Lacunary interpolation by the Spline Function of Fractional Degree
DOI:
https://doi.org/10.29304/jqcm.2023.15.3.1282Keywords:
Lacunary interpolation, Spline functions, fractional differential equationsAbstract
In almost all fields of numerical analysis, spline functions are the most effective tool for polynomials employed as the fundamental method of approximation theory. Additionally, existence, uniqueness, and error boundaries are required for the spline creation in the g-spline interpolation issue. Mathematics, physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics have all shown an interest in fractional differential equations, also work in social sciences including economics, finance, and dietary supplements. It is crucial to find both close and accurate solutions of fractional differential equations. To find solutions of fractional differential equations, several analytical and numerical techniques have been developed. In this paper, we extend the five-degree spline (0,4) lacunary interpolation on uniform meshes. The outcomes, uniqueness and error boundaries for generalize (0,4) Lacunary interpolation using five- degree splines. These generalizes outperform the usage of the (0,4) five splines for interpolation.
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References
[2] Daftardar-Gejji, V. “Fractional Calculus and Fractional Differential Equations”; Springer, Gateway East,2019 .
[3] Athassawat Kammanee, “Numerical Solution of Fractional Differential Equations with variable Coefficients by Taylor Basis Functions”, Kyungpook Mathematical Journal, KYUNGPOOK Math. J.,Vol.61,2021,pp.383-393, doi.org/10.5666/KMJ.2021.61.2.383.
[4] Deo, S.G., Lakshmlkantham, V. and Raghavendra, V. "Textbook of Ordinary Differential Equations", 2nd Edition, Megraw Hill Education, 1997.
[5] EL Tarazi, M. N. and Karaballi, A. A.,” On Even-Degree Splines with Application to quadratures”, Journal of approximation theory, Vol. 60, No.2,pp. 157-167, 1990, doi.org/10.1016/0021-9045(90)90080-A.
[6] Ghosh, U. Sarkar, S. and Das, S. "Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type", American Journal of Mathematical Analysis, Vol. 3, No. 3, pp. 72-84,2015, DOI: 10.12691/ajma-3-3-3.
[7] Birkhoff, G. and Priver, A., “Hermite interpolation errors for derivatives”, J Math Phys. Vol. 46, pp. 440-447, 1967 , doi.org/10.1002/sapm1967461440.
[8] Varma, A. and Howell, G., “Best error bounds for derivatives in two point Birkhoff interpolation problems”, J. Approx. Theory. Vol. 38, pp. 258-268, 1983 , doi.org/10.1016/0021-9045(83)90132-6.
[9] Lipschutz, S., “Theory and problem of linear algebra”, Schaum Publishing Co (McGraw-Hill) 1st.ed. (1968).
[10] Maleknejad, Khosrow and Rashidinia, Jalil and Jalilian, Hamed, “Quintic Spline functions and Fredholm integral equation”, Computational Methods for Differential Equations , Vol.9,no.1,2021,pp. 211-224,doi. 10.22034/CMDE.2019.31983.1492.
[11] Faraidun K. Hamasalh,” Fractional Polynomial Spline for solving Differential Equations of Fractional Order”, Math. Sci. Lett. 4, No. 3, pp.291-296, 2015, http://dx.doi.org/10.12785/msl/040312 .
[12] Faraidun K. Hamasalh , Seaman S. Hamasalh, “Spline Fractional Polynomial for Computing Fractional Differential Equations”, Journal of University of Babylon for Pure and Applied Sciences , Vol.30, No.2,pp.68-79,2022, https://doi.org/10.29196/jubpas.v30i2.4185.
[13] Faraidun K. Hamasalh, Amina H. Ali, “Stability Analysis of Some Fractional Differential Equations by Special type of Spline Function”, Journal Zanko Sulaimani-Part A.,Vol.19, no.1,pp.191-196,2017,doi.org/10.17656/jzs.10596 .
[14] Srivastava, R., “On Lacunary Interpolation through g-Splines”, International Journal of Innovative Research in Science, Engineering and Technology, Vol. 4, No. 6, pp. 4667-4670, 2015, DOI:10.15680/IJIRSET.2015.0406118.
[15] Svante, W., “Spline function in data analysis”, Technimetrics, American Statistical Association, Vol. 16, No. 1, pp. 1-11, 1974, doi.org/10.2307/1267485.
[16] Abbas Y. Albayati, Rostam K. Saeed, Faraidun K. Hamasalh, “The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems”, Mathematics and Statistics Journal, in USA , Vol.5,no.2,pp.123-129,2009, doi.org/10.3844/jmssp.2009.123.129.
[17] Faraidun K. Hamasalh and Mizhda Abbas Headayat, The Applications of Non-polynomial Spline to the Numerical Solution for the Fractional Differential equations, AIP Conference Proceedings 2334, 060014 (2021); https://doi.org/10.1063/5.0042319.
