1. Use the implicit method to solve for the temperature distribution for TWO time steps of...

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1. Use the implicit method to solve for the temperature distribution for TWO time steps of a long, thin rod with a length of 30 cm. Assume the value of thermal diffusivity to be 0.63 cm²/s. Assume Ax = 6 cm and At=4 s. At t=0, the temperature of the rod is T₁ = 6 °C, T₂ = 22 °C, T3 = 16 °C, T4 = 3 °C. x=0 T₁ T₂ T3 T4 Boundary conditions are fixed for all times: T(x=0) = 22 °C and the other edge is insulated. Use MATLAB online solver to solve any set of simultaneous linear equations. 2. Given A and b, to solve the set of simultaneous linear equations represented by Ax=b, Student A employs the following scheme: (i) determines the inverse of A using LU decomposition, and (ii) determines x using x= A-¹b. Student B utilizes Gauss Elimination to solve Ax = b. For a matrix of size n, derive an expression for the number of FLOPS for each of the students. If you are not able to derive an expression with n, for a 25% penalty, determine the number of FLOPs for a value of n greater than 4. Provide detailed counting of the number of operations in a tabular format. As explained, the exact numerical solution (for x) is not required. 1. Use the implicit method to solve for the temperature distribution for TWO time steps of a long, thin rod with a length of 30 cm. Assume the value of thermal diffusivity to be 0.63 cm²/s. Assume Ax = 6 cm and At=4 s. At t=0, the temperature of the rod is T₁ = 6 °C, T₂ = 22 °C, T3 = 16 °C, T4 = 3 °C. x=0 T₁ T₂ T3 T4 Boundary conditions are fixed for all times: T(x=0) = 22 °C and the other edge is insulated. Use MATLAB online solver to solve any set of simultaneous linear equations. 2. Given A and b, to solve the set of simultaneous linear equations represented by Ax=b, Student A employs the following scheme: (i) determines the inverse of A using LU decomposition, and (ii) determines x using x= A-¹b. Student B utilizes Gauss Elimination to solve Ax = b. For a matrix of size n, derive an expression for the number of FLOPS for each of the students. If you are not able to derive an expression with n, for a 25% penalty, determine the number of FLOPs for a value of n greater than 4. Provide detailed counting of the number of operations in a tabular format. As explained, the exact numerical solution (for x) is not required.

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