Variable-order fractional wave equation: Analysis, numerical approximation, and fast algorithm.

Wenlin Qiu https://orcid.org/0000-0003-1144-5315

Jinhong Jia Chuanting Jiang Yiqun Li Mengmeng Liu Wenlin Qiu Variable-order fractional wave equation: Analysis, numerical approximation, and fast algorithm. 2026 109786 158 Commun. Nonlinear Sci. Numer. Simul. https://doi.org/10.1016/j.cnsns.2026.109786 db/journals/cnsns/cnsns158.html#JiaJLLQ26 streams/journals/cnsns

Kexin Li Hao Zhang Omid Nikan Wenlin Qiu A novel semi-implicit finite difference approach for the Sobolev equation with generalized Burgers-type nonlinear term. 181-195 2026 244 Math. Comput. Simul. https://doi.org/10.1016/j.matcom.2025.12.024 db/journals/mcs/mcs244.html#LiZNQ26 streams/journals/mcs

Mengmeng Liu Xiangcheng Zheng Kexin Li Wenlin Qiu High-order compact difference method for convection-reaction-subdiffusion equation with variable exponent and coefficients. 2026 March 1677-1699 101 Numer. Algorithms 3 https://doi.org/10.1007/s11075-025-02060-6 db/journals/na/na101.html#LiuZLQ26 streams/journals/na

Wenlin Qiu Kexin Li Yiqun Li Hao Zhang The optimal error analysis of nonuniform L1 method for the variable-exponent subdiffusion model. 2026 January abs/2601.06773 CoRR https://doi.org/10.48550/arXiv.2601.06773 db/journals/corr/corr2601.html#abs-2601-06773 streams/journals/corr

Wenlin Qiu Xiangcheng Zheng Numerical analysis of high-order methods for variable-exponent fractional diffusion-wave equation. 53 2025 November 105 J. Sci. Comput. 2 https://doi.org/10.1007/s10915-025-03080-z db/journals/jscic/jscic105.html#QiuZ25 streams/journals/jscic

Hao Chen Mahmoud A. Zaky Xiangcheng Zheng Ahmed S. Hendy Wenlin Qiu Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel. 677-718 2025 February 98 Numer. Algorithms 2 https://doi.org/10.1007/s11075-024-01811-1 db/journals/na/na98.html#ChenZZHQ25 streams/journals/na

Tao Guo Yiqun Li Wenlin Qiu Compact difference method for Euler-Bernoulli beams and plates with nonlinear nonlocal strong damping. 2025 May abs/2505.02132 CoRR https://doi.org/10.48550/arXiv.2505.02132 db/journals/corr/corr2505.html#abs-2505-02132 streams/journals/corr

Wenlin Qiu Xiangcheng Zheng Tao Guo Xu Xiao Finite difference method for nonlinear damped viscoelastic Euler-Bernoulli beam model. 2025 May abs/2505.02517 CoRR https://doi.org/10.48550/arXiv.2505.02517 db/journals/corr/corr2505.html#abs-2505-02517 streams/journals/corr

Wenlin Qiu Xiangcheng Zheng Numerical analysis for subdiffusion problem with non-positive memory. 2025 May abs/2505.04924 CoRR https://doi.org/10.48550/arXiv.2505.04924 db/journals/corr/corr2505.html#abs-2505-04924 streams/journals/corr

Yingwen Guo Yinnian He Wenlin Qiu Xiangcheng Zheng Long-time error estimate and decay of finite element method to a generalized viscoelastic flow. 2025 June abs/2506.22782 CoRR https://doi.org/10.48550/arXiv.2506.22782 db/journals/corr/corr2506.html#abs-2506-22782 streams/journals/corr

Mengmeng Liu Jie Ma Wenlin Qiu Xiangcheng Zheng Multiscale solution decomposition of nonlocal-in-time problems with application in numerical computation. 2025 September abs/2509.17020 CoRR https://doi.org/10.48550/arXiv.2509.17020 db/journals/corr/corr2509.html#abs-2509-17020 streams/journals/corr

Hao Zhang Kexin Li Wenlin Qiu Two ADI compact difference methods for variable-exponent diffusion wave equations. 2025 September abs/2509.21316 CoRR https://doi.org/10.48550/arXiv.2509.21316 db/journals/corr/corr2509.html#abs-2509-21316 streams/journals/corr

Hao Zhang Kexin Li Wenlin Qiu Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent. 2025 October abs/2510.12188 CoRR https://doi.org/10.48550/arXiv.2510.12188 db/journals/corr/corr2510.html#abs-2510-12188 streams/journals/corr

Xiangyi Peng Lisen Ding Wenlin Qiu An Efficient Space-Time Two-Grid Compact Difference Scheme for the Two-Dimensional Viscous Burgers' Equation. 2025 October abs/2510.15274 CoRR https://doi.org/10.48550/arXiv.2510.15274 db/journals/corr/corr2510.html#abs-2510-15274 streams/journals/corr

Jinhong Jia Chuanting Jiang Yiqun Li Mengmeng Liu Wenlin Qiu Variable-order fractional wave equation: Analysis, numerical approximation, and fast algorithm. 2025 November abs/2511.06014 CoRR https://doi.org/10.48550/arXiv.2511.06014 db/journals/corr/corr2511.html#abs-2511-06014 streams/journals/corr

Mingchao Zhao Wenlin Qiu Finite element approximation and analysis of damped viscoelastic hyperbolic integrodifferential equations with L1 kernel. 108993 2024 151 Appl. Math. Lett. https://doi.org/10.1016/j.aml.2024.108993 db/journals/appml/appml151.html#ZhaoQ24

Xiangyi Peng Wenlin Qiu Ahmed S. Hendy Mahmoud A. Zaky Temporal Second-Order Fast Finite Difference/Compact Difference Schemes for Time-Fractional Generalized Burgers' Equations. 52 2024 May 99 J. Sci. Comput. 2 https://doi.org/10.1007/s10915-024-02514-4 db/journals/jscic/jscic99.html#PengQHZ24

Wenlin Qiu Yanping Chen 0002 Xu Xiao Xiangcheng Zheng Discrete L1 remainder stability of first and second order schemes for a Volterra integro-differential equation. 12-27 2024 219 Math. Comput. Simul. https://doi.org/10.1016/j.matcom.2023.12.012 db/journals/mcs/mcs219.html#QiuCXZ24

Xiangcheng Zheng Yiqun Li Wenlin Qiu Local Modification of Subdiffusion by Initial Fickian Diffusion: Multiscale Modeling, Analysis, and Computation. 1534-1557 2024 22 Multiscale Model. Simul. 4 https://doi.org/10.1137/24m1635612 db/journals/mmas/mmas22.html#ZhengLQ24 streams/journals/mmas

Xiangcheng Zheng Yiqun Li Wenlin Qiu Local modification of subdiffusion by initial Fickian diffusion: Multiscale modeling, analysis and computation. 2024 abs/2401.16885 CoRR https://doi.org/10.48550/arXiv.2401.16885 db/journals/corr/corr2401.html#abs-2401-16885

Xiangcheng Zheng Hong Wang 0018 Wenlin Qiu Numerical approximation for variable-exponent fractional diffusion-wave equation. 2024 abs/2406.02941 CoRR https://doi.org/10.48550/arXiv.2406.02941 db/journals/corr/corr2406.html#abs-2406-02941

Meihui Zhang Mengmeng Liu Wenlin Qiu Xiangcheng Zheng Generalizing subdiffusive Black-Scholes model by variable exponent: Model transformation and numerical approximation. 2024 abs/2411.13913 CoRR https://doi.org/10.48550/arXiv.2411.13913 db/journals/corr/corr2411.html#abs-2411-13913 streams/journals/corr

Wenlin Qiu Tao Guo Yiqun Li Xu Guo Xiangcheng Zheng A multiscale Abel kernel and application in viscoelastic problem. 2024 abs/2411.16078 CoRR https://doi.org/10.48550/arXiv.2411.16078 db/journals/corr/corr2411.html#abs-2411-16078 streams/journals/corr

Wenlin Qiu Xiangcheng Zheng Kassem Mustapha 0001 Numerical approximations for a hyperbolic integrodifferential equation with a non-positive variable-sign kernel and nonlinear-nonlocal damping. 2024 abs/2412.07394 CoRR https://doi.org/10.48550/arXiv.2412.07394 db/journals/corr/corr2412.html#abs-2412-07394 streams/journals/corr

Wenlin Qiu Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels. 43 2023 49 Adv. Comput. Math. 3 https://doi.org/10.1007/s10444-023-10050-2 db/journals/adcm/adcm49.html#Qiu23

Man Luo 0004 Wenlin Qiu Omid Nikan Zakieh Avazzadeh Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity. 127655 2023 440 Appl. Math. Comput. https://doi.org/10.1016/j.amc.2022.127655 db/journals/amc/amc440.html#LuoQNA23

Tao Guo Da Xu Wenlin Qiu Efficient third-order BDF finite difference scheme for the generalized viscous Burgers' equation. 108570 2023 June 140 Appl. Math. Lett. https://doi.org/10.1016/j.aml.2023.108570 db/journals/appml/appml140.html#GuoXQ23

Wenlin Qiu Xu Xiao Kexin Li Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity. 106804 2023 116 Commun. Nonlinear Sci. Numer. Simul. https://doi.org/10.1016/j.cnsns.2022.106804 https://www.wikidata.org/entity/Q113877773 db/journals/cnsns/cnsns116.html#QiuXL23

Tao Guo Omid Nikan Wenlin Qiu Da Xu Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics. 106989 2023 117 Commun. Nonlinear Sci. Numer. Simul. https://doi.org/10.1016/j.cnsns.2022.106989 db/journals/cnsns/cnsns117.html#GuoNQX23

Yang Cao Mahmoud A. Zaky Ahmed S. Hendy Wenlin Qiu Optimal error analysis of space-time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel. 115287 2023 431 J. Comput. Appl. Math. https://doi.org/10.1016/j.cam.2023.115287 db/journals/jcam/jcam431.html#CaoZHQ23

Leijie Qiao Wenlin Qiu Da Xu Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions. 205-231 2023 205 Math. Comput. Simul. https://doi.org/10.1016/j.matcom.2022.10.001 db/journals/mcs/mcs205.html#QiaoQX23

Xiangyi Peng Da Xu Wenlin Qiu Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers' equation. 702-726 2023 June 208 Math. Comput. Simul. https://doi.org/10.1016/j.matcom.2023.02.004 db/journals/mcs/mcs208.html#PengXQ23

Hao Chen Omid Nikan Wenlin Qiu Zakieh Avazzadeh Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers' type nonlinearity. 248-266 2023 209 Math. Comput. Simul. https://doi.org/10.1016/j.matcom.2023.02.014 db/journals/mcs/mcs209.html#ChenNQA23

Leijie Qiao Wenlin Qiu Da Xu Crank-Nicolson ADI finite difference/compact difference schemes for the 3D tempered integrodifferential equation associated with Brownian motion. 1083-1104 2023 July 93 Numer. Algorithms 3 https://doi.org/10.1007/s11075-022-01454-0 db/journals/na/na93.html#QiaoQX23

Tao Guo Omid Nikan Zakieh Avazzadeh Wenlin Qiu Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems. 2022 September 41 Comput. Appl. Math. 6 https://doi.org/10.1007/s40314-022-01934-y db/journals/cam/cam41.html#GuoNAQ22

Leijie Qiao Jing Guo Wenlin Qiu Fast BDF2 ADI methods for the multi-dimensional tempered fractional integrodifferential equation of parabolic type. 89-104 2022 123 Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2022.08.014 https://www.wikidata.org/entity/Q113878653 db/journals/cma/cma123.html#QiaoGQ22

Jun Zhou 0019 Da Xu Wenlin Qiu Leijie Qiao An accurate, robust, and efficient weak Galerkin finite element scheme with graded meshes for the time-fractional quasi-linear diffusion equation. 188-195 2022 124 Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2022.08.022 https://www.wikidata.org/entity/Q113878652 db/journals/cma/cma124.html#ZhouXQQ22

Leijie Qiao Bo Tang Da Xu Wenlin Qiu High-order orthogonal spline collocation method with graded meshes for two-dimensional fractional evolution integro-differential equation. 1305-1324 2022 99 Int. J. Comput. Math. 7 https://doi.org/10.1080/00207160.2021.1984435 https://www.wikidata.org/entity/Q115314066 db/journals/ijcm/ijcm99.html#QiaoTXQ22

Qiong Huang 0004 Ren-jun Qi Wenlin Qiu The efficient alternating direction implicit Galerkin method for the nonlocal diffusion-wave equation in three dimensions. 3067-3087 2022 October 68 J. Appl. Math. Comput. 5 https://doi.org/10.1007/s12190-021-01652-4 db/journals/jamc/jamc68.html#HuangQQ22

Jing Guo Qing Pan Da Xu Wenlin Qiu A spectral order method for solving the nonlinear fourth-order time-fractional problem. 4645-4667 2022 December 68 J. Appl. Math. Comput. 6 https://doi.org/10.1007/s12190-022-01719-w db/journals/jamc/jamc68.html#GuoPXQ22

Wenlin Qiu Da Xu Jun Zhou 0019 Jing Guo An efficient Sinc-collocation method via the DE transformation for eighth-order boundary value problems. 114136 2022 408 J. Comput. Appl. Math. https://doi.org/10.1016/j.cam.2022.114136 db/journals/jcam/jcam408.html#QiuXZG22

Wenlin Qiu Xu Xiao Kexin Li Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity. 2022 abs/2203.12226 CoRR https://doi.org/10.48550/arXiv.2203.12226 db/journals/corr/corr2203.html#abs-2203-12226

Hao Chen Mahmoud A. Zaky Ahmed S. Hendy Wenlin Qiu A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel. 2022 abs/2209.00211 CoRR https://doi.org/10.48550/arXiv.2209.00211 db/journals/corr/corr2209.html#abs-2209-00211

Xiangyi Peng Da Xu Wenlin Qiu Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers' equation. 2022 abs/2209.00217 CoRR https://doi.org/10.48550/arXiv.2209.00217 db/journals/corr/corr2209.html#abs-2209-00217

Wenlin Qiu Optimal error estimate of accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels. 2022 abs/2209.00229 CoRR https://doi.org/10.48550/arXiv.2209.00229 db/journals/corr/corr2209.html#abs-2209-00229

Wenlin Qiu Global behavior of temporal discretizations for Volterra integrodifferential equations with certain nonsmooth kernels. 2022 abs/2209.00235 CoRR https://doi.org/10.48550/arXiv.2209.00235 db/journals/corr/corr2209.html#abs-2209-00235

Xu Xiao Wenlin Qiu Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs. 2022 abs/2209.04997 CoRR https://doi.org/10.48550/arXiv.2209.04997 db/journals/corr/corr2209.html#abs-2209-04997

Wenlin Qiu Da Xu Jing Guo Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation. 125693 2021 392 Appl. Math. Comput. https://doi.org/10.1016/j.amc.2020.125693 https://www.wikidata.org/entity/Q112880132 db/journals/amc/amc392.html#QiuXG21

Leijie Qiao Wenlin Qiu Da Xu A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem. 137-145 2021 102 Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2021.10.014 db/journals/cma/cma102.html#QiaoQX21

Xuehua Yang Wenlin Qiu Haixiang Zhang 0003 Liang Tang An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation. 233-247 2021 102 Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2021.10.021 db/journals/cma/cma102.html#YangQZT21

Huifa Jiang Da Xu Wenlin Qiu Jun Zhou 0019 An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile-immobile equation. 2020 39 Comput. Appl. Math. 4 https://doi.org/10.1007/s40314-020-01345-x db/journals/cam/cam39.html#JiangXQZ20

Xuan Zheng Hongbin Chen 0002 Wenlin Qiu A Crank-Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity. 2020 39 Comput. Appl. Math. 4 https://doi.org/10.1007/s40314-020-01337-x https://www.wikidata.org/entity/Q114219337 db/journals/cam/cam39.html#ZhengCQ20

Wenlin Qiu Da Xu Haifan Chen Jing Guo An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile-immobile equation in two dimensions. 3156-3172 2020 80 Comput. Math. Appl. 12 https://doi.org/10.1016/j.camwa.2020.11.003 db/journals/cma/cma80.html#QiuXCG20

Shufang Hu Wenlin Qiu Hongbin Chen 0002 A backward Euler difference scheme for the integro-differential equations with the multi-term kernels. 1254-1267 2020 97 Int. J. Comput. Math. 6 https://doi.org/10.1080/00207160.2019.1613529 https://www.wikidata.org/entity/Q127936754 db/journals/ijcm/ijcm97.html#HuQC20

Wenlin Qiu Da Xu Hongbin Chen 0002 A formally second-order BDF finite difference scheme for the integro-differential equations with the multi-term kernels. 2055-2073 2020 97 Int. J. Comput. Math. 10 https://doi.org/10.1080/00207160.2019.1677896 https://www.wikidata.org/entity/Q127108857 db/journals/ijcm/ijcm97.html#QiuXC20

Wenlin Qiu Da Xu Jing Guo Jun Zhou 0019 A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. 39-58 2020 85 Numer. Algorithms 1 https://doi.org/10.1007/s11075-019-00801-y https://www.wikidata.org/entity/Q127281107 db/journals/na/na85.html#QiuXGZ20

Wenlin Qiu Hongbin Chen 0002 Xuan Zheng An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations. 298-314 2019 166 Math. Comput. Simul. https://doi.org/10.1016/j.matcom.2019.05.017 https://www.wikidata.org/entity/Q127729881 db/journals/mcs/mcs166.html#QiuCZ19

Zakieh Avazzadeh Yang Cao Haifan Chen Hao Chen Hongbin Chen 0002 Yanping Chen 0002 Lisen Ding Jing Guo Tao Guo Xu Guo Yingwen Guo Yinnian He Ahmed S. Hendy Shufang Hu Qiong Huang 0004 Jinhong Jia Chuanting Jiang Huifa Jiang Kexin Li Yiqun Li Mengmeng Liu Man Luo 0004 Jie Ma Kassem Mustapha 0001 Omid Nikan Qing Pan Xiangyi Peng Ren-jun Qi Leijie Qiao Bo Tang Liang Tang Hong Wang 0018 Xu Xiao Da Xu Xuehua Yang Mahmoud A. Zaky Haixiang Zhang 0003 Hao Zhang Meihui Zhang Mingchao Zhao Xiangcheng Zheng Xuan Zheng Jun Zhou 0019