How to Solve Model Differential Equations Assignment Using MATLAB

July 26, 2025

Dr. Kevin Marshall

Canada

Differential Equations

has over 9 years of experience in mathematical modeling and MATLAB programming. He earned his Ph.D. in Applied Mathematics from Lakehead University, Canada.

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Differential equations are essential for modeling systems that change over time, making them invaluable in fields like physics, biology, and engineering. When used with MATLAB, these equations become even more effective, offering students a practical way to simulate and analyze complex real-world problems. Many university assignments now require not just solving equations but also implementing them in code, which is where many students seek professional help. This blog explores a unique application—modeling stress levels of astronauts aboard the International Space Station (ISS) when new visitors arrive. The situation presents a dynamic interaction between two groups: the existing crew and the newcomers, each affecting the other’s workload and stress. Such a system is ideal for differential equation modeling, as it involves continuous changes influenced by external conditions and internal feedback. We walk through the entire process: understanding the problem, translating it into a mathematical model, and implementing it using MATLAB's ODE-solving capabilities. By following this guide, students can gain a clearer understanding of how to approach similar assignments. Whether you're new to modeling or looking for structured guidance, this example offers a practical foundation for anyone needing help with differential equations assignment in an applied context.

SIMIODE-style Modeling for Students

Modeling with differential equations often follows a process-oriented approach where the main goal is to describe a dynamic system using mathematical relationships. Inspired by SIMIODE-style problems, the stress modeling case we explore here is particularly relevant to collaborative environments like the ISS, where individual and group dynamics significantly affect performance. This scenario involves two distinct groups—astronauts who are already stationed on the ISS and new visitors who have just arrived. Both groups contribute to and experience varying levels of stress. Our task is to model how these stress levels evolve over time, based on work schedules, task intensity, and mutual influence.

How to Solve Model Differential Equations Assignment Using MATLAB

Understanding the Problem Setup

To begin, it is essential to understand the dynamics between the two groups. The astronauts, already familiar with the station, follow a structured schedule filled with tasks, research, exercise, and rest. On the other hand, the visitors, while trained, are still adapting to the environment. They are expected to assist the astronauts and manage their own assigned duties.

This scenario demands a model that can handle the evolving stress levels in both groups. Stress and capabilities are the two core quantities we’ll focus on, with the primary emphasis on stress levels over time. We treat both astronauts and visitors as single entities representing average group behavior. The model should incorporate a daily work schedule, simulate different productivity levels, and account for the effects of minor schedule deviations on stress accumulation.

Modeling Strategy and Assumptions

Our initial modeling strategy focuses on simplicity and clarity. We define two primary stress functions: sa(t) for astronauts and sv(t) for visitors. The rate of change of stress for each group depends on their current stress levels, the intensity of the work they are doing, and whether they are in a working or resting period.

The general form of the differential equations is as follows:

dsa/dt = αa * W(t) * sa(t) * sv(t) – βa * sa(t) * (1 – W(t))

dsv/dt = αv * W(t) * sa(t) * sv(t) – βv * sv(t) * (1 – W(t))

Here, αa and αv are constants that determine how quickly stress accumulates during work, and βa and βv determine how quickly stress reduces during rest. The function W(t) switches between 1 and 0 depending on whether the group is working or resting at time t. The multiplication of sa(t) and sv(t) reflects the mutual influence of the two groups' stress levels. We assume that visitors are more prone to stress and recover more slowly, so their α and β values will reflect this.

Designing the Work Schedule

To simulate realistic stress accumulation and relief, we define a daily work schedule for astronauts and visitors. A typical day might look like this:

07:00 to 11:00 → Work
11:00 to 12:30 → Break
12:30 to 17:30 → Work
17:30 to 19:00 → Break
19:00 to 21:30 → Work
21:30 to 07:00 → Sleep/rest

From this, we define the work 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 evaluates whether the current time falls within any of the scheduled work periods. The mod operation ensures that time is treated cyclically across multiple days.

Building the Initial MATLAB Model

We begin our simulation by focusing only on the astronauts. We use simple constants for stress buildup and relief. Using MATLAB's ode and solve functionalities, we simulate stress over a 3–4 day period. Here’s how we set up the initial model:

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 graph will show the evolution of stress over time. When W(t) is 1 (working period), stress increases; during resting hours, it decreases. By adjusting the constants, we aim to find values that allow the astronauts' stress to stabilize over time rather than spiral upward.

Adding Visitor Interaction

With the astronaut-only model working, we introduce the visitors. These individuals are more susceptible to stress and less effective at destressing. We assign separate α and β values and define a system of differential equations that now models both groups:

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")

This code simulates the mutual stress dynamics between the astronauts and visitors. The first element in the solution vector corresponds to astronaut stress, while the second represents visitor stress. By plotting both together, we can see how each group influences the other. Adjusting the parameters helps us analyze under what conditions both groups reach a steady stress level or whether things become unstable.

Analyzing the Results

The simulations reveal a lot about the dynamics on the ISS. For instance, if stress accumulates faster than it is relieved, even a few visitors can dramatically raise stress levels across the crew. However, tweaking the constants or adjusting the work schedule can lead to more stable results. This sensitivity to parameters is an essential part of the modeling process and gives insight into how small interventions (like more rest periods or better task delegation) can significantly impact stress outcomes.

Future Enhancements

Although the model described here provides a good starting point, there are several areas for improvement. One idea is to create distinct W(t) functions for astronauts and visitors, reflecting different work intensities or rest periods. Another improvement could involve tracking individual stress responses using agent-based modeling rather than group averages.

Introducing productivity or capability functions tied to stress levels could provide a more nuanced understanding of performance over time. Lastly, adding random elements to represent unpredictable events on the ISS could make the model even more realistic.

Conclusion

Modeling stress using differential equations in MATLAB provides a practical and insightful way to apply mathematical concepts to real-life scenarios. This approach helps students understand how systems evolve over time by translating physical or psychological conditions into mathematical relationships.

For example, modeling astronaut stress on the ISS allows learners to explore how different work schedules and group interactions affect overall stress levels. By defining assumptions, formulating equations, and implementing them using MATLAB's ODE solvers, students gain valuable experience in both theory and computation.

This process enhances their ability to break down complex situations into manageable mathematical models and equips them with skills that are widely applicable in engineering, physics, and applied sciences. MATLAB makes it easier to visualize how changes in parameters influence the system, offering an interactive and flexible way to experiment with different outcomes.

For those who find this transition from theory to code challenging, seeking help with MATLAB assignment can be extremely beneficial. Expert guidance ensures students not only meet their academic goals but also truly grasp the modeling process, allowing them to refine, extend, and analyze models with confidence in future projects.