How to Solve Differential Equations Modeling Assignment Using MATLAB SCUDEM
Mathematical modeling using differential equations is a powerful approach to simulate and understand real-world systems, particularly in dynamic and high-stakes environments like space missions. By capturing how quantities change over time, differential equations help explain complex interactions, such as the psychological and physical stress experienced by astronauts during long-term missions. MATLAB, known for its robust computational capabilities, offers a flexible platform for building, testing, and refining such models.
In this blog, we focus on a scenario inspired by SCUDEM-style challenges, where students are asked to develop a model that describes the stress levels of astronauts and newly arrived visitors aboard the International Space Station (ISS). These kinds of problems are perfect examples of how mathematical modeling bridges theory and application. We’ll walk through the complete process—from interpreting a problem statement and making modeling assumptions to implementing the system of differential equations in MATLAB and visualizing the results. For students who find these concepts challenging, getting help with differential equations assignment tasks can make a significant difference. Understanding how to structure models and solve them computationally builds valuable skills. Whether you're preparing for a competition or completing coursework, mastering this modeling workflow will deepen your understanding of both mathematics and real-world problem solving.
Understanding the Challenge
We consider a scenario inspired by a SCUDEM problem where astronauts are working on the ISS and a new group of visitors arrives to collaborate on tasks and experiments. The visitors, while trained, are not accustomed to the ISS environment and thereby contribute to increased stress for everyone onboard. The challenge is to model the stress dynamics of both groups using a system of differential equations.
We define two variables to represent the stress of astronauts (sa) and visitors (sv), and construct a mathematical model based on the assumption that stress increases during work and decreases during rest. We also assume that the stress of one group affects the other. Our model also takes into account the daily work schedule and explores how different productivity patterns impact long-term stress.
Planning the Mathematical Model
To begin with, we assume that the change in stress is proportional to the amount of current stress and whether an individual is working or resting. For simplicity, capability is incorporated into the stress equations through parameters. We define the stress equations for astronauts and visitors as follows:
Where W(t) is a binary function that returns 1 when the crew is working and 0 when they are resting. Parameters αa and αv represent the rate of stress accumulation, while βa and βv represent the rate of stress reduction. The system models mutual influence: as visitor stress rises, astronaut stress may also rise, and vice versa. This gives us a dynamic and interdependent stress system to simulate.
Creating the Work Function W(t)
To represent the ISS working schedule, we define a work function that reflects periods of activity and rest. Based on a general daily routine, we implement the function W(t) as follows:
function w = W(t)
normT = mod(floor(t),24) + (t - floor(t));
if (normT >= 7) && (normT <= 11)
w = 1;
elseif (normT >= 12.5) && (normT <= 17.5)
w = 1;
elseif (normT >= 19) && (normT <= 21.5)
w = 1;
else
w = 0;
end
end
This function uses a 24-hour clock and returns 1 during work periods and 0 otherwise. It repeats daily to simulate long-term stress dynamics.
Implementing the Model in MATLAB
We start by modeling the astronauts alone to test for a stable stress pattern. We define the parameters and solve the ODE using MATLAB’s numerical solver:
alphaA = 0.3;
betaA = 2;
initValA = 0.3;
dsa = @(t, y) alphaA * y * W(t) - betaA * y * (1 - W(t));
initF = ode(ODEFcn = dsa, InitialTime = 7, InitialValue = initValA);
sol = solve(initF, 7, 100);
plot(sol.Time, sol.Solution, '-o')
This simulation models how astronaut stress evolves over several days. The parameters can be adjusted to reflect more or less stressful environments. Once the behavior looks reasonable, we add the visitors into the system.
alphaV = 0.9;betaV = 1.5;
initValV = 0.5;
ds = @(t, y) [
alphaA * y(1) * y(2) * W(t) - betaA * y(1) * (1 - W(t));
alphaV * y(1) * y(2) * W(t) - betaV * y(2) * (1 - W(t))
];
F = ode(ODEFcn = ds, InitialTime = 7, InitialValue = [initValA; initValV]);
sol = solve(F, 7, 100);
plot(sol.Time, sol.Solution, '-o')
In the plot, one curve represents astronaut stress, and the other represents visitor stress. This visualization helps assess whether the stress levels stabilize or escalate uncontrollably.
Analyzing the Results
The model’s output depends heavily on the parameter values. For instance, increasing α values or reducing β values leads to faster stress accumulation or slower recovery, possibly causing system instability. Experiment with different parameters to simulate various scenarios, such as consistent high workload, rest followed by intensive work, or small schedule changes. This allows analysis of which schedule minimizes stress and maximizes productivity. Observing the system’s sensitivity to these factors gives insight into optimal scheduling and workload management strategies.
Possible Improvements
This initial model can be extended in several ways to make it more realistic and applicable to broader scenarios. Some possibilities include:
- Incorporating task complexity with variable workloads.
- Modeling individual crew members instead of groups.
- Making capability a dynamic variable influenced by stress.
- Using empirical data to refine parameter values.
These enhancements can make the model suitable for more complex simulations in aerospace operations, mental health studies, or industrial team dynamics.
Extending the Model Further
Advanced users can explore modeling productivity, fatigue thresholds, or schedule optimization. Other additions could include environmental effects or integrating machine learning to estimate parameters from observed data. The flexibility of MATLAB allows all these improvements and more. With its robust ODE solving features, visualization tools, and scripting capabilities, MATLAB is ideal for expanding your basic model into a full-fledged simulation system.
Final Thoughts
This blog has shown a practical approach to solving SCUDEM-style challenges using MATLAB. We began by carefully understanding the problem, then developed a basic differential equations model to simulate the stress levels of astronauts and visitors aboard the ISS. By incorporating a work function that reflects daily schedules, we were able to simulate how stress changes over time and analyze different scenarios using MATLAB's ODE solvers.
Although the initial model is quite simple, it establishes a strong foundation for building more advanced simulations. With further customization, students can add complexity such as varying task difficulty, dynamic capability, or even real-world data for parameter tuning.
This kind of modeling not only sharpens analytical thinking but also enhances technical skills in MATLAB—skills that are valuable for both academic success and real-world applications. Whether you're participating in a math modeling competition or just working on a class project, mastering this technique is essential. If you're looking for help with MATLAB assignment tasks like this, understanding how to break down complex systems into solvable components using differential equations is a great start. With the right tools and support, any student can gain confidence in tackling mathematical models and interpreting their outcomes.