An Explicit Finite Difference Scheme for Time-Fractional Stochastic Traveling Wave Equations Driven by Multiplicative White Noise

Mojad, Priti; Gavhane, Babasaheb; Gaikwad, Shrikisan; Takale, Kalyanrao; Shirore, Chetan

doi:10.30473/coam.2026.77231.1394

Articles in Press

Document Type : Research Article

Authors

¹ Department of Mathematics, New Arts Commerce and Science College, Ahmednagar, affiliated to Savitribai Phule Pune University, Pune, India

² Department of Mathematics, K. J. Somaiya College of Arts, Commerce and Science, Kopargaon, India

³ Department of Mathematics, New Arts Commerce and Science College, Ahmednagar, India

⁴ Department of Computer Science, Jaykranti College of Computer Science and Management, Pune, India

⁵ Department of Mathematics, K.R.T. Arts, B.H. Commerce and A.M. Science College, Nashik, India

https://doi.org/10.30473/coam.2026.77231.1394

Abstract

This work addresses the numerical approximation of the time-fractional stochastic traveling wave equation (TFSTWE) with a Caputo fractional derivative of order α ∈ (1, 2] driven by multiplicative space-time white noise. An explicit finite difference method (FDM) is constructed by discretizing the space domain. For the first time step, a mean-square stability bound and an error estimate of order O(∆t^2-^α + ∆x²) are derived under a restriction on ∆t that depends explicitly on ∆x, the diffusion coefficient K, and the fractional order α, while the noise intensity σ enters the mean-square growth constant; stability and convergence at later time levels are supported numerically. Comprehensive numerical experiments demonstrate the effectiveness and reliability of the proposed method, and the Python programming language is used to plot the approximate mean solution profiles in two and three dimensions. The outcomes confirm the suitability of the method for accurately solving TFSTWEs under stochastic perturbations and support the theoretical analysis.

Highlights

An explicit FDM is proposed for time-fractional stochastic traveling wave equations.
Caputo derivative of order α ∈ (1, 2] with multiplicative space-time white noise.
Conditional mean-square stability proved via mathematical induction.
Established convergence order O(Δx² + Δt²) for the explicit finite difference scheme.
2D and 3D mean solution profiles computed in Python across noise intensities σ ∈ {0,1,2,3}.

Keywords

Main Subjects

Numerical Methods

References

[1] Bayer, C., Qiu, J., Yao, Y. (2022). “Pricing options under rough volatility with backward SPDEs”. SIAM Journal on Financial Mathematics, 13, 179–212. https://doi.org/10.1137/20M1357639

[2] Chen, L., Hu, G., Hu, Y., Hu, J. (2017). “Space-time fractional diffusions in Gaussian noisy environment”. Stochastics, 89, 171–206. https://doi.org/10.1080/17442508.2016.1146282

[3] Chukwuyem, N. J., Nkonyeasua, N. I., Apanapudor, J. S. (2025). “Crank-Nicolson finite element method for the time fractional stochastic wave equation”. Mathematics and Statistics, 13(3), 154–162. https://doi.org/10.13189/ms.2025.130304

[4] Darehmiraki, M. (2018). “An efficient solution for stochastic fractional partial differential equations with additive noise by a meshless method”. International Journal of Applied and Computational Mathematics, 4, 1–13. https://doi.org/10.1007/ s40819-017-0455-9

[5] Demir, I. (2025). “Refinements of the Jensen inequality and estimates of the Jensen gap based on interval-valued functions”. Mathematical Methods in the Applied Sciences, 48, 12567–12576. https://doi.org/10.1002/mma.11046

[6] Ghode, K., Takale, K., Gaikwad, S. (2021). “New technique for solving time fractional wave equation: Python”. Journal of Mathematical and Computational Science, 11, 5327– 5343. https://www.scik.org/index.php/jmcs/article/view/6048

[7] Goufo, F. D., Nieto, J. J. (2018). “Attractors for fractional differential problems of transition to turbulent flows”. Journal of Computational and Applied Mathematics, 339, 329– 342. https://doi.org/10.1016/j.cam.2017.08.026

[8] Hoult, J., Yan, Y. (2024). “Numerical approximation for a stochastic fractional differential equation driven by integrated multiplicative noise”. Mathematics, 12(3), Art. no. 365. https://doi.org/10.3390/math12030365

[9] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. (2006). Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Amsterdam: Elsevier.

[10] Li, J., Zhang, J., Jiang, Y. (2025). “A finite difference method for Caputo generalized time fractional diffusion equations”. Fractal and Fractional, 10, Art. no. 19. https://doi.org/10.3390/fractalfract10010019

[11] Ma, J., Ogunsolu, M., Qiu, J., Sezer, D. (2023). “Credit risk pricing in a consumption-based equilibrium framework with incomplete accounting information”. Mathematical Finance, 33, 666–708. https://doi.org/10.1111/mafi.12386

[12] Mirzaee, F., Samadyar, N. (2020). “Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations”. Engineering with Computers, 36, 1673–1686. https://doi.org/10.1007/s00366-019-00789-y

[13] Mirzaee, F., Sayevand, K., Rezaei, S., Samadyar, N. (2021). “Finite difference and spline approximation for solving fractional stochastic advection-diffusion equation”. Iranian Journal of Science and Technology, Transactions A: Science, 45, 607–617. https: //doi.org/10.1007/s40995-020-01036-6

[14] Moghaddam, B. P., Babaei, A., Dabiri, A., Galhano, A. (2024). “Fractional stochastic partial differential equations: Numerical advances and practical applications—A state of the art review”. Symmetry, 16, Art. no. 563. https://doi.org/10.3390/sym16050563

[15] Mojad, P., Gavhane, B., Gaikwad, S., Takale, K. (2025). “Crank-Nicolson approximation to space-time fractional stochastic traveling wave equation with additive white noise”. Boletim da Sociedade Paranaense de Matemática, 43, 1–15. https://doi.org/10.5269/ bspm.77257

[16] Roth, C. H. (2002). “Difference methods for stochastic partial differential equations”. ZAMM Journal of Applied Mathematics and Mechanics, 82, 821–830. https://doi. org/10.1002/1521-4001(200211)82:11/12<821::AID-ZAMM821>3.0.CO;2-L

[17] Sababheh, M., Choi, D. (2016). “A complete refinement of Young’s inequality”. Journal of Mathematical Analysis and Applications, 440(1), 379–393. https://doi.org/10.1016/j.jmaa.2016.03.049

[18] Singh, A. K., Mehra, M. (2023). “Difference methods for stochastic space fractional diffusion equation driven by additive space–time white noise via Wong–Zakai approximation”. Journal of Mathematical Chemistry, 61, 47–74. https://doi.org/10.1007/ s10910-022-01408-w

[19] Singh, S., Singh, S., Aggarwal, A. (2023). “A new spline technique for the time fractional diffusion-wave equation”. MethodsX, 10, Art. no. 102007. https://doi.org/10.1016/ j.mex.2023.102007

[20] Song, E. (2023). “Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation”. Journal of Biological Physics, 49, 521–533. https://doi.org/10.1007/s10867-023-09644-0

[21] Song, J., Song, X., Xu, F. (2020). “Fractional stochastic wave equation driven by Gaussian noise rough in space”. Bernoulli, 26(4), 2699–2726. https://doi.org/10.3150/ 20-BEJ1204

[22] Sun, Z., Wu, X. (2006). “A fully discrete difference scheme for a diffusion-wave system”. Applied Numerical Mathematics, 56(2), 193–209. https://doi.org/10.1016/j.apnum.2005.03.003

[23] Week, B., Nuismer, S. L., Harmon, L. J., Krone, S. M. (2021). “A white noise approach to evolutionary ecology”. Journal of Theoretical Biology, 521, Art. no. 110660. https: //doi.org/10.1016/j.jtbi.2021.110660

[24] Zhang, Z., Karniadakis, G. E. (2017). “Numerical methods for stochastic partial differential equations with white noise”. Applied Mathematical Sciences, 196. https://doi. org/10.1007/978-3-319-57511-7

Control and Optimization in Applied Mathematics

An Explicit Finite Difference Scheme for Time-Fractional Stochastic Traveling Wave Equations Driven by Multiplicative White Noise

References

References

Articles in Press, Corrected Proof
Available Online from 24 September 2026

An Explicit Finite Difference Scheme for Time-Fractional Stochastic Traveling Wave Equations Driven by Multiplicative White Noise

References

References

Articles in Press, Corrected Proof Available Online from 24 September 2026

Articles in Press, Corrected Proof
Available Online from 24 September 2026