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The Pixel Array Method

Bijl, L.C. van der (2019) The Pixel Array Method.

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Abstract:The Pixel array method is a method that was proposed only a few years ago. It is used for finding all solutions of a set of equations within a certain bound. The pixel array method has a few strong properties. For example, it does not return any false negatives when used. However, because of this property, the error can be quite large. In this paper, a formal definition of the pixel array method will be given with some illustrating examples. The method will then be adjusted such that it is also capable of returning the steady state solutions of a differential equation, this new method will be referred to as the PASS method. MATLAB is used for calculating the steady state answers of discrete approximations of the heat equation and the Fisher-KPP equation with given boundary conditions such that these can be compared to their actual solutions. The heat equation is approximated quite well by the PASS method and we see that the PASS solutions converge to the actual solutions as we increase the resolution. However, the PASS approximation of the Fisher-KPP method does not converge as nicely. Even though this equations only has $2$ solutions, the PASS method returns hundreds and the amount keeps growing when increasing the resolution. Most of these PASS solutions are for boundary conditions that do not actually have a steady state solution. This is due to the tolerance, which has to be higher than some threshold in order for it to be valid. Our conclusion is therefore that the pixel array method is a nice method with strong properties. However, further research is needed with respect to this tolerance and the possibilities of lowering the threshold in order for it to give meaningful results for relations that change rapidly.
Item Type:Essay (Bachelor)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:31 mathematics
Programme:Applied Mathematics BSc (56965)
Link to this item:http://purl.utwente.nl/essays/79679
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