{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:17:27Z","timestamp":1760059047195,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T00:00:00Z","timestamp":1747267200000},"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 paper introduces a novel numerical approach for solving fractional stochastic differential equations (FSDEs) using bilinear time-series models, driven by the Caputo\u2013Katugampola (C-K) fractional derivative. The C-K operator generalizes classical fractional derivatives by incorporating an additional parameter, enabling the enhanced modeling of memory effects and hereditary properties in stochastic systems. The primary contribution of this work is the development of an efficient numerical framework that combines bilinear time-series discretization with the C-K derivative to approximate solutions for FSDEs, which are otherwise analytically intractable due to their nonlinear and memory-dependent nature. We rigorously analyze the impact of fractional-order dynamics on system behavior. The bilinear time-series framework provides a computationally efficient alternative to traditional methods, leveraging multiplicative interactions between past observations and stochastic innovations to model complex dependencies. A key advantage of our approach is its flexibility in handling both stochasticity and fractional-order effects, making it suitable for applications in a famous nuclear physics model. To validate the method, we conduct a comparative analysis between exact solutions and numerical approximations, evaluating convergence properties under varying fractional orders and discretization steps. Our results demonstrate robust convergence, with simulations highlighting the superior accuracy of the C-K operator over classical fractional derivatives in preserving system dynamics. Additionally, we provide theoretical insights into the stability and error bounds of the discretization scheme. Using the changes in the number of simulations and the operator parameters of Caputo\u2013Katugampola, we can extract some properties of the stochastic fractional differential model, and also note the influence of Brownian motion and its formulation on the model, the main idea posed in our contribution based on constructing the fractional solution of a proposed fractional model using known bilinear time series illustrated by application in nuclear physics models.<\/jats:p>","DOI":"10.3390\/sym17050764","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T06:10:44Z","timestamp":1747289444000},"page":"764","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Solving Fractional Stochastic Differential Equations via a Bilinear Time-Series Framework"],"prefix":"10.3390","volume":"17","author":[{"given":"Rami","family":"Alkhateeb","sequence":"first","affiliation":[{"name":"Department of Allied Sciences, AL-Ahliyya Amman University, Amman 19328, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ma\u2019mon","family":"Abu Hammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, AL-Zaytoonah University of Jordan, Queen Alia Airport St. 594, Amman 11942, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3114-7788","authenticated-orcid":false,"given":"Basma","family":"AL-Shutnawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tafila Technical University, P.O. Box 179, Tafila 66110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nabil","family":"Laiche","sequence":"additional","affiliation":[{"name":"Applied Mathematics Department, University of Seville, 41012 Seville, Spain"},{"name":"Dynamical Systems and Control Laboratory, Oum El Bouaghi University, Oum El Bouaghi 04000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7621-6711","authenticated-orcid":false,"given":"Zouaoui","family":"Chikr El Mezouar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,15]]},"reference":[{"key":"ref_1","unstructured":"Rothman, P. (2012). Nonlinear Time Series Analysis of Economic and Financial Data, Springer."},{"key":"ref_2","unstructured":"Rao, T.S., and Gabr, M.M. 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