Multiquadric radial basis functions combined with compact discretization to estimate solutions of two dimensions nonlinear elliptic type partial differential equations are presented. The scattered grid network with continuously varying step sizes helps tune the solution accuracies depending upon the location of high oscillation. The radial basis functions employing a nine-point grid network are used to improve the functional evaluations by compact formulation, and it saves memory space and computing time. A detailed description of convergence theory is presented to estimate the error bounds. The analysis is based on a strongly connected graph of the Jacobian matrix, and their monotonicity occurred in the scheme. It is shown that the present strategy improves the approximate solution values for the elliptic equations exhibiting a sharp changing character in a thin zone. Numerical simulations for the convection-diffusion equation, Graetz-Nusselt equation, Schr¨odinger equation, Burgers equation, and Gelfand-Bratu equation are reported to illustrate the utility of the new algorithm.
[1] V. Bayona, M. Moscoso, M. Carretero, and M. Kindelan, RBF-FD formulas and convergence properties, J. Comput. Phys., 229 (2010), 8281–8295.
[2] S. Banei and K. Shanazari, On the convergence analysis and stability of the RBF-adaptive method for the forward- backward heat problem in 2D, Appl. Numer. Math., 159 (2021), 297—310.
[3] D. Britz, Digital simulation in electrochemistry, Springer, Berlin, 2005.
[4] R. Feng and J. Duan, High accurate finite differences based on RBF interpolation and its application in solving differential equations, J. Sci. Comput., 76 (2018), 1785–1812.
[5] A. I. Fedoseyev, M. J. Friedman, and E. J. Kansa, Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method, Int. J. Bifurc. Chaos, 10(2) (2000), 481–492.
[6] J. H. Ferziger and M. Peric, Computational methods for fluid dynamics, Springer, Berlin, 2002.
[7] P. Henrici, Discrete variable methods in ordinary differential equations, John Wiley and Sons, New York, 1962.
[8] N. J. Higham, Accuracy and stability of numerical algorithms, SIAM, Philadelphia, 2002.
[9] L. Jianyu, L. Siwei, Q. Yingjian, and H. Yaping, Numerical solution of elliptic partial differential equation using radial basis function neural networks, Neural Networks, 16 (2003), 729—734.
[10] N. Jha and N. Kumar, A fourth-order accurate quasi-variable meshes compact finite-difference scheme for two- space dimensions convection-diffusion problems, Adv. Differ. Equ., 64 (2017), 1–13.
[11] N. Jha and L. K. Bieniasz, A fifth (six) order accurate, three-point compact finite difference scheme for the numerical solution of sixth order boundary value problems on geometric meshes, J. Sci. Comput., 64 (2015), 898–913.
[12] N. Jha and B. Singh, Fourth-order compact scheme based on quasi-variable mesh for three-dimensional mildly nonlinear stationary convection-diffusion equations, Numer. Methods Partial Differ. Equ. (2020), 1–27.
[13] N. Jha and N. Kumar, An exponential expanding meshes sequence and finite difference method adopted for two- dimensional elliptic equations, Int. J. Model. Simul. Sci., 7(2) (2016), 1–17.
[14] A. Komech and E. Kopylova, Stationary Schro¨dinger equation, in Dispersion Decay and Scattering Theory, John Wiley and Sons, 2012.
[15] H. Liu, B. Xing, Z. Wang, and L. Li, Legendre neural network method for several classes of singularly perturbed differential equations based on mapping and piecewise optimization technology, Neural Process. Lett., 51 (2020), 2891–913.
[16] M. Malekzadeh, S. Hamzehei-Javaran, and S. Shojaee, Improvement of numerical manifold method using nine- node quadrilateral and ten-node triangular elements along with complex Fourier RBFs in modeling free and forced vibrations, J. Appl. Comput. Mech., 7(4) (2021), 2049–2063.
[17] W. F. Mitchell, A collection of 2D elliptic problems for testing adaptive grid refinement algorithms, Appl. Math. Comput., 220 (2013), 350–364.
[18] R. K. Mohanty, G. Manchanda, and A. Khan, Operator compact exponential approximation for the solution of the system of 2D second order quasilinear elliptic partial differential equations, Adv. Differ. Equ., 47 (2019), 1–36.
[19] A. Mohebbi and M. Saffarian, Implicit RBF meshless method for the solution of two-dimensional variable order fractional cable equation, J. Appl. Comput. Mech., 6(2) (2020), 235–247.
[20] A. Mohsen, A simple solution of the Bratu problem, Comput. Math. Appl., 67 (2014), 26–33.
[21] A. D. Polyanin, Handbook of linear partial differential equations for engineers and scientists, Chapman and Hall/CRC, Boca Raton, 2001.
[22] G. So¨derlind, I. Fekete, and I. Farag´o, On the zero-stability of multistep methods on smooth nonuniform grids, BIT Numer. Math., 58 (2018), 1125–1143.
[23] G. I. Shishkin, and L. P. Shishkina, Difference methods for singular perturbation problems, CRC Press, 2010.
[25] J. W. Thomas, Numerical partial differential equations: conservation laws and elliptic equations, Springer, New York, 1999.
[26] C. M. T. Tien, N. Mai-Duy, C. D. Tran, and T. T. Cong, A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations, Comput. Math. Appl., 72 (2016), 2364–2387.
[27] R. S. Varga, Matrix Iterative Analysis, Springer, Berlin, 2000.
[28] G. B. Wright and B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions, J. Comput. Phys., 212 (2006), 99–123.
[29] D. M. Young, Iterative solution of large linear systems, Academic Press, New York, 1971.