{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T06:43:49Z","timestamp":1780728229706,"version":"3.54.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T00:00:00Z","timestamp":1739491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T00:00:00Z","timestamp":1739491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s11075-025-02026-8","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T05:16:46Z","timestamp":1739510206000},"page":"883-914","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A novel meshless approach based on new fractional spaces for solving nonlinear Riesz space distributed order reaction-diffusion equations with non-smooth solutions and stability analysis"],"prefix":"10.1007","volume":"101","author":[{"given":"Lin","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhong","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Du","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Biao","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,2,14]]},"reference":[{"key":"2026_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2023.105991","volume":"265","author":"L Liu","year":"2023","unstructured":"Liu, L., Chen, S., Feng, L., Zhu, J., Zhang, J., Zheng, L., Xie, C.: A novel distributed order time fractional model for heat conduction, anomalous diffusion, and viscoelastic flow problems. Comput. Fluids. 265, 105991 (2023)","journal-title":"Comput. Fluids."},{"key":"2026_CR2","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1016\/j.matcom.2023.07.021","volume":"215","author":"N Li","year":"2024","unstructured":"Li, N., Wang, X., Xu, H., Qi, H.: Numerical study on radiative mhd flow of viscoelastic fluids with distributed-order and variable-order space fractional operators. Math. Comput. Simulat. 215, 291\u2013305 (2024)","journal-title":"Math. Comput. Simulat."},{"key":"2026_CR3","volume":"427","author":"Q Yu","year":"2022","unstructured":"Yu, Q., Turner, I., Liu, F., Vegh, V.: The application of the distributed-order time fractional bloch model to magnetic resonance imaging. Appl. Math. Comput. 427, 127188 (2022)","journal-title":"Appl. Math. Comput."},{"key":"2026_CR4","doi-asserted-by":"crossref","unstructured":"Feng, L., Turner, I., Moroney, T., Liu, F.: Moroney and F. Liu, An investigation of space distributed-order models for simulating anomalous transport in a binary medium. Appl. Math. Comput. 434, 127423 (2022)","DOI":"10.1016\/j.amc.2022.127423"},{"key":"2026_CR5","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","volume":"64","author":"HG Sun","year":"2018","unstructured":"Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. 64, 213\u2013231 (2018)","journal-title":"Commun. Nonlinear Sci."},{"key":"2026_CR6","doi-asserted-by":"crossref","unstructured":"Abbaszadeh, M., Bagheri\u00a0Salec, A., Kamel Abd Al-Khafaji, S.: A reduced-order jacobi spectral collocation method for solving the space-fractional fitzhugh-nagumo models with application in myocardium. Eng. Computation. 40, 2980\u20133008 (2023)","DOI":"10.1108\/EC-06-2023-0254"},{"key":"2026_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.camwa.2022.03.021","volume":"113","author":"Y Zhang","year":"2022","unstructured":"Zhang, Y., Fan, Y., Li, Y.: General linear and spectral galerkin methods for the nonlinear two-sided space distributed-order diffusion equation. Comput. Math. Appl. 113, 1\u201312 (2022)","journal-title":"Comput. Math. Appl."},{"key":"2026_CR8","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/j.camwa.2023.09.032","volume":"151","author":"M Yousuf","year":"2023","unstructured":"Yousuf, M., Furati, K.M., Khaliq, A.Q.M.: A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations. Comput. Math. Appl. 151, 116\u2013126 (2023)","journal-title":"Comput. Math. Appl."},{"issue":"5","key":"2026_CR9","doi-asserted-by":"publisher","first-page":"382","DOI":"10.3390\/fractalfract7050382","volume":"7","author":"M Zhang","year":"2023","unstructured":"Zhang, M., Shen, M., Chen, H.: An Implicit Numerical Method for the Riemann-Liouville Distributed-Order Space Fractional Diffusion Equation. Fractal Fract. 7(5), 382 (2023)","journal-title":"Fractal Fract."},{"key":"2026_CR10","doi-asserted-by":"publisher","first-page":"1850073","DOI":"10.1142\/S0217984918500732","volume":"32","author":"R Cheng","year":"2018","unstructured":"Cheng, R., Sun, F., Wei, Q., Wang, J.: Numerical treatment for solving two-dimensional space-fractional advection-dispersion equation using meshless method. Mod. Phys. Lett. 32, 1850073 (2018)","journal-title":"Mod. Phys. Lett."},{"key":"2026_CR11","doi-asserted-by":"publisher","first-page":"259","DOI":"10.3390\/axioms10040259","volume":"10","author":"M Raei","year":"2021","unstructured":"Raei, M., Cuomo, S.: An efficient localized meshless method based on the space-time gaussian rbf for high-dimensional space fractional wave and damped equations. Axioms. 10, 259 (2021)","journal-title":"Axioms."},{"key":"2026_CR12","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1016\/j.camwa.2022.10.020","volume":"140","author":"T Jiang","year":"2023","unstructured":"Jiang, T., Wang, X.C., Ren, J.L., Huang, J.J., Yuan, J.Y.: A high-efficient accurate coupled mesh-free scheme for 2D\/3D space-fractional convection-diffusion\/Burgers\u2019 problems. Comput. Math. Appl. 140, 260\u2013281 (2023)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"2026_CR13","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1002\/num.22933","volume":"39","author":"X Chen","year":"2023","unstructured":"Chen, X., Chen, J., Liu, F., Sun, Z.Z.: A fourth-order accurate numerical method for the distributed-order Riesz space fractional diffusion equation. Numer. Meth. Part. D. E. 39(2), 1266\u20131286 (2023)","journal-title":"Numer. Meth. Part. D. E."},{"key":"2026_CR14","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s00466-020-01917-y","volume":"67","author":"HG Sun","year":"2021","unstructured":"Sun, H.G., Wang, Z.Y., Nie, J.Y., Zhang, Y., Xiao, R.: Generalized finite difference method for a class of multidimensional space-fractional diffusion equations. Comput. Mech. 67, 17\u201332 (2021)","journal-title":"Comput. Mech."},{"key":"2026_CR15","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.enganabound.2023.05.046","volume":"155","author":"S Fouladi","year":"2023","unstructured":"Fouladi, S., Mohammadi-Firouzjaei, H.: Local discontinuous galerkin method for the riesz space distributed-order sobolev equation. Eng. Anal. Bound. Elem. 155, 38\u201347 (2023)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"2026_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s12190-023-01954-9","volume":"70","author":"Y Wang","year":"2024","unstructured":"Wang, Y., Huang, J., Li, H.: A second-order difference scheme for two-dimensional two-sided space distributed-order fractional diffusion equations with variable coefficients. J. Appl. Math. Comput. 70, 1\u201326 (2024)","journal-title":"J. Appl. Math. Comput."},{"key":"2026_CR17","doi-asserted-by":"publisher","first-page":"10583","DOI":"10.1016\/j.enganabound.2024.105838","volume":"166","author":"JL Ren","year":"2024","unstructured":"Ren, J.L., Wang, Y.C., Jiang, T., Jiang, R.R., Wang, D.S.: A novel semi-implicit WLS scheme for time-memory nonlinear behavior in 2D variable-order TF-NLSEs. Eng. Anal. Bound. Elem. 166, 10583 (2024)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"2026_CR18","doi-asserted-by":"publisher","first-page":"101","DOI":"10.2298\/TSCI23S1101S","volume":"27","author":"M Shakeel","year":"2023","unstructured":"Shakeel, M., Khan, M.N., Ahmad, I., Ahmad, H., Jarasthitikulchai, N., Sudsutad, W.: Local meshless collocation scheme for numerical simulation of space fractional pde. Therm. Sci. 27, 101\u2013109 (2023)","journal-title":"Therm. Sci."},{"key":"2026_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2023.109023","volume":"296","author":"T Jiang","year":"2024","unstructured":"Jiang, T., Liu, Y.H., Li, Q., Ren, J.L., Wang, D.S.: An accelerated novel meshless coupled algorithm for non-local nonlinear behavior in 2d\/3d space-fractional gpes. Comput. Phys. Commun. 296, 109023 (2024)","journal-title":"Comput. Phys. Commun."},{"key":"2026_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.camwa.2023.12.007","volume":"156","author":"J Yang","year":"2024","unstructured":"Yang, J., Li, Y., Liu, Z.: A finite difference\/kansa method for the two-dimensional time and space fractional bloch-torrey equation. Comput. Math. Appl. 156, 1\u201315 (2024)","journal-title":"Comput. Math. Appl."},{"key":"2026_CR21","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1137\/0719003","volume":"19","author":"M Luskin","year":"1982","unstructured":"Luskin, M., Rannacher, R.: On the smoothing property of the galerkin method for parabolic equations. SIAM J. Numer. Anal. 19, 93\u2013113 (1982)","journal-title":"SIAM J. Numer. Anal."},{"key":"2026_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-019-01117-8","volume":"82","author":"MA Zaky","year":"2020","unstructured":"Zaky, M.A., Hendy, A.S., Mac\u00edas-D\u00edaz, J.E.: Semi-implicit galerkin-legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions. J. Sci. Comput. 82, 1\u201327 (2020)","journal-title":"J. Sci. Comput."},{"key":"2026_CR23","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1016\/j.apnum.2020.05.002","volume":"156","author":"AS Hendy","year":"2020","unstructured":"Hendy, A.S., Zaky, M.A.: Global consistency analysis of l1-galerkin spectral schemes for coupled nonlinear space-time fractional schr\u00f6dinger equations. Appl. Numer. Math. 156, 276\u2013302 (2020)","journal-title":"Appl. Numer. Math."},{"key":"2026_CR24","doi-asserted-by":"crossref","unstructured":"Du, H., Chen, Z.: A new method of solving the Riesz fractional advection-dispersion equation with nonsmooth solution. Appl. Math. Lett. 109022 (2024)","DOI":"10.1016\/j.aml.2024.109022"},{"key":"2026_CR25","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1016\/j.enganabound.2021.03.003","volume":"127","author":"M Hosseininia","year":"2021","unstructured":"Hosseininia, M., Heydari, M.H., Ghaini, F.M., Avazzadeh, Z.: A meshless technique based on the moving least squares shape functions for nonlinear fractal-fractional advection-diffusion equation. Eng. Anal. Bound. Elem. 127, 8\u201317 (2021)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"2026_CR26","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Wang, J., Zhang, L.: Basic theory of fractional differential equations. World scientific. (2016)","DOI":"10.1142\/10238"},{"key":"2026_CR27","volume-title":"The analysis of fractional differential equations","author":"J Morel","year":"2004","unstructured":"Morel, J., Takens, F., Teissier, B.: The analysis of fractional differential equations. Lecture Notes in Mathematics, Springer, Berlin (2004)"},{"key":"2026_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-28769-0","volume-title":"Theoretical numerical analysis","author":"K Atkinson","year":"2005","unstructured":"Atkinson, K., Han, W.: Theoretical numerical analysis. Springer, Berlin (2005)"},{"key":"2026_CR29","unstructured":"Adams R.A., Fournier, J.J.: Sobolev spaces. Elsevier. (2003)"},{"key":"2026_CR30","unstructured":"Cui, M., Lin, Y.: Nonlinear numerical analysis in reproducing kernel space. Nova Science Publishers, Inc. (2009)"},{"key":"2026_CR31","unstructured":"Chen, Z., Jiang, W.: Developments of reproducing kernel methods for sovling fractional differential equations. Harbin institute of technology press. (2020)"},{"key":"2026_CR32","unstructured":"Ding, G.: Introduction to Banach spaces. Science Press. (1984)"},{"key":"2026_CR33","volume-title":"Principle of numerical analysis","author":"B Wu","year":"2003","unstructured":"Wu, B., Wang, D., Ding, X., Li, D.: Principle of numerical analysis. Science Press, Beijing (2003)"},{"key":"2026_CR34","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1016\/j.matcom.2021.06.004","volume":"190","author":"RM Hafez","year":"2021","unstructured":"Hafez, R.M., Zaky, M.A., Hendy, A.S.: A novel spectral galerkin\/petrov-galerkin algorithm for the multi-dimensional space-time fractional advection-diffusion-reaction equations with nonsmooth solutions. Math. Comput. Simulat. 190, 678\u2013690 (2021)","journal-title":"Math. Comput. Simulat."},{"key":"2026_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108004","volume":"130","author":"H Du","year":"2022","unstructured":"Du, H., Chen, Z.: A new meshless method of solving 2d fractional diffusion-wave equations. Appl. Math. Lett. 130, 108004 (2022)","journal-title":"Appl. Math. Lett."},{"key":"2026_CR36","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods: fundamentals in single domains. Springer Science & Business Media. (2007)","DOI":"10.1007\/978-3-540-30728-0"},{"key":"2026_CR37","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/0021-9045(73)90050-6","volume":"7","author":"R Carlson","year":"1973","unstructured":"Carlson, R., Hall, C.: Error bounds for bicubic spline interpolation. J. Approx. Theory. 7, 41\u201347 (1973)","journal-title":"J. Approx. Theory."},{"issue":"6","key":"2026_CR38","doi-asserted-by":"publisher","first-page":"1303","DOI":"10.1007\/s11425-013-4716-8","volume":"57","author":"J Huang","year":"2014","unstructured":"Huang, J., Nie, N., Tang, Y.: A second order finite difference-spectral method for space fractional diffusion equations. Sci. China Math. 57(6), 1303\u20131317 (2014)","journal-title":"Sci. China Math."},{"issue":"2","key":"2026_CR39","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1002\/num.20169","volume":"23","author":"VJ Ervin","year":"2007","unstructured":"Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in $$\\mathbb{R} ^{d}$$. Numer. Methods Partial Differ. Equ. 23(2), 256\u2013281 (2007)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"2026_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.enganabound.2024.105932","volume":"169","author":"M Abbaszadeh","year":"2024","unstructured":"Abbaszadeh, M., Azis, M.I., Dehghan, M.: A mesh-free method using Pascal polynomials for analyzing space-fractional PDEs in irregular biological geometries. Eng. Anal. Bound. Elem. 169, 105932 (2024)","journal-title":"Eng. Anal. Bound. Elem."},{"issue":"8","key":"2026_CR41","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1080\/00207160.2012.667561","volume":"89","author":"W Jiang","year":"2012","unstructured":"Jiang, W., Chen, Z., Zhang, C.P.: Chebyshev collocation method for solving singular integral equation with cosecant kernel. Int. J. Comput. Math. 89(8), 975\u2013982 (2012)","journal-title":"Int. J. Comput. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02026-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-025-02026-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02026-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T08:08:19Z","timestamp":1770624499000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-025-02026-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,14]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["2026"],"URL":"https:\/\/doi.org\/10.1007\/s11075-025-02026-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,14]]},"assertion":[{"value":"24 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 February 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}