Development of radial basis functionmethods for solving the viscous Burgersequation with applications to diffusivecoagulation-fragmentation equations

This paper develops a novel numerical scheme for solving the two-dimensional generalized Burgers equation, with direct applications to diffusive coagulation-fragmentation models. The method combines a second-order Crank-Nicolson/leapfrog time-stepping scheme with a shifted surface spline radial basis function collocation method (RBFCM) for spatial discretization. This approach delivers exceptional computational efficiency while maintaining conditional stability—a property rigorously established through a novel matrix-based analysis that employs Lagrange polynomial approximations of the radial basis functions. We derive a priori error estimates demonstrating $\mathcal{O}(\tau h^m)$ convergence, revealing that uniform node distributions significantly enhance accuracy compared to scattered configurations. Comprehensive numerical experiments, including challenging coagulation-fragmentation test cases, validate the theoretical framework and demonstrate the method's robustness, accuracy, and practical effectiveness across diverse problem regimes. 2000 Mathematics Subject Classification: 65M12; 65D12; 35L65; 82C22; 65N15.