{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T21:34:31Z","timestamp":1777584871593,"version":"3.51.4"},"reference-count":61,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,5]],"date-time":"2024-05-05T00:00:00Z","timestamp":1714867200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article aims to provide a comprehensive review of the latest advancements in numerical methods and practical implementations in the field of fractional stochastic partial differential equations (FSPDEs). This type of equation integrates fractional calculus, stochastic processes, and differential equations to model complex dynamical systems characterized by memory and randomness. It introduces the foundational concepts and definitions essential for understanding FSPDEs, followed by a comprehensive review of the diverse numerical methods and analytical techniques developed to tackle these equations. Then, this article highlights the significant expansion in numerical methods, such as spectral and finite element methods, aimed at solving FSPDEs, underscoring their potential for innovative applications across various disciplines.<\/jats:p>","DOI":"10.3390\/sym16050563","type":"journal-article","created":{"date-parts":[[2024,5,6]],"date-time":"2024-05-06T15:05:01Z","timestamp":1715007901000},"page":"563","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Fractional Stochastic Partial Differential Equations: Numerical Advances and Practical Applications\u2014A State of the Art Review"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4957-9028","authenticated-orcid":false,"given":"Behrouz Parsa","family":"Moghaddam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan 1477893855, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6980-9786","authenticated-orcid":false,"given":"Afshin","family":"Babaei","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Mazandaran, Babolsar 4741613534, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4407-4314","authenticated-orcid":false,"given":"Arman","family":"Dabiri","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Mechatronics, Southern Illinois University, Edwardsville, IL 62026, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8262-1369","authenticated-orcid":false,"given":"Alexandra","family":"Galhano","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias Naturais, Engenharias e Tecnologias, Universidade Lus\u00f3fona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,5]]},"reference":[{"key":"ref_1","first-page":"2645","article-title":"Noise-driven signal study of power systems based on stochastic partial differential equations","volume":"23","author":"Chen","year":"2023","journal-title":"J. Comput. Methods Sci. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"100924","DOI":"10.1016\/j.ejcon.2023.100924","article-title":"On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems","volume":"75","author":"Lamoline","year":"2024","journal-title":"Eur. J. Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1319719","DOI":"10.3389\/fmars.2023.1319719","article-title":"Efficient 3D real-time adaptive AUV sampling of a river plume front","volume":"10","author":"Berild","year":"2023","journal-title":"Front. Mar. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111176","DOI":"10.1016\/j.automatica.2023.111176","article-title":"Solving nonlinear filtering problems with correlated noise based on Hermite\u2013Galerkin spectral method","volume":"156","author":"Sun","year":"2023","journal-title":"Automatica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"108833","DOI":"10.1016\/j.ymssp.2022.108833","article-title":"Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations","volume":"171","author":"Zhang","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1111\/mafi.12233","article-title":"A regularity structure for rough volatility","volume":"30","author":"Bayer","year":"2020","journal-title":"Math. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1080\/1350486X.2020.1758176","article-title":"Limit Order Books, Diffusion Approximations and Reflected SPDEs: From Microscopic to Macroscopic Models","volume":"27","author":"Hambly","year":"2020","journal-title":"Appl. Math. Financ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1111\/mafi.12386","article-title":"Credit risk pricing in a consumption-based equilibrium framework with incomplete accounting information","volume":"33","author":"Ma","year":"2023","journal-title":"Math. Financ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1137\/20M1357639","article-title":"Pricing Options under Rough Volatility with Backward SPDEs","volume":"13","author":"Bayer","year":"2022","journal-title":"Siam J. Financ. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1137\/19M1254489","article-title":"A stochastic partial differential equation model for limit order book dynamics","volume":"12","author":"Cont","year":"2021","journal-title":"Siam J. Financ. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Week, B., Nuismer, S.L., Harmon, L.J., and Krone, S.M. (2021). A white noise approach to evolutionary ecology. J. Theor. Biol., 521.","DOI":"10.1016\/j.jtbi.2021.110660"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"881885","DOI":"10.3389\/fphy.2022.881885","article-title":"Mass-Conservation Increases Robustness in Stochastic Reaction-Diffusion Models of Cell Crawling","volume":"10","author":"Moreno","year":"2022","journal-title":"Front. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3185","DOI":"10.1007\/s11538-019-00613-0","article-title":"Multiscale Stochastic Reaction-Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations","volume":"81","author":"Kang","year":"2019","journal-title":"Bull. Math. Biol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s10867-023-09644-0","article-title":"Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation","volume":"49","author":"Song","year":"2023","journal-title":"J. Biol. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Guivant, J., Narula, K., Kim, J., Li, X., and Khan, S. (2023). Compressed Gaussian Estimation under Low Precision Numerical Representation. Sensors, 23.","DOI":"10.20944\/preprints202305.1082.v1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3277","DOI":"10.1007\/s00122-019-03424-y","article-title":"Flexible modelling of spatial variation in agricultural field trials with the R package INLA","volume":"132","author":"Selle","year":"2019","journal-title":"Theor. Appl. Genet."},{"key":"ref_17","unstructured":"Pardoux, E. (1976). Stochastic Hilbert Space Integrals, Internal Publications, University Paris-Dauphine."},{"key":"ref_18","unstructured":"Bensoussan, A. (1978). Stochastic Control of Partially Observable Systems, Cambridge University Press."},{"key":"ref_19","unstructured":"Temam, R. (1984). Navier\u2013Stokes Equations: Theory and Numerical Analysis, North-Holland."},{"key":"ref_20","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.cam.2017.08.026","article-title":"Attractors for fractional differential problems of transition to turbulent flows","volume":"339","author":"Goufo","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press.","DOI":"10.1017\/CBO9781139170666"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.jsv.2016.10.013","article-title":"Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation","volume":"388","author":"Dabiri","year":"2017","journal-title":"J. Sound Vib."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3431","DOI":"10.1038\/srep03431","article-title":"Measuring memory with the order of fractional derivative","volume":"3","author":"Du","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_25","first-page":"207","article-title":"Application of variable-order fractional calculus in solid mechanics","volume":"7","author":"Moghaddam","year":"2019","journal-title":"Appl. Eng. Life Soc. Sci. Part A"},{"key":"ref_26","unstructured":"Cont, R. (2004). Financial Modeling with Jump Processes, Chapman and Hall\/CRC."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Fallahgoul, H., Focardi, S., and Fabozzi, F. (2016). Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application, Academic Press.","DOI":"10.1016\/B978-0-12-804248-9.50002-4"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.cnsns.2017.04.001","article-title":"The role of fractional calculus in modeling biological phenomena: A review","volume":"51","author":"Ionescu","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Awrejcewicz, J., and Tenreiro Machado, J.A. (2019). Entropy in Dynamic Systems. Entropy, 21.","DOI":"10.3390\/e21090896"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Tudor, C. (2022). Stochastic Partial Differential Equations with Additive Gaussian Noise\u2013Analysis And Inference, World Scientific Publishing Company.","DOI":"10.1142\/13089"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"031021","DOI":"10.1115\/1.4055366","article-title":"Symmetry Breaking in an Experimental Annular Combustor Model With Deterministic Electroacoustic Feedback and Stochastic Forcing","volume":"145","author":"Humbert","year":"2023","journal-title":"J. Eng. Gas Turbines Power"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4250","DOI":"10.1109\/TGRS.2016.2538820","article-title":"Spherical Symmetry of Complex Stochastic Models in Multivariate High-Resolution PolSAR Images","volume":"54","author":"Pralon","year":"2016","journal-title":"IEEE Trans. Geosci. Remote. Sens."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s12210-013-0234-4","article-title":"Stochastic models of chiral symmetry breaking in autocatalytic networks with anomalous fluctuations","volume":"24","author":"Longo","year":"2013","journal-title":"Rendiconti Lincei"},{"key":"ref_34","first-page":"185","article-title":"Symmetries and martingales in a stochastic model for the Navier-Stokes equation","volume":"162","author":"Lassalle","year":"2016","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Liu, Y., Khan, M., and Yan, Y. (2016). Fourier spectral methods for some linear stochastic space-fractional partial differential equations. Mathematics, 4.","DOI":"10.3390\/math4030045"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3173","DOI":"10.1137\/16M1096451","article-title":"Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations","volume":"55","author":"Li","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_37","first-page":"290","article-title":"Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises","volume":"24","author":"Liu","year":"2018","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Babaei, A., Jafari, H., and Banihashemi, S. (2020). A collocation approach for solving time-fractional stochastic heat equation driven by an additive noise. Symmetry, 12.","DOI":"10.3390\/sym12060904"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"107742","DOI":"10.1016\/j.cnsns.2023.107742","article-title":"Discrete Chebyshev polynomials for the numerical solution of stochastic fractional two-dimensional Sobolev equation","volume":"130","author":"Heydari","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"115441","DOI":"10.1016\/j.cam.2023.115441","article-title":"A numerical solution for a quasi solution of the time-fractional stochastic backward parabolic equation","volume":"437","author":"Nasiri","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1007\/s11075-018-0613-0","article-title":"Numerical solutions to time-fractional stochastic partial differential equations","volume":"82","author":"Zou","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"110346","DOI":"10.1016\/j.chaos.2020.110346","article-title":"Numerical study for time fractional stochastic semi linear advection diffusion equations","volume":"141","author":"Sweilam","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"103274","DOI":"10.1016\/j.probengmech.2022.103274","article-title":"Nonlinear random vibrations of micro-beams with fractional viscoelastic core","volume":"69","author":"Loghman","year":"2022","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1016\/j.camwa.2016.07.021","article-title":"Solution of stochastic nonlinear time fractional PDEs using polynomial chaos expansion combined with an exponential integrator","volume":"73","author":"Hosseini","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1007\/s40995-020-01036-6","article-title":"Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation","volume":"45","author":"Mirzaee","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s40819-017-0455-9","article-title":"An Efficient Solution for Stochastic Fractional Partial Differential Equations with Additive Noise by a Meshless Method","volume":"4","author":"Darehmiraki","year":"2018","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1090\/mcom\/3397","article-title":"Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise","volume":"88","author":"Gunzburger","year":"2018","journal-title":"Math. Comput."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4135","DOI":"10.1016\/j.camwa.2018.03.019","article-title":"A Galerkin finite element method for time-fractional stochastic heat equation","volume":"75","author":"Zou","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"091001","DOI":"10.1115\/1.4043725","article-title":"Analysis and Numerical Solutions for Fractional Stochastic Evolution Equations with Almost Sectorial Operators","volume":"14","author":"Ding","year":"2019","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1007\/s00366-019-00789-y","article-title":"Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic Advection-Diffusion equations","volume":"36","author":"Mirzaee","year":"2020","journal-title":"Eng. Comput."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.enganabound.2021.03.009","article-title":"Numerical solution of two-dimensional stochastic time-fractional Sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods","volume":"127","author":"Mirzaee","year":"2021","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"107371","DOI":"10.1016\/j.cnsns.2023.107371","article-title":"Strong convergence of a fractional exponential integrator scheme for finite element discretization of time-fractional SPDE driven by fractional and standard Brownian motions","volume":"125","author":"Noupelah","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_53","unstructured":"Podlubny, I. (2010). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"\u00d8ksendal, B. (2003). Stochastic Differential Equations, Springer. [6th ed.].","DOI":"10.1007\/978-3-642-14394-6"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Argun, A., Callegari, A., and Volpe, G. (2021). Simulation of Complex Systems, IOP Publishing.","DOI":"10.1088\/978-0-7503-3843-1ch14"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Parzen, E. (1999). Stochastic Processes, SIAM.","DOI":"10.1137\/1.9781611971125"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Lord, G., Powell, C., and Shardlow, T. (2014). An Introduction to Computational Stochastic PDEs, Cambridge University Press.","DOI":"10.1017\/CBO9781139017329"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1137\/1010093","article-title":"Fractional Brownian motions, fractional noises and applications","volume":"10","author":"Mandelbrot","year":"1968","journal-title":"Siam Rev."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1023\/A:1016547232119","article-title":"Time fractional diffusion: A discrete random walk approach","volume":"29","author":"Gorenflo","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.spl.2016.09.026","article-title":"Inverse source problem for time-fractional diffusion with discrete random noise","volume":"120","author":"Tuan","year":"2017","journal-title":"Stat. Probab. Lett."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"045002","DOI":"10.1088\/1361-6420\/ab532c","article-title":"An inverse random source problem in a stochastic fractional diffusion equation","volume":"36","author":"Niu","year":"2020","journal-title":"Inverse Probl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/563\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:39:50Z","timestamp":1760107190000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/563"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,5]]},"references-count":61,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["sym16050563"],"URL":"https:\/\/doi.org\/10.3390\/sym16050563","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,5]]}}}