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]]>After figuring that the previous attempt was a challenge we moved on to altering the boundaries of the path differently. We took a look at altering the boundary where instead of it oscillating too much, it would just be one bump. Therefore, we decided to make the boundaries -L+A(y^2+2πy) and L+A(y^2+2πy), where the amplitude, A, we looked at was ¼. Finally, we ran into challenges with this one as well and one next step we took was to look at a profile that is an even extension of a linear function of y. While we were juggling between profiles, Jenn let us know that we had to account for a variable very important in convolution. Therefore, next week we will take a look at the profiles to make sure all the variables needed are accounted for.

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]]> P → ( r ( ( k / β ) – δ_1) ) / ( b k / β ) S → 0 F → δ_1 / β

We found that:

1) There will never be a case in which the radical in the eigenvalue is a real number and the entire eigenvalue is negative.

2) For almost all cases the radical will be imaginary and therefore the entire eigenvalue will be negative

3) The only way that the radical would be real is if δ_1 is greater than (4 k^2 / β^2) / (r+4 k /β)

4) For the third eigenvalue to be negative, δ_1 >= k /β which is very unlikely to occur because β would have to be very small

5) It could be possible that the first two eigenvalues are positive if δ_1 >= k \ β

6) Our questions

– How are all three eigenvalues sometimes negative?

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]]>The post Scroungers Report appeared first on Computing Change with Partial Differential Equations.

]]>- Producers find food on their own
- Scroungers eat the left overs of the producers or they run into food on their own (however, the latter is less likely)
- Food has logistic growth and is eaten at a rate corresponding with the producers and scroungers

We’re moving towards simplifying the model. In doing this, we can take a step back and really try to understand the minute details of the models.

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]]>The post Streaming through Week 2 appeared first on Computing Change with Partial Differential Equations.

]]>For the remainder of the week, our team took a further look at convolution and its relevance to the Stream Function. This process involved us taking several partial derivatives of the Stream Function with respect to different variables. Once we took a look at this, we decided to take the Laplacian of the Stream Function and it surprisingly yielded something that had some relation to the velocity u(x,y). Moreover, some of the literature that was provided demonstrated some u (velocity) that we are trying to prove is a steady state solution to the partial differential equation that we are working with. As of now we are continuing to prove that the function u(x,y) (velocity) is a steady state solution to the differential and we are doing so by using a Dirac function.

Part of the process this week was to gain a better understanding of what the three functions (Stream Function, velocity, and vorticity) mean in relation to each other and how to move forward in our project.

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]]>- Further practice with Mathematica
- Learn to use LaTeX for typing mathematics
- Start work on individual research problems

**Our Research Teams: **

- Pattern Formation in Animal Foraging (#scroungers) with Nessy Tania
- Caira Anderson
- Issa Susa

- Vortex Patch Dynamics (#vortexpatches) with Jen Beichman
- Victoria Camarena
- Christian Madrigal
- Sasha Shrouder

**Suggested Readings: **

- Pattern Formation
- Vortex Patch Dynamics

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]]>The post Our Ideas for Improving Smith College and Higher Education appeared first on Computing Change with Partial Differential Equations.

]]>- Provide inter-student mentor options, or at least have lunch presentations/meetings with students where they address a lot of the insecurities people have about their skills and smarts and share information about how to navigate their new environments as college students, STEM students, and Smith students. Maybe along with presentation of major, a student panel in the beginning of the semester.

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]]>The post Week 1: PDE and Mathematical Computing Minicourses appeared first on Computing Change with Partial Differential Equations.

]]>- Learn basics of PDEs and computing with Mathematica
- Form research groups
- Write MathFest presentation abstracts

**Mathematica Materials: **

Mathematica Day 1: Mathematica Notebook

Mathematica Day 2: Solving ODEs and steady state solutions

Mathematica Day 3: Fourier Series Demo

Mathematica Day 3: Heat Separation

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