Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation

The generalized multiquadric radial basis function (φ_j=(x-x_j)²+c²]^β) has the exponent β and shape parameter c that play an important role in the accuracy of the approximation. In this study, we present a trigonometric variable shape parameter and exponent strategy and apply it to function interpolations and linear boundary value problems. Several numerical experiments with the uniformly spaced nodes show that the inverse multiquadric radial basis function (β = −0.5) with the trigonometric variable shape parameter c strategy results in the best accuracy for the one-dimensional interpolations; the trigonometric variable shape parameters and exponent strategy produces the best accuracy for the two-dimensional interpolations and linear boundary value problems. For the non-uniformly spaced nodes, the random variable shape parameter c and exponent β strategy produces the best accuracy for the two-dimensional boundary value problem.