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Professor of Mathematics Xiaoyan Liu discusses numerical methods for solving systems of integral equations with cardinal splines. Liu spoke in the Presidents’ Dining Room Tuesday for the weekly faculty lecture sponsored by the La Verne Academy. / photo by Noel Cabrera

Professor of Mathematics Xiaoyan Liu discusses numerical methods for solving systems of integral equations with cardinal splines. Liu spoke in the Presidents’ Dining Room Tuesday for the weekly faculty lecture sponsored by the La Verne Academy. / photo by Noel Cabrera

Kaila Williams
Staff Writer

It was an afternoon of integral equations and cardinal splines as Professor of Mathematics Xiaoyan Liu presented her sabbatical research Tuesday in the President’s Dining Room.

Liu went on sabbatical during the fall 2014 semester and wrote research papers regarding Fredholm and Volterra integral equations, Bézier curves, and the EH interpolation spline.

Liu began the lecture by explaining the approximation theory and spline functions, which were the basis of her research papers.

“Approximation theory is the research of how to use simpler functions … to approximate complicated or unknown functions,” Liu said.

Liu used the example of a company that wants to predict profit numbers in five years. To find that unknown number, you would use the approximation theory to estimate the profit.

Liu went on to explain that spline functions are piece-wised functions, meaning that they are made up of pieces to make one whole function. There are different kinds of splines functions like polynomial splines, trigonometric splines, exponential splines and more.

One research paper, “Numerical Methods for Solving Systems of Fredholm Integral Equations with Cardinal Splines,” was presented at the International Conference on Nonlinear Problems in Aviation and Aerospace World Congress conference in Narvik, Norway. The focus of this research paper was to find alternative methods for solving systems of integral equations.

Liu drew graphs and lines on a whiteboard to further give explanations of the research to the audience.

“It was interesting when she explained splines and gave an example of the spline function she was using,” said Sasha Webb, senior math major. “I liked that she gave the history behind the function as well.”

Webb took Liu’s calculus two class during her freshman year.

Liu presented another paper, “Row Convergence of Padé Approximation,” at Mahidol University in Thailand, which took a look at French mathematician Henri Padé’s Padé table. This paper was published in the journal Methods and Applications of Analysis.

Along with the research, Liu shared photos of her trips to Norway and Thailand for the conferences.

Junior math major Rafael Gutierrez was impressed that Liu presented her work abroad.

“As a student, I often find myself thinking that the only thing professors do is teach, when in reality, most of their time is taken up doing research and attending conferences,” Gutierrez said.

“A Kind of QT – Bézier Curves and its Applications,” another one of Liu’s papers, focused on Bézier curves with two shape parameters that can be altered by changing the values of the parameters.

This work was published in the book Advances in Future Manu­facturing Engineering: Pro­ceedings of the 2014 Inter­national Conference on Future Manufacturing Engineering.

Liu also wrote another paper, “Solving Systems of Volterra Integral Equations with Cardinal Splines.”

This was similar to the previous paper she worked on.

Liu also had her article, “The EH Interpolation Spline and Its Approximation,” published in the journal Abstract and Applied Analysis. This article focused on a new interpolation spline, EH interpolation spline, which is like the cubic Hermite interpolation spline.

“It isn’t often that we see people from the math department giving these lectures so it was cool to see one given,” Webb said.

Liu has 26 papers published in various mathematics journals. Two collaborated papers have been submitted for publication and Liu has more research papers in progress.

Kaila Williams can be reached at kaila.williams@laverne.edu.

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