Solving PDEs with a Hybrid Radial Basis Function: Power-Generalized Multiquadric Kernel

Radial Basis Function (RBF) kernels are key functional forms for advanced solutions of higher-order partial differential equations (PDEs). In the present study, a hybrid kernel was developed for meshless solutions of PDEs widely seen in several engineering problems. This kernel, Power-Generalized Multiquadric - Power-GMQ, was built up by vanishing the dependence of e, which is significant since its selection induces severe problems regarding numerical instabilities and convergence issues. Another drawback of e-dependency is that the optimal e value does not exist in perpetuity. We present the Power-GMQ kernel which combines the advantages of Radial Power and Generalized Multiquadric RBFs in a generic formulation. Power-GMQ RBF was tested in higher-order PDEs with particular boundary conditions and different domains. RBF-Finite Difference (RBF-FD) discretization was also implemented to investigate the characteristics of the proposed RBF. Numerical results revealed that our proposed kernel makes similar or better estimations as against to the Gaussian and Multiquadric kernels with a mild increase in computational cost. Gauss-QR method may achieve better accuracy in some cases with considerably higher computational cost. By using Power-GMQ RBF, the dependency of solution on e was also substantially relaxed and consistent error behavior were obtained regardless of the selected e accompanied.