[1] M. Abbaszadeh and H. Amjadian, Second-order finite difference/spectral element formulation for solving the fractional advection-diffusion equation, Communications on Applied Mathematics and Computation, 2 (2020), pp. 653–669.
[2] M. Abbaszadeh and M. Dehghan, An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate, Numerical Algorithms, 75 (2017), pp. 173– 211.
[3] 
,
Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction– diffusion problem on complex domains, Computers & Mathematics with Applications, 79 (2020), pp. 876–888.
[4] 
,
A meshless numerical investigation based on the RBF-QR approach for elasticity problems, AUT Journal of Mathematics and Computing, 1 (2020), pp. 1–15.
[5] 
,
Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection–diffusion equation, Engineering with Computers, 37 (2021), pp. 873–889.
[6] 
,
The proper orthogonal decomposition modal spectral element method for two-dimensional viscoelastic equation, Thin-Walled Structures, 161 (2021), p. 107429.
[7] M. Abbaszadeh, M. Dehghan, A. Khodadadian, and T. Wick, Legendre spectral element method
(LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models, Applicable Analysis, 101 (2022), pp. 2279–2294.
[8] M. Abbaszadeh, M. Dehghan, and Y. Zhou, Alternating direction implicit-spectral element method
(ADI-SEM) for solving multi-dimensional generalized modified anomalous sub-diffusion equation, Computers & Mathematics with Applications, 78 (2019), pp. 1772–1792.
[9] M. Abbaszadeh, A. Ghoreyshi, and M. Dehghan, A finite block method framework for nonlinear fractional integro-differential equations, Mathematical Methods in the Applied Sciences, (2025), pp. 1–19.
[10] M. Abbaszadeh, A. Khodadadian, M. Dehghan, and T. Wick, Analysis of a Legendre spectral element method (LSEM) for the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models, arXiv preprint arXiv:1904.06263, (2019).
[11] M. Abbaszadeh, A. B. Salec, and A. S. Jebur, Integrated radial basis function technique to simulate the nonlinear system of time fractional distributed-order diffusion equation with graded time-mesh discretization, Engineering Analysis with Boundary Elements, 156 (2023), pp. 57–69.
[12] W. Abd-Elhameed, Y. Youssri, and A. Atta, Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials, Boundary Value Problems, 2024 (2024), p. 102.
[13] S. Adeeb, Introduction to Solid Mechanics and Finite Element Analysis using Mathematica, Kendall Hunt, Dubuque, Iowa, USA, 2011.
[14] N. Ahmadi, S. Wang, and G. Karniadakis, Pharmacometrics modeling via physics-informed neural networks: Integrating time-variant absorption rates and fractional calculus for enhancing prediction accuracy, arXiv preprint arXiv:2412.21076, (2024).
[15] A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations, Differential Equations, 46 (2010), pp. 660–666.
[16] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, Journal of Computational Physics, 280 (2015), pp. 424–438.
[17] A. A. Alikhanov and C. Huang, A high-order L2 type difference scheme for the time-fractional diffusion equation, Applied Mathematics and Computation, 411 (2021), p. 126545.
[18] J. Angulo, M. Ruiz-Medina, V. Anh, and W. Grecksch, Fractional diffusion and fractional heat equation, Advances in Applied Probability, 32 (2000), pp. 1077–1099.
[19] K. E. Atkinson, An Introduction to Numerical Analysis, John Wiley & Sons, New York, 2008.
[20] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3, World Scientific, Toh Tuck Link, Singapore, 2012.
[21] D. Baleanu, S. S. Sajjadi, A. Jajarmi, and O. Defterli¨ , On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: a new fractional analysis and control, Advances in Difference Equations, 2021 (2021), p. 234.
[22] A. Bhrawy and M. Zaky, Numerical algorithm for the variable-order Caputo fractional functional differential equation, Nonlinear Dynamics, 85 (2016), pp. 1815–1823.
[23] A. H. Bhrawy and M. Zaky, An improved collocation method for multi-dimensional space–time variableorder fractional Schr¨odinger equations, Applied Numerical Mathematics, 111 (2017), pp. 197–218.
[24] A. H. Bhrawy and M. A. Zaky, A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations, Journal of Computational Physics, 281 (2015), pp. 876– 895.
[25] A. H. Bhrawy, M. A. Zaky, and D. Baleanu, New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method, Rom. Rep. Phys, 67 (2015), pp. 340–349.
[26] G. W. Bohannan, Analog fractional order controller in temperature and motor control applications, Journal of Vibration and Control, 14 (2008), pp. 1487–1498.
[27] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, New York, 2011.
[28] W. Bu, A. Xiao, and W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, Journal of Scientific Computing, 72 (2017), pp. 422–441.
[29] R. L. Burden and J. D. Faires, Numerical Analysis, Brooks Cole, Boston, Massachusetts, USA, 2010.
[30] K. Burrage, A. Cardone, R. D’Ambrosio, and B. Paternoster, Numerical solution of time fractional diffusion systems, Applied Numerical Mathematics, 116 (2017), pp. 82–94.
[31] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Science & Business Media, New York, 1988.
[32] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, Germany, 2006.
[33] J. Cao, C. Li, and Y. Chen, High-order approximation to Caputo derivatives and Caputo-type advectiondiffusion equations (II), Fractional Calculus and Applied Analysis, 18 (2015), pp. 735–761.
[34] M. C. Caputo and D. F. Torres, Duality for the left and right fractional derivatives, Signal Processing, 107 (2015), pp. 265–271.
[35] C. M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics, 227 (2007), pp. 886–897.
[36] H. Chen, H. Qiao, W. Wei, and J. Li, Time fractional diffusion equation based on Caputo fractional derivative for image denoising, Optics & Laser Technology, 168 (2024), p. 109855.
[37] W. Chen, L. Ye, and H. Sun, Fractional diffusion equations by the Kansa method, Computers & Mathematics with Applications, 59 (2010), pp. 1614–1620.
[38] M. Cui, Compact finite difference method for the fractional diffusion equation, Journal of Computational Physics, 228 (2009), pp. 7792–7804.
[39] S. Das, Analytical solution of a fractional diffusion equation by variational iteration method, Computers & Mathematics with Applications, 57 (2009), pp. 483–487.
[40] M. Dehghan and M. Abbaszadeh, The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations, Journal of Computational Physics, 351 (2017), pp. 478–510.
[41] 
,
An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch–Torrey equations, Applied Numerical Mathematics, 131 (2018), pp. 190– 206.
[42] 
,
A finite difference/finite element technique with error estimate for space fractional tempered diffusionwave equation, Computers & Mathematics with Applications, 75 (2018), pp. 2903–2914.
[43] 
,
A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed-order fractional damped diffusion-wave equation, Mathematical Methods in the Applied Sciences, 41 (2018), pp. 3476–3494.
[44] M. Dehghan, M. Abbaszadeh, and A. Mohebbi, The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions, Computers & Mathematics with Applications, 68 (2014), pp. 212–237.
[45] 
,
An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein– Gordon equations, Engineering Analysis with Boundary Elements, 50 (2015), pp. 412–434.
[46] 
,
Analysis of a meshless method for the time fractional diffusion-wave equation, Numerical Algorithms, 73 (2016), pp. 445–476.
[47] 
,
Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation, Applied Mathematical Modelling, 40 (2016), pp. 3635–3654.
[48] M. Dehghan, N. Shafieeabyaneh, and M. Abbaszadeh, Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method, Numerical Methods for Partial Differential Equations, 37 (2021), pp. 360–382.
[49] M. Delkhosh and K. Parand, A new computational method based on fractional Lagrange functions to solve multi-term fractional differential equations, Numerical Algorithms, (2021), pp. 1–38.
[50] W. Deng, Finite element method for the space and time fractional Fokker–Planck equation, SIAM Journal on Numerical Analysis, 47 (2009), pp. 204–226.
[51] L. Diening, P. Harjulehto, P. Hast¨ o, and M. Ruzicka¨ , Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg, Germany, 2011.
[52] H. Ding, The development of higher-order numerical differential formulas of Caputo derivative and their applications (I), Computers & Mathematics with Applications, 84 (2021), pp. 203–223.
[53] 
,
The construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application, Communications in Nonlinear Science and Numerical Simulation, 123 (2023), p. 107272.
[54] H. Ding and C. Li, High-order compact difference schemes for the modified anomalous subdiffusion equation, Numerical Methods for Partial Differential Equations, 32 (2016), pp. 213–242.
[55] 
,
High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation, Communications in Nonlinear Science and Numerical Simulation, 120 (2023), p. 107160.
[56] H. Ding and Q. Yi, The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I), Communications in Nonlinear Science and Numerical Simulation, 110 (2022), p. 106394.
[57] E. H. Doha, A. H. Bhrawy, and S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multiterm fractional orders differential equations, Applied Mathematical Modelling, 35 (2011), pp. 5662–5672.
[58] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Computers & Mathematics with Applications, 62 (2011), pp. 2364–2373.
[59] M. Fardi, B. Azarnavid, and S. Mohammadi, Numerical simulation of nonlinear fractional integrodifferential equations on two-dimensional regular and irregular domains: RBF partition of unity, Computers & Mathematics with Applications, 181 (2025), pp. 21–43.
[60] Y. Feng, L. Li, J. G. Liu, and T. Tang, Some Gr¨onwall inequalities for a class of discretizations of time fractional equations on nonuniform meshes, SIAM Journal on Numerical Analysis, 62 (2024), pp. 2196–2221.
[61] L. L. Ferras, N. J. Ford, M. L. Morgado, and M. Rebelo´ , A numerical method for the solution of the time-fractional diffusion equation, International Conference on Computational Science and Its Applications, (2014), pp. 117–131.
[62] G. H. Gao, A. A. Alikhanov, and Z. Z. Sun, The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations, Journal of Scientific Computing, 73 (2017), pp. 93–121.
[63] G. H. Gao and Z. Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, Journal of Computational Physics, 230 (2011), pp. 586–595.
[64] G. H. Gao, Z. Z. Sun, and H. W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, Journal of Computational Physics, 259 (2014), pp. 33–50.
[65] W. Gautschi, Numerical Analysis, Springer Science & Business Media, New York, NY, USA, 2011.
[66] A. Ghoreyshi, M. Abbaszadeh, M. A. Zaky, and M. Dehghan, An accurate and robust numerical method for solving distributed-order space–time fractional PDEs, Zeitschrift fu¨r angewandte Mathematik und Physik, 76 (2025), p. 242.
[67] 
,
Finite block method for nonlinear time-fractional partial integro-differential equations: Stability, convergence, and numerical analysis, Applied Numerical Mathematics, 214 (2025), pp. 82–103.
[68] 
,
Two high-order numerical schemes based on the Lagrange polynomials for solving a distributed-order time-fractional partial integro-differential equation on non-rectangular domains, Journal of Applied Mathematics and Computing, (2025), pp. 1–41.
[69] M. Giona, S. Cerbelli, and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), pp. 449–453.
[70] A. Golbabai, O. Nikan, and M. Molavi Arabshahi, Numerical approximation of time fractional advection-dispersion model arising from solute transport in rivers, TWMS Journal of Pure and Applied Mathematics, 10 (2019).
[71] D. Gottlieb and J. S. Hesthaven, Spectral methods for hyperbolic problems, Journal of Computational and Applied Mathematics, 128 (2001), pp. 83–131.
[72] S. Guo, L. Mei, C. Li, Z. Zhang, and Y. Li, Semi-implicit Hermite–Galerkin spectral method for distributed-order fractional-in-space nonlinear reaction–diffusion equations in multidimensional unbounded domains, Journal of Scientific Computing, 85 (2020), pp. 1–27.
[73] M. Haghi, M. Ilati, and M. Dehghan, A fourth-order compact difference method for the nonlinear timefractional fourth-order reaction–diffusion equation, Engineering with Computers, 39 (2023), pp. 1329–1340.
[74] X. Hu and L. Zhang, A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Computer Physics Communications, 182 (2011), pp. 1645–1650.
[75] C. C. Ji and Z. Z. Sun, A high-order compact finite difference scheme for the fractional sub-diffusion equation, Journal of Scientific Computing, 64 (2015), pp. 959–985.
[76] 
,
The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation, Applied Mathematics and Computation, 269 (2015), pp. 775–791.
[77] S. Jiang, J. Zhang, Q. Zhang, and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Communications in Computational Physics, 21 (2017), pp. 650– 678.
[78] Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, Journal of Computational and Applied Mathematics, 235 (2011), pp. 3285–3290.
[79] B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA Journal of Numerical Analysis, 35 (2015), pp. 561–582.
[80] B. Jin, R. Lazarov, and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis, 36 (2016), pp. 197–221.
[81] V. John, Finite Element Methods for Incompressible Flow Problems, vol. 51, Springer, Cham, Switzerland, 2016.
[82] E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates, Computers & Mathematics with Applications, 19 (1990), pp. 127–145.
[83] 
,
Multiquadrics—A scattered data approximation scheme with applications to computational fluiddynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & Mathematics with Applications, 19 (1990), pp. 147–161.
[84] I. Karatay, N. Kale, and S. R. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank-Nicholson method, Fractional Calculus and Applied Analysis, 16 (2013), pp. 892–910.
[85] G. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, 2005.
[86] E. Kharazmi, M. Zayernouri, and G. E. Karniadakis, Petrov–Galerkin and spectral collocation methods for distributed order differential equations, SIAM Journal on Scientific Computing, 39 (2017), pp. A1003– A1037.
[87] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, 2006.
[88] M. Lakestani, M. Dehghan, and S. Irandoust-Pakchin, The construction of operational matrix of fractional derivatives using B-spline functions, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), pp. 1149–1162.
[89] S. Larsson, M. Racheva, and F. Saedpanah, Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity, Computer Methods in Applied Mechanics and Engineering, 283 (2015), pp. 196–209.
[90] M. Lei, M. Li, P. Wen, and C. Bailey, Moving boundary analysis in heat conduction with multilayer composites by finite block method, Engineering Analysis with Boundary Elements, 89 (2018), pp. 36–44.
[91] C. Li and A. Chen, Numerical methods for fractional partial differential equations, International Journal of Computer Mathematics, 95 (2018), pp. 1048–1099.
[92] C. Li, R. Wu, and H. Ding, High-order approximation to Caputo derivatives and Caputo-type advection– diffusion equations (I), Communications in Applied and Industrial Mathematics, 6 (2014), pp. e–536.
[93] C. Li and F. Zeng, Finite difference methods for fractional differential equations, International Journal of Bifurcation and Chaos, 22 (2012), p. 1230014.
[94] 
,
Finite element methods for fractional differential equations, Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis, (2013), pp. 49–68.
[95] 
,
Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, FL, USA, 2015.
[96] H. Li, J. Cao, and C. Li, High-order approximation to Caputo derivatives and Caputo-type advection– diffusion equations (III), Journal of Computational and Applied Mathematics, 299 (2016), pp. 159–175.
[97] M. Li, M. Lei, A. Munjiza, and P. Wen, Frictional contact analysis of functionally graded materials with Lagrange finite block method, International Journal for Numerical Methods in Engineering, 103 (2015), pp. 391–412.
[98] M. Li, L. Meng, P. Hinneh, and P. Wen, Finite block method for interface cracks, Engineering Fracture Mechanics, 156 (2016), pp. 25–40.
[99] M. Li, A. Monjiza, Y. Xu, and P. Wen, Finite block Petrov–Galerkin method in transient heat conduction, Engineering Analysis with Boundary Elements, 60 (2015), pp. 106–114.
[100] M. Li and P. Wen, Finite block method for transient heat conduction analysis in functionally graded media, International Journal for Numerical Methods in Engineering, 99 (2014), pp. 372–390.
[101] M. Li, X. Xiong, and Y. Wang, A numerical evaluation and regularization of Caputo fractional derivatives, Journal of Physics: Conference Series, 290 (2011), p. 012011.
[102] S. Li and H. Ding, Numerical analysis of two-dimensional time-fractional Allen-Cahn equation on a new non-uniform mesh construction strategy, Journal of Scientific Computing, 104 (2025), p. 81.
[103] 
,
A relaxed step-ratio constraint for time-fractional Cahn–Hilliard equations: Analysis and computation, arXiv preprint arXiv:2508.17178, (2025).
[104] X. Li, S. Liu, W. Huang, and P. Wen, Finite block method with automatic differentiation algorithm for Reissner plate nonlinear analysis, Engineering Analysis with Boundary Elements, 179 (2025), p. 106354.
[105] X. Li and C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Communications in Computational Physics, 8 (2010), p. 1016.
[106] Y. Li, F. Liu, I. W. Turner, and T. Li, Time-fractional diffusion equation for signal smoothing, Applied Mathematics and Computation, 326 (2018), pp. 108–116.
[107] H. L. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reactionsubdiffusion equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 1112–1133.
[108] H. L. Liao, W. McLean, and J. Zhang, A discrete Gr¨onwall inequality with applications to numerical schemes for subdiffusion problems, SIAM Journal on Numerical Analysis, 57 (2019), pp. 218–237.
[109] Y. Lin, X. Li, and C. Xu, Finite difference/spectral approximations for the fractional Cable equation, Mathematics of Computation, 80 (2011), pp. 1369–1396.
[110] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 (2007), pp. 1533–1552.
[111] F. Liu, S. Shen, V. Anh, and I. Turner, Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, The Proceedings of ANZIAM, 46 (2004), pp. C488–C504.
[112] G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and their Programming, Springer, Dordrecht, The Netherlands, 2005.
[113] J. Liu, T. Wang, and T. Zhang, A second-order finite difference scheme for the multi-dimensional nonlinear time-fractional Schr¨odinger equation, Numerical Algorithms, 92 (2023), pp. 1153–1182.
[114] Q. X. Liu, Y. Gu, P. Zhuang, F. Liu, and Y. Nie, An implicit RBF meshless approach for time fractional diffusion equations, Computational Mechanics, 48 (2011), pp. 1–12.
[115] Y. Liu, Z. Fang, H. Li, and S. He, A mixed finite element method for a time-fractional fourth-order partial differential equation, Applied Mathematics and Computation, 243 (2014), pp. 703–717.
[116] Y. Liu, M. Zhang, H. Li, and J. Li, High-order local discontinuous Galerkin method combined with WSGDapproximation for a fractional subdiffusion equation, Computers & Mathematics with Applications, 73 (2017), pp. 1298–1314.
[117] C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM Journal on Scientific Computing, 38 (2016), pp. A2699–A2724.
[118] F. Mainardi, The time fractional diffusion-wave equation, Radiophysics and Quantum Electronics, 38 (1995), pp. 13–24.
[119] 
,
Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, London, UK, 2022.
[120] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection– dispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004), pp. 65–77.
[121] 
,
Finite difference approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56 (2006), pp. 80–90.
[122] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339 (2000), pp. 1–77.
[123] K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
[124] A. Mohebbi, M. Abbaszadeh, and M. Dehghan, A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, Journal of Computational Physics, 240 (2013), pp. 36–48.
[125] 
,
The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schr¨odinger equation arising in quantum mechanics, Engineering Analysis with Boundary Elements, 37 (2013), pp. 475–485.
[126] 
,
The meshless method of radial basis functions for the numerical solution of time fractional telegraph equation, International Journal of Numerical Methods for Heat & Fluid Flow, 24 (2014), pp. 1636–1659.
[127] R. Mokhtari and F. Mostajeran, A high order formula to approximate the Caputo fractional derivative, Communications on Applied Mathematics and Computation, 2 (2020), pp. 1–29.
[128] M. Molavi Arabshahi, J. Rashidinia, and S. Tanoomand, An efficient spectral collocation method based on the generalized Laguerre polynomials to multi-term time fractional diffusion-wave equations, AIP Advances, 14 (2024).
[129] 
,
Numerical solving of multi-term time fractional diffusion-wave equations using shifted Gegenbauer spectral collocation method, Computational Methods for Differential Equations, 13 (2025), pp. 815–827.
[130] H. Nasir, B. Gunawardana, et al., A second order finite difference approximation for the fractional diffusion equation, International Journal of Applied Physics and Mathematics, 3 (2013), p. 237.
[131] R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (b), 133 (1986), pp. 425–430.
[132] O. Nikan, Z. Avazzadeh, and J. T. Machado, An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model, Journal of King Saud University-Science, 33 (2021), p. 101243.
[133] O. Nikan, S. M. Molavi-Arabshai, and H. Jafari, Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves, Discret. Contin. Dyn. Syst. S, 14 (2021), pp. 3685–3701.
[134] K. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, San Diego, California, USA, 1974.
[135] G. Pang, L. Lu, and G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM Journal on Scientific Computing, 41 (2019), pp. A2603–A2626.
[136] A. T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, Journal of Computational Physics, 54 (1984), pp. 468–488.
[137] P. Pedregal, Functional Analysis, Sobolev Spaces, and Calculus of Variations, Springer, Cham, Switzerland, 2024.
[138] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Elsevier, San Diego, California, USA, 1998.
[139] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. V. Jara, Matrix approach to discrete fractional calculus II: Partial fractional differential equations, Journal of Computational Physics, 228 (2009), pp. 3137–3153.
[140] A. Quarteroni, R. Sacco, and F. Saleri, Numerical mathematics, vol. 37, Springer Science & Business Media, New York, NY, USA, 2006.
[141] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, vol. 23, Springer Science & Business Media, Berlin, Germany, 2008.
[142] P. Rajput, N. Srivastava, and V. K. Singh, A high order numerical method for the variable order time-fractional reaction-subdiffusion equation, Chinese Journal of Physics, 85 (2023), pp. 431–444.
[143] M. Ramezani and R. Mokhtari, A novel high-order finite-difference method for the time-fractional diffusion equation with smooth/nonsmooth solutions, Bulletin of the Iranian Mathematical Society, 48 (2022), pp. 3987–4013.
[144] 
,
Numerical solution of distributed-order fractional diffusion equations using a high-order temporal scheme, Communications on Applied Mathematics and Computation, (2025), pp. 1–15.
[145] M. Ramezani, R. Mokhtari, and G. Haase, Some high order formulae for approximating Caputo fractional derivatives, Applied Numerical Mathematics, 153 (2020), pp. 300–318.
[146] M. Ran and C. Zhang, New compact difference scheme for solving the fourth-order time fractional subdiffusion equation of the distributed order, Applied Numerical Mathematics, 129 (2018), pp. 58–70.
[147] S. S. Ray and R. Bera, Analytical solution of a fractional diffusion equation by Adomian decomposition method, Applied Mathematics and Computation, 174 (2006), pp. 329–336.
[148] B. Ross, The development of fractional calculus 1695–1900, Historia Mathematica, 4 (1977), pp. 75–89.
[149] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics, (1997), pp. 15–67.
[150] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional-order differential equations, Computers & Mathematics with Applications, 59 (2010), pp. 1326–1336.
[151] 
,
Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method, Numerical Methods for Partial Differential Equations: An International Journal, 26 (2010), pp. 239–252.
[152] 
,
A tau approach for solution of the space fractional diffusion equation, Computers & Mathematics with Applications, 62 (2011), pp. 1135–1142.
[153] S. Sabermahani, Y. Ordokhani, and S. Yousefi, Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations, Computational and Applied Mathematics, 37 (2018), pp. 3846–3868.
[154] S. Sabermahani, Y. Ordokhani, and S. A. Yousefi, Fractional-order general Lagrange scaling functions and their applications, BIT Numerical Mathematics, 60 (2020), pp. 101–128.
[155] G. Sayyar, S. M. Hosseini, and F. Mostajeran, A high-order scheme for time-space fractional diffusion equations with Caputo-Riesz derivatives, Computers & Mathematics with Applications, 104 (2021), pp. 34–43.
[156] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), pp. 134–144.
[157] M. Shafiq, M. Abbas, K. M. Abualnaja, M. Huntul, A. Majeed, and T. Nazir, An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana–Baleanu derivative, Engineering with Computers, 38 (2022), pp. 901–917.
[158] J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM Journal on Scientific Computing, 15 (1994), pp. 1489–1505.
[159] J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, vol. 41, Springer Science & Business Media, Berlin, 2011.
[160] S. Shen, F. Liu, V. Anh, and I. Turner, Detailed analysis of a conservative difference approximation for the time fractional diffusion equation, Journal of Applied Mathematics and Computing, 22 (2006), pp. 1–19.
[161] A. Shokri and M. Dehghan, A Not-a-Knot meshless method using radial basis functions and predictor– corrector scheme to the numerical solution of improved Boussinesq equation, Computer Physics Communications, 181 (2010), pp. 1990–2000.
[162] 
,
A meshless method using radial basis functions for the numerical solution of two—dimensional complex Ginzburg–Landau equation, Computer Modeling in Engineering and Sciences, 84 (2012), p. 333.
[163] F. Song, C. Xu, and G. E. Karniadakis, Computing fractional Laplacians on complex-geometry domains: Algorithms and simulations, SIAM Journal on Scientific Computing, 39 (2017), pp. A1320–A1344.
[164] J. Stoer, R. Bulirsch, R. Bartels, W. Gautschi, and C. Witzgall, Introduction to Numerical
Analysis, vol. 1993, Springer, 1980.
[165] M. Stynes, E. O’Riordan, and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM Journal on Numerical Analysis, 55 (2017), pp. 1057–1079.
[166] Z. Z. Sun and G. H. Gao, Fractional Differential Equations: Finite Difference Methods, Walter de Gruyter GmbH & Co KG, 2020.
[167] Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56 (2006), pp. 193–209.
[168] N. Sweilam, M. Khader, and A. Mahdy, Crank-Nicolson finite difference method for solving timefractional diffusion equation, Journal of Fractional Calculus and Applications, 2 (2012), pp. 1–9.
[169] N. H. Sweilam and M. Khader, A chebyshev pseudo-spectral method for solving fractional-order integrodifferential equations, The ANZIAM Journal, 51 (2010), pp. 464–475.
[170] C. Tadjeran, M. M. Meerschaert, and H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, Journal of Computational Physics, 213 (2006), pp. 205–213.
[171] V. Thomee´ , Galerkin Finite Element Methods for Parabolic Problems, vol. 25, Springer Science & Business Media, Berlin, Germany, 2007.
[172] W. Tian, H. Zhou, and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Mathematics of Computation, 84 (2015), pp. 1703–1727.
[173] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
[174] 
,
Approximation Theory and Approximation Practice, extended edition, SIAM, Philadelphia, Pennsylvania, USA, 2019.
[175] J. Wang, F. X. Sun, Y. M. Cheng, and A. Huang, Error estimates for the interpolating moving leastsquares method, Applied Mathematics and Computation, 245 (2014), pp. 321–342.
[176] Y. Wang, G. Wang, L. Bu, and L. Mei, Two second-order and linear numerical schemes for the multidimensional nonlinear time-fractional Schr¨odinger equation, Numerical Algorithms, 88 (2021), pp. 419–451.
[177] Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, Journal of Computational Physics, 277 (2014), pp. 1–15.
[178] P. Wen, P. Cao, and T. Korakianitis, Finite block method in elasticity, Engineering Analysis with Boundary Elements, 46 (2014), pp. 116–125.
[179] W. Wyss, The fractional diffusion equation, Journal of Mathematical Physics, 27 (1986), pp. 2782–2785.
[180] S. Yan, F. Zhao, C. Li, and L. Zhao, High order WSGL difference operators combined with Sinc-Galerkin method for time fractional Schro¨dinger equation, International Journal of Computer Mathematics, 97 (2020), pp. 2259–2286.
[181] X. Yan, S. Liu, P. Wen, J. Sladek, and V. Sladek, Homogeneous and functionally graded piezoelectric structure analysis with finite block method, Composite Structures, 365 (2025), p. 119188.
[182] J. Yang, Z. Wang, O. Adetoro, P. Wen, and C. Bailey, The thermal analysis of cutting/grinding processes by meshless finite block method, Engineering Analysis with Boundary Elements, 100 (2019), pp. 68– 79.
[183] X. Yang, H. Zhang, and D. Xu, Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation, Journal of Computational Physics, 256 (2014), pp. 824–837.
[184] S. Yeganeh, R. Mokhtari, and J. S. Hesthaven, A local discontinuous Galerkin method for twodimensional time fractional diffusion equations, Communications on Applied Mathematics and Computation, 2 (2020), pp. 689–709.
[185] Q. Yi, A. Chen, and H. Ding, High order difference method for fractional convection equation, Mathematics and Computers in Simulation, 234 (2025), pp. 286–298.
[186] M. Zayernouri and G. E. Karniadakis, Exponentially accurate spectral and spectral element methods for fractional ODEs, Journal of Computational Physics, 257 (2014), pp. 460–480.
[187], Fractional spectral collocation method, SIAM Journal on Scientific Computing, 36 (2014), pp. A40–A62.
[188] 
,
Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, Journal of Computational Physics, 293 (2015), pp. 312–338.
[189] F. Zeng and C. Li, A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation, Applied Numerical Mathematics, 121 (2017), pp. 82–95.
[190] F. Zeng, C. Li, F. Liu, and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM Journal on Scientific Computing, 35 (2013), pp. A2976–A3000.
[191] 
,
Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM Journal on Scientific Computing, 37 (2015), pp. A55–A78.
[192] N. Zhang, W. Deng, and Y. Wu, Finite difference/element method for a two-dimensional modified fractional diffusion equation, Advances in Applied Mathematics and Mechanics, 4 (2012), pp. 496–518.
[193] X. Zhang, Y. Feng, Z. Luo, and J. Liu, A spatial sixth-order numerical scheme for solving fractional partial differential equation, Applied Mathematics Letters, 159 (2025), p. 109265.
[194] X. Zhang, P. Huang, X. Feng, and L. Wei, Finite element method for two-dimensional time-fractional Tricomi-type equations, Numerical Methods for Partial Differential Equations, 29 (2013), pp. 1081–1096.
[195] Y. Zhang and H. Ding, An efficient high-order numerical algorithm for the time fractional Fokker–Planck equations, International Journal of Computer Mathematics, 98 (2021), pp. 357–366.
[196] Y. N. Zhang and Z. Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, Journal of Computational Physics, 230 (2011), pp. 8713–8728.
[197] Y. N. Zhang, Z. Z. Sun, and H. L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, Journal of Computational Physics, 265 (2014), pp. 195–210.
[198] Y. N. Zhang, Z. Z. Sun, and H. W. Wu, Error estimates of Crank–Nicolson-type difference schemes for the subdiffusion equation, SIAM Journal on Numerical Analysis, 49 (2011), pp. 2302–2322.
[199] X. Zhao and Z. Z. Sun, A box-type scheme for fractional sub-diffusion equation with neumann boundary conditions, Journal of Computational Physics, 230 (2011), pp. 6061–6074.
[200] H. Zheng, J. Xiong, Y. Yuan, and P. Wen, Mixed-mode dynamic stress intensity factors by variation technique with finite block method, Engineering Analysis with Boundary Elements, 106 (2019), pp. 27–33.
[201] H. Zhong and H. Ding, Caputo-tempered subdiffusion problems: Blended nonuniform L1 and compact difference formulations, Mathematical Methods in the Applied Sciences, (2025), pp. 1–20.
[202] Y. Zhou, W. Huang, J. Yang, and P. Wen, Galerkin finite block method with Lagrange multipliers method for cracked solids in functionally graded materials, Engineering Analysis with Boundary Elements, 163 (2024), pp. 606–615.
[203] H. Zhu and C. Xu, A highly efficient numerical method for the time-fractional diffusion equation on unbounded domains, Journal of Scientific Computing, 99 (2024), p. 47.
[204] P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, Journal of Applied Mathematics and Computing, 22 (2006), pp. 87–99.
[205] 
,
Finite difference approximation for two-dimensional time fractional diffusion equation, Journal of Algorithms & Computational Technology, 1 (2007), pp. 1–16.