[1] Brunner, H., Han, H., Yin, D.: The maximum principle for time-fractional diffusion equations and its application. Numer. Funct. Anal. Optim. 36(10), 1307-1321 (2015)
[2] Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 54(4), 937-954 (2014)
[3] Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34(4), A2145-A2172 (2012)
[4] Cen, D., Vong, S.: The tracking of derivative discontinuities for delay fractional equations based on fitted L1 method. Comput. Meth. Appl. Mat. 23(3), 591-601 (2023)
[5] Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience Publishers, New York (1953)
[6] Doha, E., Bhrawy, A., Ezz-Eldien, S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36(10), 4931-4943 (2012)
[7] Doha, E., Bhrawy, A., Hafez, R.: On shifted Jacobi spectral method for high-order multi-point boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 17(10), 3802-3810 (2012)
[8] Du, N., Wang, H.: A fast finite element method for space-fractional dispersion equations on bounded domains in \mathbb{R} ^2. SIAM J. Sci. Comput. 37(3), A1614-A1635 (2015)
[9] Ford, N.J., Xiao, J., Yan, Y.: A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14, 454-474 (2011)
[10] Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. Springer, Germany (2014)
[11] Gracia, J.L., O’Riordan, E., Stynes, M.: A fitted scheme for a Caputo initial-boundary value problem. J. Sci. Comput. 76, 583-609 (2018)
[12] Guo, L., Wu, H., Yu, X., Zhou, T.: Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations. https://doi.org/10.48550/arXiv.2203.08501 (2022)
[13] Hao, Z., Moongyu, P., Cai, Z.: Neural network method for integral fractional Laplace equations. East Asian J. Appl. Math. 13, 95-118 (2023)
[14] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4(2), 251-257 (1991)
[15] Ilic, M., Liu, F., Turner, I., Anh, V.: Numerical approximation of a fractional-in-space diffusion equation. I. Fract. Calc. Appl. Anal. 8(3), 323-341 (2005)
[16] Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 142, 1-8 (2012)
[17] Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235(11), 3285-3290 (2011)
[18] Ke, R., Ng, M., Sun, H.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303, 203-211 (2015)
[19] Li, C., Li, Z., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85, 1-27 (2020)
[20] Lin, X., Ng, M., Sun, H.: A splitting preconditioner for Toeplitz-like linear systems arising from fractional diffusion equations. SIAM J. Matrix Anal. Appl. 38(4), 1580-1614 (2017)
[21] Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191(1), 12-20 (2007)
[22] Liu, M., Cai Z., Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs. Comput. Math. Appl. 113, 103-116 (2022)
[23] Lu, L., Meng, X., Mao, Z., Karniadakis, G.E.: DeepXDE: a deep learning library for solving differential equations. SIAM Rev. 63(1), 208-228 (2019)
[24] Ma, L., Li, R., Zeng, F., Karniadakis, G.E.: Bi-orthogonal fPINN: a physics-informed neural network method for solving time-dependent stochastic fractional PDEs. Commun. Comput. Phys. 34(4), 1133-1176 (2023)
[25] Ma, Z., Hou J., Zhu W., Peng Y., Li Y.: PMNN: physical model-driven neural network for solving time-fractional differential equations. Chaos Solitons Fractals 177, 114238 (2023)
[26] Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65-77 (2004)
[27] Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56(1), 80-90 (2006)
[28] Ou, C., Cen, D., Vong, S., Wang, Z.: Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations. Appl. Numer. Math. 177, 34-57 (2022)
[29] Ou, C., Cen, D., Wang, Z., Vong, S.: Fitted schemes for Caputo-Hadamard fractional differential equations. Numer. Algorithms 97(1), 1-30 (2023)
[30] Pang, G., Lu, L., Karniadakis, G.E.: fPINNs: fractional physics-informed neural networks. SIAM J. Sci. Comput. 41(4), A2603-A2626 (2019)
[31] Podlubny, I.: Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)
[32] Qu, H., Liu, X., Lu, X., Ur Rahman, M., She, Z.: Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order. Chaos Solitons Fractals 156, 111856 (2022)
[33] Qu, H., Liu, X., She, Z.: Neural network method for fractional-order partial differential equations. Neurocomputing 414, 225-237 (2020)
[34] Qu, H., She, Z., Liu, X.: Neural network method for solving fractional diffusion equations. Appl. Math. Comput. 391(1), 125635 (2021)
[35] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686-707 (2018)
[36] Roop, J., John, P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \mathbb{R} ^2. J. Comput. Appl. Math. 193(1), 243-268 (2006)
[37] Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426-447 (2011)
[38] Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms. Analysis and Applications. Springer, Germany (2011)
[39] Sivalingam, S.M., Govindaraj, V.: Physics informed neural network based scheme and its error analysis for \psi-Caputo type fractional differential equations. Phys. Scripta 99(9), 096002 (2024)
[40] Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057-1079 (2017)
[41] Sugandha, A., Trilok, M., Shivi, A., Kamlesh, T., Phalguni, G.: Applications of fractional calculus in computer vision: a survey. Neurocomputing 489, 407-428 (2022)
[42] Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703-1727 (2015)
[43] Wang, Q., Ma, J., Yu, S., Tan, L.: Noise detection and image denoising based on fractional calculus. Chaos Solitons Fractals 131, 109463 (2020)
[44] Yadav, P., Jahan, S., Nisar, K.S.: Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics. Ain Shams Eng. J. 15(1), 102299 (2024)
[45] Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200-218 (2010)
[46] Ye, Y., Li, Y., Fan, H., Liu, X., Zhang, H.: SLeNN-ELM: a shifted Legendre neural network method for fractional delay differential equations based on extreme learning machine. Netw. Heterog. Media 18(1), 494-512 (2023)
[47] Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40-A62 (2014)
[48] Zeng, F., Liu, F., Li, C.: A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52(6), 2599-2622 (2014)
[49] Zhao, T., Li, C., Li, D.: Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative. Fract. Calc. Appl. Anal. 26, 2903-2927 (2023)
