University of Qom Civil Infrastructure Researches 2783-140X 11 1 2025 05 22 Developing Relationships for Determining the Optimal Shape Parameter in the Multiquadric Meshless Method Developing Relationships for Determining the Optimal Shape Parameter in the Multiquadric Meshless Method 69 86 3460 10.22091/cer.2025.12181.1598 FA Hanieh Talebi Kalan Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran 0009-0006-1178-9045 Sara Mohsenzadeh Golafzani Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran 0009-0006-6332-6229 Reza Babaee Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran 0000-0002-3068-3602 Ehsan Jabbari Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran 0000-0002-6345-8567 Journal Article 2025 01 22 The accuracy and efficiency of the Multiquadric Radial Basis Function (MQ-RBF) method are highly sensitive to the choice of the shape parameter. This study aims to identify optimal relationships for determining the shape parameter in solving common partial differential equations (PDEs) encountered in water engineering. To this end, the procedure for reconstructing and solving PDEs using the MQ-RBF method is first outlined. Then, optimal values for the shape parameter are derived based on domain length and the number of computational centers. Based on these findings, empirical formulas for the optimal shape parameter are proposed, which significantly reduce computational cost. The performance of the proposed formulas is compared with exact solutions and existing empirical relations. Results show that, unlike previous approaches, the new formulas offer high accuracy, with negligible errors across different examples-sometimes approaching zero. Moreover, compared to optimization-based techniques, the proposed method dramatically improves computational speed by eliminating the need for iterative algorithms. The study also investigates stability conditions, showing that for diffusion, advection-diffusion, and Burgers’ equations, the maximum allowable time step depends on the diffusion coefficient, flow velocity, and Reynolds number. The accuracy and efficiency of the Multiquadric Radial Basis Function (MQ-RBF) method are highly sensitive to the choice of the shape parameter. This study aims to identify optimal relationships for determining the shape parameter in solving common partial differential equations (PDEs) encountered in water engineering. To this end, the procedure for reconstructing and solving PDEs using the MQ-RBF method is first outlined. Then, optimal values for the shape parameter are derived based on domain length and the number of computational centers. Based on these findings, empirical formulas for the optimal shape parameter are proposed, which significantly reduce computational cost. The performance of the proposed formulas is compared with exact solutions and existing empirical relations. Results show that, unlike previous approaches, the new formulas offer high accuracy, with negligible errors across different examples-sometimes approaching zero. Moreover, compared to optimization-based techniques, the proposed method dramatically improves computational speed by eliminating the need for iterative algorithms. The study also investigates stability conditions, showing that for diffusion, advection-diffusion, and Burgers’ equations, the maximum allowable time step depends on the diffusion coefficient, flow velocity, and Reynolds number. https://cer.qom.ac.ir/article_3460_4babf7576642f832ed4d6272aa165018.pdf