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Proceedings of the 2019 Undergraduate Mathematics Day

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Partial differential equations (PDEs) are useful for describing a wide variety of natural phenomena, but analytical solutions of these PDEs can often be difficult to obtain. As a result, many numerical approaches have been developed. Some of these numerical approaches are based on the particular solutions. Derivation of these particular solutions are challenging. This work is about how the Laplace operator can be written in a more convenient form when it is applied to radial basis functions and then use this form to derive the (closed-form) particular solution of the Poisson’s equation in 3D with the oscillatory radial function in the forcing term.


Partial differential equations, Radial basis functions, Particular solution, Oscillatory radial basis functions




This paper was presented Saturday, Nov. 2, 2019, as part of Undergraduate Mathematics Day at the University of Dayton. Launched in 2003, Undergraduate Mathematics Day is held in odd-numbered years and alternates with the Biennial Alumni Career Seminar. The conference coincides with the annual Schraut Memorial Lecture, named Kenneth “Doc” Schraut, a mathematics faculty member from 1940 to 1978 and department chair from 1954 to 1970.

The 2019 Schraut lecturer was Tommy Ratliff, professor of mathematics at Wheaton College in Norton, Massachusetts, who presented the lecture “So How Do You Detect a Gerrymander?” His recent research hastaken up mathematical questions related to redistricting and gerrymandering, and he has been involved with the Metric Geometry and Gerrymandering Group based at Tufts University and Massachusetts of Technology.

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Mathematics Commons