[1] Abd-Elhameed, W.M., Atta, A.G., Youssri, Y.H. (2022). “Shifted fifth-kind Chebyshev polynomials Galerkin based procedure for treating fractional diffusion-wave equation”, International Journal of Modern Physics C, 33(8), 2250102, doi:10.1142/S0129183122501026.
[2] Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H. (2022). “A fast Galerkin approach for solving the fractional Rayleigh–Stokes problem via sixth-kind Chebyshev polynomials”, Mathematics, 10(11), 1843, doi:10.3390/math10111843.
[3] Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H. (2022). “Modal shifted fifth-kind Chebyshev Tau integral approach for solving heat conduction equation”, Fractal and Fractional, 6(11), 619, doi:10.3390/fractalfract6110619.
[4] Atta, A.G., Youssri, Y.H. (2022). “Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel”. Computational and Applied Mathematics, 41(8), 381, doi:10.1007/s40314-022-01904-1.
[5] Bayona, V., Flyer, N., Fornberg, B. (2019). “On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries”, Journal of Computational Physics, 380, 378-399, doi:10.1016/j.jcp. 2018.12.036.
[6] Bayona, V., Flyer, N., Fornberg, B., Barnett, G.A. (2016). “On the role of polynomials in RBFFD approximations: I. Interpolation and accuracy”, Journal of Computational Physics, 321, 21- 38, doi:10.1016/j.jcp. 2016.05.005.
[7] Fasshauer, G.E. (2007). “Mesh-free Approximation Methods with MATLAB, volume 44, World Scientific Publishers, Singapore, doi:10.1142/9789812706331.
[8] Fasshauer, G.E., Zhang, J.G. (2007). “On choosing 'optimal' shape parameters for RBF approximation”, Numerical Algorithms, 45, 345-368, doi:10.1007/s11075-007-9072-8.
[9] Fornberg, B., Flyer, N. (2015).“A primer on radial basis functions with applications to the geosciences”, SIAM, Philadelphia, doi:10.1137/1.9781611974041.
[10] Fornberg, B., Flyer, N. (2015). “Solving PDEs with radial basis functions”, Acta Numerica, 24, 215-258, doi:10.1017/S096249291500005X.
[11] Fornberg, B., Piret, C. (2008). “On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere”, Journal of Computational Physics, 227, 2758-2780, doi:10.1016/j.jcp.2007.11.013.
[12] Fornberg, B., Bayona, V., Flyer, N., Barnett, G.A. (2017). “On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs”, Journal of Computational Physics, 332, 257-273, doi:10.1016/j.jcp.2016.12.041.
[13] Haghighi, D., Abbasbandy, S., Shivanian, E. (2023). “Study of the Fragile Points Method for solving two-dimensional linear and nonlinear wave equations on complex and cracked domains”, Engineering Analysis with Boundary Elements, 146, 44-55, 10.1016/j.enganabound.2022.09.036.
[14] Hardy, R.L. (1971). “Multiquadric equations of topography and other irregular surfaces”, Journal of Geophysical Research, 76, 1905-1915, doi:10.1029/JB076i008p01905.
[15] Hosseinzadeh, N., Shivanian, E., Fairooz, M.Z., Chegini, T.G. (2025). “A robust RBF-FD technique combined with polynomial enhancements for valuing European options in jump-diffusion frameworks”, International Journal of Dynamics and Control, 13, 212, 10.1007/s40435-025-01722-6.
[16] Kansa, E.J. (1990). “Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics I: Surface approximations and partial derivative estimates”, Computers & Mathematics with Applications, 19, 127-145, doi:10.1016/0898-1221(90)90270-T.
[17] Kansa, E.J. (1990). “Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations”, Computers & Mathematics with Applications, 19, 147-161. doi:10.1016/0898-1221(90)90271-K.
[18] Moscoso, M., Bayona, V., Kindelan, M. (2011). “Optimal constant shape parameter for multiquadric based RBF-FD method”, Journal of Computational Physics, 230, 7384-7399, doi:10.1016/j.jcp.2011.06.025.
[19] Moscoso, M., Bayona, V., Kindelan, M. (2012). “Optimal variable shape parameter for multiquadric based RBF-FD method”, Journal of Computational Physics, 231, 2466-2481, doi:10.1016/j.jcp.2011.12.036.
[20] Phys, J.C. (2006). “Scattered node compact finite difference-type formulas generated from radial basis functions”, Engineering Analysis with Boundary Elements, 212, 99-123, doi:
10.1016/j.jcp.2005.05.030.
[21] Rahimi, A., Shivanian, E. (2023). “An efficient RBF-FD method using polyharmonic splines alongside polynomials for the numerical solution of two-dimensional PDEs held on irregular domains and subject to Dirichlet and Robin boundary conditions”, International Journal of Nonlinear Analysis and Applications, (1), doi:
10.22075/IJNAA.2023.28181.3826.
[22] Schaback, R. (1995). “Error estimates and condition numbers for radial basis function interpolants”, Advances in Computational Mathematics, 3, 251-264, doi:10.1007/BF02123482.
[23] Shirzadi, M., Dehghan, M., Foroush Bastani, A. (2021). “A trustable shape parameter in the kernel-based collocation method with application to pricing financial options”, Engineering Analysis with Boundary Elements, 126, 108-117, doi:10.1016/j.enganabound.2021.02.005.
[24] Shivanian, E., Khodabandehlo, H.R., Abbasbandy, S. (2022). “Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix”, Engineering with Computers, 38, 204-219, doi:10.1007/s00366-020-01114-1.
[25] Shivanian, E., Jafarabadi, A., Chegini, T.G., Dinmohammadi, A. (2025). “Analysis of a time-dependent source function for the heat equation with nonlocal boundary conditions through a local meshless procedure”, Computational & Applied Mathematics, 44, 282, 10.1007/s40314-025-03246-3.
[26] Shivanian, E., Hajimohammadi, Z., Baharifard, F., Parand, K., Kazemi, R. (2023). “A novel learning approach for different profile shapes of convecting–radiating fins based on shifted Gegenbauer LSSVM”, New Mathematics and Natural Computation, 19 (1), 195-215, 10.1142/S1793005723500060.