Solving Poisson Equations by the MN-Curve Approach

In this paper, we adopt the choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Luh's choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolation. Here, we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be very successful, not only by its easy accessibility but also by its striking accuracy and efficiency. Our emphases are on the highly reliable prediction of the optimal value of the shape parameter and the extremely small approximation errors of the numerical solutions to the differential equations. We hope that our approach can be accepted by both mathematicians and non-mathematicians.

Automated summary

In this paper, we propose an easily accessible approach to solving differential equations using a choice theory of shape parameters. Our approach is characterized by its high accuracy and efficiency in predicting optimal shape parameter values and achieving extremely small approximation errors in numerical solutions.