Journals Information
Mathematics and Statistics Vol. 13(3), pp. 154 - 162
DOI: 10.13189/ms.2025.130304
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Crank-Nicolson Finite Element Method for the Time Fractional Stochastic Wave Equation
Nwankwo Jude Chukwuyem 1,*, Njoseh Ignatius Nkonyeasua 2, Joshua Sarduana Apanapudor 2
1 Department of Mathematics, University of Delta, Nigeria
2 Department of Mathematics, Delta State University, Nigeria
ABSTRACT
The Crank-Nicolson Finite Element Method (CNFEM) provides a robust and efficient framework for solving time-fractional stochastic wave equations, which are essential in modeling dynamic systems influenced by both memory effects and random perturbations. These equations often arise in fields such as geophysics, engineering, and financial mathematics, where processes exhibit fractional-order characteristics and stochastic influences. In our approach, we construct a numerical scheme that integrates the Crank-Nicolson method with the finite element technique to achieve an accurate approximation of the solution. The time-fractional derivative is managed using the Caputo definition, ensuring the incorporation of non-local temporal effects that characterize fractional processes. The stochastic component is introduced through white noise or noise perturbations, which are essential for modeling real-world uncertainties. By leveraging CNFEM, we ensure stability in time discretization while offering flexibility in spatial discretization, making it particularly useful for handling complex and irregular computational domains. A rigorous analysis of the proposed numerical scheme is conducted to examine its convergence and stability. The scheme is shown to be unconditionally stable, meaning it does not impose restrictive conditions on the time step or spatial mesh size, thereby enhancing computational efficiency. The numerical implementation of our method is carried out using MAPLE 18, a powerful symbolic and numerical computation tool, which aids in performing high-precision calculations and symbolic manipulations. By applying CNFEM to time-fractional stochastic wave equation, our study provides a reliable and efficient numerical strategy for simulating complex dynamical systems. The results demonstrate the method's capability in capturing both the fractional-order memory effects and stochastic behaviors inherent in these equations, making it a valuable tool for researchers in applied mathematics and computational science.
KEYWORDS
Finite Element Method, Stochastic Wave Equation, Fractional Stochastic Processes, Crank-Nicolson Finite Element Method, Noise Term
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Nwankwo Jude Chukwuyem , Njoseh Ignatius Nkonyeasua , Joshua Sarduana Apanapudor , "Crank-Nicolson Finite Element Method for the Time Fractional Stochastic Wave Equation," Mathematics and Statistics, Vol. 13, No. 3, pp. 154 - 162, 2025. DOI: 10.13189/ms.2025.130304.
(b). APA Format:
Nwankwo Jude Chukwuyem , Njoseh Ignatius Nkonyeasua , Joshua Sarduana Apanapudor (2025). Crank-Nicolson Finite Element Method for the Time Fractional Stochastic Wave Equation. Mathematics and Statistics, 13(3), 154 - 162. DOI: 10.13189/ms.2025.130304.