a) Write the centered finite difference equation that approximates this equation at the general node "j". Assume a uniform mesh spacing of h. (b) Write the difference equation at j = 0, 1, 2 and 3, making sure to use centered finite difference to approximate the derivative BC at x = 0 and replacing any boundary values with the given BCs. (c) Using the relaxation method, write the system matrix that describes the boundary value problem above. REPLACE h with an appropriate numerical value. (d) What noticeable feature does this matrix system have? How would you solve it for the unknown values y? (Note that you do not have to solve the system matrix). (e) Clearly describe the process you would use to implement the shooting method on this problem to find the solution of the ODE. You must explicitly include responses to the following 2 items (1) The set of 1st order equation you must solve. (ii) Clearly state the boundary conditions, constraint and adjustable parameter
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