Improved Space-Time GMQ-RBF Collocation Using Symmetric Variable Shape and Exponent Parameters

This paper introduces a novel approach to enhance the efficiency and accuracy of space-time radial basis function (RBF) collocation methods for solving partial differential equations (PDEs). We employ generalized multiquadric RBFs (GMQ-RBFs) equipped with symmetric variable shape parameters (SVSP) and symmetric variable exponents (SVE). Our method is compared against traditional RBF collocation techniques using constant shape parameters and exponents. Numerical results demonstrate that the proposed SVSP and SVE strategy outperforms conventional methods, resulting in significantly reduced condition numbers, lower interpolation errors, and faster convergence to the exact solution. The effectiveness of our approach is validated through solving the telegraph equation, underscoring its potential for broader applications in PDEs.