A higher order stable numerical approximation for time‐fractional non‐linear Kuramoto–Sivashinsky equation based on quintic B‐‐spline

• Published in: Mathematical methods in the applied sciences 2024-10, Vol.47 (15), p.11953-11975
• Main Authors: Choudhary, Renu, Singh, Satpal, Das, Pratibhamoy, Kumar, Devendra
• Format: Article
• Language: English
• Subjects: Accuracy; Approximation; Burgers equation; Caputo derivative; computational cost; convergence; Derivatives; higher order splines; L1 scheme; L1‐2 scheme; Linear equations; Mathematical analysis; Numerical analysis; Spline functions; stability analysis; time varying layer; time‐fractional Kuramoto–Sivashinsky equation
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9778

This article deals with designing and analyzing a higher order stable numerical analysis for the time‐fractional Kuramoto–Sivashinsky (K‐S) equation, which is a fourth‐order non‐linear equation. The fractional derivative of order γ∈(0,1)$$ \upgamma \in \left(0,1\right) $$ present in the considered problem is taken into Caputo sense and approximated using the L1−2$$ L1-2 $$ scheme. In space direction, the discretization process uses quintic B$$ \mathfrak{B} $$‐spline functions to approximate the derivatives and the solution of the problem. The present approach is unconditionally stable and is convergent with rate of accuracy O(h2+k2)$$ \mathcal{O}\left({h}^2+{k}^2\right) $$, where h$$ h $$ and k$$ k $$ denote the space and time step sizes, respectively. We have also noted that the linearized version of the K‐S equation leads the rate of accuracy to O(h2+k3−γ)$$ \mathcal{O}\left({h}^2+{k}^{3-\upgamma}\right) $$. The present approach is also highly effective for the time‐fractional Burgers' equation. We have shown that the present approach provides better accuracy than the L1$$ L1 $$ scheme with the same computational cost for several linear/non‐linear problems, with classical as well as fractional time derivatives.