AN EFFECTIVE COMPUTATIONAL SCHEME FOR SOLVING VARIOUS MATHEMATICAL FRACTIONAL DIFFERENTIAL MODELS VIA NON-DYADIC HAAR WAVELETS FRACTIONAL DIFFERENTIAL MODELS VIA NON-DYADIC HAAR WAVELETS
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Abstract
The Non-dyadic Haar wavelet (Haar scale-3) collocation approach is used in this article to generate numerical solutions of fractional differential equations. The nonlinear fractional ordinary differential equations are linearized using the Quasilinearisation technique. The Haar scale-3 wavelet approach works on transforming the set of differential calculations into a set of linear algebraic equations. The reliability of the numerical solution can be improved by raising the degree of resolution, and error analysis can be done. The numerical examples are solved to test the method's simplicity and flexibility. The outcomes of numerical examples are compatible with the exact solution and provide better results than previous results existing in literature. This means that the procedure used here is consistent, reliable, and convenient to use.
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Licensee MJS, Universiti Malaya, Malaysia. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
References
Almeida, R., & Bastos, N. R. O. (2016). A numerical method to solve higher-order fractional differential equations. Mediterranean Journal of Mathematics, 13(3), 1339–1352.
Amin, R., Shah, K., Asif, M., Khan, I., & Ullah, F. (2021). An efficient algorithm for numerical solution of fractional integrodifferential equations via Haar wavelet. Journal of Computational and Applied Mathematics, 381, 113028.
Arora, G., Kumar, R., & Kaur, H. (2020). Scale-3 Haar wavelet and Quasi-linearisation based hybrid technique for solution of coupled space-time fractional-Burgers’ equation. Pertanika Journal of Science & Technology, 28(2), 579–607.
Baleanu, D., & Shiri, B. (2018). Collocation methods for fractional differential equations involving non-singular kernel. Chaos, Soliton & Fractals, 116, 136–145.
Chen, Y., Yi, M., & Yu, C. (2012). Error analysis for numerical solution of fractional differential equation by Haar wavelets method. Journal of Computational Science, 3(5), 367–373.
Das, S. (2011). Functional fractional calculus. Springer.
Gowrisankar, A., & Uthayakumar, R. (2016). Fractional calculus on fractal interpolation for a sequence of data with countable iterated function system. Mediterranean Journal of Mathematics, 13(6), 3887–3906.
Kobra, R., & Mohsen, R. (2021). Fractional-order Boubaker wavelets method for solving fractional Riccati differential equations. Applied Numerical Mathematics, 168, 221–234.
Kumar, R., & Bakhtawar, S. (2022). An improved algorithm based on Haar scale 3 wavelets for the numerical solution of Integro-differential equations. Mathematics in Engineering, Science & Aerospace (MESA), 13(2), 617–633.
Kumar, R., & Gupta, J. (2022). Numerical analysis of linear and non-linear dispersive equation using Haar scale-3 wavelet. Mathematics in Engineering, Science & Aerospace (MESA), 13(4), 993–1006.
Li, C., & Zeng, F. (2015). Numerical methods for fractional calculus. Chapman & Hall/CRC.
Miller, K. S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley.
Mittal, R. C., & Pandit, S. (2018a). New scale-3 Haar wavelets algorithm for numerical simulation of second order ordinary differential equations. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 89(4), 799–808.
Mittal, R. C., & Pandit, S. (2018b). Quasi-linearized scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems. Engineering Computations (Swansea, Wales), 35(5), 1907–1931.
Odibat, Z., & Momani, S. (2008). Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order. Chaos, Solitons & Fractals, 36(1), 167–174.
Oldham, K. B., & Spanier, J. (1974). The fractional calculus: Theory and applications of differentiation and integration to arbitrary order. Academic Press.
Podlubny, I. (1999). Fractional differential equations. Academic Press.
Saeed, U., & Rehman, M. U. (2013). Haar wavelet–Quasi-linearization technique for fractional nonlinear differential equations. Applied Mathematics and Computation, 220, 630–648.
Shah, F. A., Abass, R., & Debnath, L. (2017). Numerical solution of fractional differential equations using Haar wavelet operational matrix method. International Journal of Applied and Computational Mathematics, 3(3), 2423–2445.
Shah, K., Khan, Z. A., Ali, A., Amin, R., Khan, H., and Khan, A. (2022). Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative. Alexandria Engineering Journal, 59(5), 3221–3231.
Shiralashetti, S. C., & Deshi, A. B. (2016). Haar wavelet collocation method for solving Riccati and fractional Riccati differential equations. Bulletin of Mathematical Sciences and Applications, 17, 46–56.
Su, H., Chen, W., Li, C., & Chen, Y. (2012). Finite difference schemes for variable-order time fractional diffusion equation. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 22(04), 1250–1085.