Comparative Numerical Solution of Fractional Spline with Continuity Equations
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Abstract
In this paper, constructed a fractional polynomial spline to compute the solution of FDEs; the spline interpolation with fractional polynomial coefficients must be constructed using the Caputo fractional derivative. For the provided spline function, error bounds were studied and a stability analysis was completed. To consider the numerical explanation for the provided method and compared, three examples were studied. The fractional spline function, which interpolates data, appears to be useful and accurate in solving unique problems, according to the research.
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References
[1] Kammanee, A. (2021). Numerical Solutions of Fractional Differential Equations with Variable Coe cients by Taylor Basis Functions. Kyungpook Mathematical Journal, 61(2), 383-393. [2] Srivastava, R. (2015). On Lacunary Interpolation through g-Splines. International Journal of Innovative Research in Science, Engineering and Technology 2015, 4, 4667-4670. [3] Wold, S. (1974). Spline functions in data analysis. Technometrics, 16(1), 1-11. [4] Mainardi, F. (2010). Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific. [5] Amin, M., Abbas, M., Iqbal, M. K., & Baleanu, D. (2019). Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations. Advances in Difference Equations, 2019(1), 1-22. [6] Jiang, Z. (2017). A New Approximation Method with High Order Accuracy. Mathematical and Computational Applications, 22(1), 11. [7] Lang, F. G., & Xu, X. P. (2014). Error Analysis for a Noisy Lacunary Cubic Spline Interpolation and a Simple Noisy Cubic Spline Quasi Interpolation. Adv. Numer. Anal., 2014, 353194-1 [8] Zahra, W. K., & Elkholy, S. M. (2012). The use of cubic splines in the numerical solution of fractional differential equations. International Journal of Mathematics and Mathematical Sciences, 2012. [9] Hamasalh, F. K., & Ali, A. H. (2019, April). On the generalized fractional cubic spline with application. In AIP Conference Proceedings (Vol. 2096, No. 1, p. 020004). AIP Publishing LLC. [10] Hamasalh, F. K., & Ali, A. H. Stability Analysis of Some Fractional Differential Equations by Special type of Spline Function. Journal of Zankoy Sulaimani, 19(1). [11] Hamasalh, F. K., & Headayat, M. A. (2021, March). The applications of non-polynomial spline to the numerical solution for fractional differential equations. In AIP Conference Proceedings (Vol. 2334, No. 1, p. 060014). AIP Publishing LLC.
[12] Milici, C., Drăgănescu, G., & Machado, J. T. (2018). Introduction to fractional differential equations (Vol. 25). Springer. [13] Hamasalh, F. K., & Muhammad, P. O. (2015). Numerical Solution of Fractional Differential Equations by using Fractional Spline Functions. Journal of Zankoy Sulaimani-Part A, 17(3), 97-110. [14] Qu, H., & Liu, X. (2015). A numerical method for solving fractional differential equations by using neural network. Advances in Mathematical Physics, 2015. [15] Birkhoff, G., & Priver, A. (1967). Hermite interpolation errors for derivatives. Journal of Mathematics and Physics, 46(1-4), 440-447. [16] Varma, A. K., & Howell, G. (1983). Best error bounds for derivatives in two point Birkhoff interpolation problems. Journal of Approximation Theory, 38(3), 258-268.
[17] Seymour, L, (1968)."Theory and problem of linear algebra", Schaum Publishing Co (McGraw-Hill) 1st.ed. [18] Maleknejad, K., Rashidinia, J., & Jalilian, H. (2021). Quintic Spline functions and Fredholm integral equation. Computational Methods for Differential Equations, 9(1), 211-224. [19] Rahimy, M. (2010). Applications of fractional differential equations. Applied Mathematical Sciences, 4(50), 2453-2461. [20] Hamasalh, F. K., & Hamzah, K. A. (2020). Quintic B-spline polynomial for Solving Bagely-Torvik Fractional Differential Problems.