Application of compact local integrated RBFs technique to solve fourth-order time-fractional diffusion-wave system

[1] M. Abbaszadeh, A. Ebrahimijahan, M. Dehghan, Application of compact local integrated RBF
(CLI-RBF) for solving transient forward and backward heat conduction problems with continuous
and discontinuous sources, Eng. Anal. Bound. Elem. 146 (2023) 733–748.
[2] M. Abbaszadeh, M. Dehghan, A Galerkin meshless reproducing kernel particle method for nu-
merical solution of neutral delay time-space distributed-order fractional damped diffusion-wave
equation, Appl. Numer. Math. 169 (2021) 44–63.
[3] M. Abbaszadeh, M. Dehghan, Numerical investigation of reproducing kernel particle Galerkin
method for solving fractional modified distributed-order anomalous sub-diffusion equation with
error estimation, Appl. Math. Comput. 392 (2021) 125718.

[4] O.P. Agrawal, A general solution for the fourth-order fractional diffusion-wave equation, Fract.
Calculus Appl. Anal. 3 (2000) 1–12.
[5] O.P. Agrawal, A general solution for the fourth-order fractional diffusionwave equation defined in
bounded domain, Comput. Struct. 79 (2001) 1497–1501.
[6] R. Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity,
J. Rheol. 27 (1983) 201–210.
[7] C.M. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation de-
scribing sub-diffusion, J. Comput. Phys. 227 (2007) 886–897.
[8] S. Chen, F. Liu, P. Zhuang, V. Anh, Finite difference approximations for the fractional Fokker-
Planck equation, Appl. Math. Model. 33 (2009) 256–273.
[9] M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys.
228 (2009) 7792–7804.
[10] M. Dehghan, B. Hooshyarfarzin, M. Abbaszadeh, Radial basis function partition of unity procedure
combined with the reduced-order method for solving Zakharov-Rubenchik equations, Eng. Anal.
Bound. Elem. 145 (2022) 93–116.
[11] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265
(2002) 229–248.
[12] A. Ebrahimijahan, M. Dehghan, M. Abbaszadeh, Simulation of plane elastostatic equations of
anisotropic functionally graded materials by integrated radial basis function based on finite differ-
ence approach, Eng. Anal. Bound. Elem. 134 (2022) 553-570.
[13] A. Ebrahimijahan, M. Dehghan, M. Abbaszadeh, Simulation of Maxwell equation based on an ADI
approach and integrated radial basis function-generalized moving least squares (IRBF-GMLS)
method with reduced order algorithm based on proper orthogonal decomposition, Eng. Anal.
Bound. Elem. 143 (2022) 397-417.
[14] A. Ebrahimijahan, M. Dehghan, M. Abbaszadeh, Numerical simulation of shallow water waves
based on generalized equal width (GEW) equation by compact local integrated radial basis function
method combined with adaptive residual subsampling technique, Nonlinear Dyn. 105 (2021) 3359–
3391.
[15] R. M. Hafez, M. A. Zaky, M. A. Abdelkawy, Jacobi spectral Galerkin method for distributed-order
fractional Rayleigh–Stokes problem for a generalized second grade fluid, Front. Phys. 7 (2020)
240.
[16] X. Hu, L. Zhang, On finite difference methods for fourth-order fractional diffusionwave and subd-
iffusion systems, Appl. Math. Comput. 218 (2012) 5019–5034.
[17] R. M. Hafez, M. A. Zaky, A. S. Hendy, A novel spectral Galerkin/petrov–Galerkin algorithm for the
multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth
solutions, Math. Comput. Simul. 190 (2021) 678–690.

[18] A. S. Hendy, M. A. Zaky, Combined Galerkin spectral/finite difference method over graded meshes
for the generalized nonlinear fractional Schr¨odinger equation, Nonlinear Dyn. 103 (2021) 2493–
2507.
[19] C. Li, Q. Yi, A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional
differential equations, J. Comput. Phys. 316 (2016) 614–631.
[20] T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the
fractional diffusion equation, J. Comput. Phys. 205 (2005) 719–736.
[21] F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J.
Comput. Appl. Math. 166 (2004) 209–219.
[22] F. Liu, C. Yang, K. Burrage, Numerical method and analytical technique of the modified anomalous
subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math. 231 (2009) 160–176.
[23] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods
for the space-time fractional advection-diffusion equation, Appl. Math. Comput. 191 (2007) 12—
20.
[24] N. Mai-Duy, T. Tran-Cong, An efficient indirect RBFN-based method for numerical solution of
PDEs, Numer. Methods Partial Differ. Equ. 21 (2005) 770–790.
[25] N. Mai-Duy, T. Tran-Cong, A compact five-point stencil based on integrated RBFs for 2D second-
order differential problems, J. Comput. Phys. 235 (2013) 302–321.
[26] S. Mohammadi Rick, R. Jalil, A. H. Refahi Sheikhani, Combination of Sinc and radial basis func-
tions for time-space fractional diffusion equations, J. Math. Model. 10 2022 315–329.
[27] A. Mohebbi, Finite difference and spectral collocation methods for the solution of semilinear
time fractional convection-reaction-diffusion equations with time delay, J. Appl. Math. Comput.
61 (2019) 635-656.
[28] X. Mu, J. Yang, H. Yao, A binary Caputo-Fabrizio fractional reproducing kernel method for the
time-fractional Cattaneo equation, J. Appl. Math. Comput. 9 (2023) 1–37.
[29] Z.M. Odibat, Computational algorithms for computing the fractional derivatives of functions, Math.
Comput. Simul. 79 (2009) 2013–2020.
[30] K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and
Integration to Arbitrary Order, Academic Press, 1974.
[31] K.B. Oldham, J. Spanier, The Fractional Calculus, New York and London, Academic Press, 1974.
[32] I. Podulbny, Fractional Differential Equations, New York, Academic Press, 1999.
[33] S.A. Sarra, Integrated multiquadric radial basis function approximation methods, Comput. Math.
Appl. 51 (2006) 1283–1296.

[34] A. Soltanpour Moghadam, M. Arabameri, M. Barfeie, Numerical solution of space-time variable
fractional order advection-dispersion equation using radial basis functions, J. Math. Model. 10
(2022) 549–562.
[35] Z.Z. Sun, X.N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer.
Math. 56 (2006) 193–209.
[36] C. Tadjeran, M.M. Meerschaert, H.P. Scheffler, A second-order accurate numerical approximation
for the fractional diffusion equation, J. Comput. Phys. 213 (2006) 205–213.
[37] N. Thai-Quang, N. Mai-Duy, C. D. Tran, T. Tran-Cong, High-order alternating direction implicit
method based on compact integrated-RBF approximations for unsteady/steady convection-diffusion
equations, CMES - Comput. Model. Eng. Sci. 89 (2012) 189–220.
[38] C. Tien, N. Mai-Duy, C.D. Tran, T. Tran-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.
[39] C. Tien, N. Thai-Quang, N. Mai-Duy, C.-D. Tran, T. Tran-Cong, A three-point coupled compact
integrated RBF scheme for second-order differential problems, CMES - Comput. Model. Eng. Sci.
104 (2015) 425–469.
[40] W. Wess, The fractional diffusion equation, J. Math. Phys. 27 (1996) 2782–2785.
[41] S.B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Com-
put. Phys. 216 (2006) 264–274.
[42] P. Zhuang, F. Liu, Implicit difference approximation for the time fractional diffusion equation, J
Appl. Math. Comput. 22 (2006) 87–99.
[43] P. Zhuang, F. Liu, V. Anh, I. Turner, New solution and analytical techniques of the implicit numer-
ical methods for the anomalous sub-diffusion equation, SIAM J. Numer. Anal. 46 (2008) 1079–
1095.