Category: Linear Algebra

1. For the 1-D bar system, assume k = 30 W/m−◦C, L = 0.4 m, α = 50◦C, β = 20◦C,
and q(x) = 4000x(L − x) W/m3 . For N = 4, write down the three equations in the
system Au = f and solve for the vector u. No coding necessary for this part. Plot
the temperature solution along the bar using the temperatures at the five nodes. Use
Python for plotting purposes.
2. Consider the following data:
xi 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ui 22.5 26.4 28.4 32.5 37.4 40.2 32.0 29.0 25.2
(a) Calculate the first-order accurate finite difference approximations to the first deriva-
tive u'(x) at all xi.
(b) Next, calculate the corresponding second-order accurate finite difference approximations.
(c) Compare the two sets of approximations by plotting the approximations against x. Again, use Python for this purpose
I need python code file and resulted graphs separately. Because i have to submit python code and report separately.

1. For the 1-D bar system, assume k = 30 W/m−◦C, L = 0.4 m, α = 50◦C, β = 20◦C,
and q(x) = 4000x(L − x) W/m3. For N = 4, write down the three equations in the
system Au = f and solve for the vector u. No coding necessary for this part. Plot
the temperature solution along the bar using the temperatures at the five nodes. Use
Python for plotting purposes.
2. Consider the following data:
xi 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ui 22.5 26.4 28.4 32.5 37.4 40.2 32.0 29.0 25.2
(a) Calculate the first-order accurate finite difference approximations to the first derivative
u′(x) at all xi.
(b) Next, calculate the corresponding second-order accurate finite difference approximations.
(c) Compare the two sets of approximations by plotting the approximations against x.
Again, use Python for this purpose.

1. For the 1-D bar system, assume k = 30 W/m−◦C, L = 0.4 m, α = 50◦C, β = 20◦C,
and q(x) = 4000x(L − x) W/m3. For N = 4, write down the three equations in the
system Au = f and solve for the vector u. No coding necessary for this part. Plot
the temperature solution along the bar using the temperatures at the five nodes. Use
Python for plotting purposes.
2. Consider the following data:
xi 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ui 22.5 26.4 28.4 32.5 37.4 40.2 32.0 29.0 25.2
(a) Calculate the first-order accurate finite difference approximations to the first derivative
u′(x) at all xi.
(b) Next, calculate the corresponding second-order accurate finite difference approximations.
(c) Compare the two sets of approximations by plotting the approximations against x.
Again, use Python for this purpose.

1. For the 1-D bar system, assume k = 30 W/m−◦C, L = 0.4 m, α = 50◦C, β = 20◦C,
and q(x) = 4000x(L − x) W/m3. For N = 4, write down the three equations in the
system Au = f and solve for the vector u. No coding necessary for this part. Plot
the temperature solution along the bar using the temperatures at the five nodes. Use
Python for plotting purposes.
2. Consider the following data:
xi 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ui 22.5 26.4 28.4 32.5 37.4 40.2 32.0 29.0 25.2
(a) Calculate the first-order accurate finite difference approximations to the first derivative
u′(x) at all xi.
(b) Next, calculate the corresponding second-order accurate finite difference approximations.
(c) Compare the two sets of approximations by plotting the approximations against x.
Again, use Python for this purpose.

Problems 1, 2, 3. Please show work and steps. Also please include explanation, I am confused and would like to properly understand how to solve.

The mathematical method of employing equations to solve for variables is used in algebra. By manipulating these variables with the use of mathematical frameworks, linear algebra elevates it to a new level. We discover that we may represent whole systems of equations as matrices in linear algebra.

Module 3 will build on our knowledge of linear programming and to represent large models efficiently and compactly using index notation. We’ll use multiperiod production planning as a concrete example, but these ideas apply very broadly
Note: In the Lecture 3 – Part 2, I provide a 3-step process for working through formulettes with indices. I believe I need to revise that method to the following for more clarity:
Identify each “for each” statement, covert to a “for all” notation
Plug in values from formulette (e.g., plug in values for indices that take on specific values)
Develop the formulette and solve the equation
Finish by adding summations and/or controlling “uncontrolled” indices
The key distinction is plugging in values earlier in the process. I realize while working through the problems, I was doing this earlier than I initially recommended. Hopefully, this new process simplifies the math and alleviates any confusion.

OVERVIEW: As the title suggests, this course is a operations research (OR) topics course geared towards solving problems in energy systems. This course focuses on mathematical programing via optimization. Optimization is one of the primary tools used in OR. In addition, this course will also cover related topics that blend statistics and simulation — the two other pillars of OR. Primary topics covered are:
Mathematical optimization techniques – linear, integer, and multi-objective programming.
Optimization modeling tools – Excel Solver
Model and results analysis – sensitivity and post-optimality analysis.
Optimization applications – military and private sector optimization applications with a focus on energy systems (electric power and fuel).
Related OR methods and applications – time series forecasting (regression models), simulation and uncertainty.
InstructionsPlease provide answers to all prompts and upload all complete homework files (written + Excel) as attachments.