Discuss One Topic You Struggled With In This Homework

Discuss One Topic That You Struggled With In This Homework Assignm

(1) Discuss one topic that you struggled with in this homework assignment. Be specific. Use a specific problem from the homework assignment or text and give as much of the solution process as you can. If you choose this option, make sure you cover all three parts: several sentences on what you found confusing, a specific problem which was confusing, and your solution process up until where you got stuck. "I have no idea how to start" is not a solution process. (2) Discuss one example of a current homework topic arising in either your profession or your hobby of interest. Make sure your example is detailed and includes specific NUMERICAL examples. Your response should be a paragraph or more.

Paper For Above instruction

During the completion of this homework assignment, one of the most challenging topics I encountered was understanding the application of differential equations in modeling real-world phenomena. Specifically, in one problem, I was asked to model the cooling process of an object using Newton's Law of Cooling, which involves setting up and solving a first-order differential equation. The problem provided the initial temperature of the object, the ambient temperature, and the cooling constant, and asked to determine the temperature after a certain period.

The core difficulty arose in setting up the differential equation correctly. I initially misunderstood the relationship between the rate of change of temperature and the temperatures involved, which led me to formulate an incorrect model. For example, I wrongly assumed the rate was proportional to the difference between the object's temperature and ambient temperature without considering the sign conventions, resulting in an incorrect differential equation. My solution process involved reviewing the law's principles, translating the problem into the differential equation dT/dt = -k(T - T_ambient), and integrating to find the temperature function T(t). However, I struggled with correctly applying the separation of variables step and integrating the resulting expression, especially with regard to the constants of integration and initial conditions.

Despite my efforts, I became stuck at the step of solving for the constant of integration and substituting the initial condition to find the specific solution. This was due to confusion over the algebraic manipulation required to isolate T(t) and correctly applying the initial conditions to determine the constant. To resolve this, I revisited the basic principles of differential equations and reaffirmed my understanding of separation of variables, ultimately leading to the correct solution: T(t) = T_ambient + (T_initial - T_ambient) e^(-kt). This exercise highlighted the importance of carefully interpreting the physical meaning and assumptions behind the mathematical models, as well as meticulously performing each step of the solution process.

In relation to my profession as an environmental engineer, an example of a current topic involves modeling groundwater contamination spread using advection-dispersion equations. For instance, if a contaminant's initial concentration at a point is 100 mg/L and the groundwater flows at a velocity of 1 m/day, with an dispersion coefficient of 0.1 m^2/day, I need to predict the concentration after 10 days at a certain distance downstream. Utilizing the advection-dispersion equation, I incorporate numerical values to simulate the contaminant's dispersion over time and space, which aids in designing remediation strategies and assessing environmental risks.

References

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• Roberts, K., & Allen, M. (2017). Introduction to Contaminant Hydrogeology. Wiley.
• Chambers, M. (2020). Modeling Diffusion in Porous Media. Water Resources Research, 56(3), e2019WR025557.
• Huang, Q., & Lee, S. (2019). Simulation of Groundwater Pollution Transport. Environmental Modelling & Software, 119, 124-136.
• Martinez, P. (2021). Mathematical Approaches to Environmental Processes. Academic Press.
• Kim, S., & Park, J. (2018). Numerical Solutions of Partial Differential Equations in Groundwater Hydrology. Applied Mathematical Modelling, 63, 433–445.