Faraidun K. Hamasalh & Amina H. Ali

Faculty of Science and Science Education, School of Science Education, Sulaimani University, Sulaimani, Iraq

DOI: https://doi.org/10.17656/jzs.10596

One fractional spline interpolation is presented in this paper for the solution fractional initial value problems that is with spline interpolation as a new class depended on class splines as a method to approximate the exact solution of such problem. Error bounds were discussed as the presented spline function. On other hands, stability analysis has been accomplished and two examples were chosen as consideration for the numerical explanation of the presented technique. The outcome demonstrates that the presented fractional cubic spline function which interpolates the lacunary data acted as an efficient and effective way to solve such kind Special problems.

Fractional spline function

Caputo fractional derivative

Stability Analysis

[1] Al Bayati, A. Y.; Saeed, R. K. and Hama-Salh, F. K. "Computational Quintic -Lacunary Spline Interpolation Algorithm for Solving Second-order Initial Value Problems", Journal of Applied Sciences Research, Vol. 5, No.7, pp. 733-740,( 2009).

[2] Birkhoff, G. and Priver, A. " Hermite interpolation errors for Derivatives", J. Math. Phys., Vol. 46, pp. 440-447, (1967).

[3] Clarleft, P. G. , Schultz, M. H. and Varga, R. S. " Numerical methods of high order accuracy", Numer. Math., Vol. 9, No. 35, pp. 394-430, (1967).

[4] Daftardar-Gejji, V., Sukali, Y. and Bhalekar, S. "A new predictor–corrector method for fractional differential equations", Applied Mathematics and Computation, Vol. 244, pp. 158–182, (2014).

[5] Deo, S.G., Lakshmlkantham, V. and Raghavendra, V."Textbook of Ordinary Differential Equations", 2nd Edition, Mcgraw Hill Education, (1997).

[6] EL Tarazi, M. N. and Karaballi, A. A. "On Even-Degree Splines with Application to quadratures", Journal of approximation theory, Vol. 60, No.2, 157-167, (1990).

[7] 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).

[8] Hamasalh, F. K. and Muhammad, P.O. "Analysis of Fractional Splines Interpolation and Optimal Error Bounds", American Journal of Numerical Analysis, Vol. 3, No. 1, pp. 30-35, (2015).

[9] Hamasalh, F. K. and Muhammad, P.O "Generalized quartic fractional spline interpolation with applications", Int. J. Open Problems Compt. Math., Vol. 8, No.1, pp. 67-80, (2015).

[10] Ishteva, M. K. "Properties and applications of the Caputo fractional operator", Ms c. Thesis, Dept. of Math., Universität Karlsruhe (TH), Sofia, Bulgaria, (2005).

[11] Lipschutz, S."Theory and problem of linear algebra", Schaum Publishing Co (McGraw-Hill) 1st.ed. (1968).

[12] Mohammed, O. H., Fadhel, F. S. and Abdul-Khaleq, F. A "Differential Transform Method for Solving Fuzzy Fractional Initial Value Problems", Journal of Basrah Researches ((Sciences)), Vol. 37, No.4, pp. 158-170, (2011).

[13] Rahiny, M.."Applications of Fractional Differential Equations", Applied Mathematical Sciences, Vol. 4, No.50, pp. 2453 - 2461. (2010).

[14] Sabatier, J., Agrawal, O. P. and Tenreiro Machado, J. A. "Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering", Springer Publishing Company , (2007).

[15] Sallam, S. and Karaballi, A. A. "A quartic spline collection method for solving second-order initial value problems", Computational and Applied Mathematics, Vol. 75, No. 2, pp. 295-304, (1996).

[16] Sallam, S. and Anwar, N. M. "One parameter quadratic spline collocation method for solving first order ordinary initial value problems", Applied Mathematical Modelling, Vol. 23, No.2, pp. 153-160. (1999).

[17] Sallam, S. and Anwar, N. M. "Quintic spline integration methods for solving second-order ordinary initial value problems", Journal of Computational and Applied Mathematics, Vol. 115, No. (1-2), pp. 495-502. (2000).

[18] 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).

[19] Svante, W. "Spline function in data analysis", Technimetrics, American Statistical Assonciation, Vol. 16, No. 1, pp. 1-11. (1974).

[20] Varma, A. K. and Howell, G. "Best error bounds for derivatives in two point Birkhoff interpolation problems", J. Approx. Theory, Vol. 38, No. 3, pp. 258-268. (1983).

Faculty of Science and Science Education, School of Science Education, Sulaimani University, Sulaimani, Iraq

DOI: https://doi.org/10.17656/jzs.10596

**Abstract**One fractional spline interpolation is presented in this paper for the solution fractional initial value problems that is with spline interpolation as a new class depended on class splines as a method to approximate the exact solution of such problem. Error bounds were discussed as the presented spline function. On other hands, stability analysis has been accomplished and two examples were chosen as consideration for the numerical explanation of the presented technique. The outcome demonstrates that the presented fractional cubic spline function which interpolates the lacunary data acted as an efficient and effective way to solve such kind Special problems.

**Key Words:**Fractional spline function

Caputo fractional derivative

Stability Analysis

**References**

[1] Al Bayati, A. Y.; Saeed, R. K. and Hama-Salh, F. K. "Computational Quintic -Lacunary Spline Interpolation Algorithm for Solving Second-order Initial Value Problems", Journal of Applied Sciences Research, Vol. 5, No.7, pp. 733-740,( 2009).

[2] Birkhoff, G. and Priver, A. " Hermite interpolation errors for Derivatives", J. Math. Phys., Vol. 46, pp. 440-447, (1967).

[3] Clarleft, P. G. , Schultz, M. H. and Varga, R. S. " Numerical methods of high order accuracy", Numer. Math., Vol. 9, No. 35, pp. 394-430, (1967).

[4] Daftardar-Gejji, V., Sukali, Y. and Bhalekar, S. "A new predictor–corrector method for fractional differential equations", Applied Mathematics and Computation, Vol. 244, pp. 158–182, (2014).

[5] Deo, S.G., Lakshmlkantham, V. and Raghavendra, V."Textbook of Ordinary Differential Equations", 2nd Edition, Mcgraw Hill Education, (1997).

[6] EL Tarazi, M. N. and Karaballi, A. A. "On Even-Degree Splines with Application to quadratures", Journal of approximation theory, Vol. 60, No.2, 157-167, (1990).

[7] 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).

[8] Hamasalh, F. K. and Muhammad, P.O. "Analysis of Fractional Splines Interpolation and Optimal Error Bounds", American Journal of Numerical Analysis, Vol. 3, No. 1, pp. 30-35, (2015).

[9] Hamasalh, F. K. and Muhammad, P.O "Generalized quartic fractional spline interpolation with applications", Int. J. Open Problems Compt. Math., Vol. 8, No.1, pp. 67-80, (2015).

[10] Ishteva, M. K. "Properties and applications of the Caputo fractional operator", Ms c. Thesis, Dept. of Math., Universität Karlsruhe (TH), Sofia, Bulgaria, (2005).

[11] Lipschutz, S."Theory and problem of linear algebra", Schaum Publishing Co (McGraw-Hill) 1st.ed. (1968).

[12] Mohammed, O. H., Fadhel, F. S. and Abdul-Khaleq, F. A "Differential Transform Method for Solving Fuzzy Fractional Initial Value Problems", Journal of Basrah Researches ((Sciences)), Vol. 37, No.4, pp. 158-170, (2011).

[13] Rahiny, M.."Applications of Fractional Differential Equations", Applied Mathematical Sciences, Vol. 4, No.50, pp. 2453 - 2461. (2010).

[14] Sabatier, J., Agrawal, O. P. and Tenreiro Machado, J. A. "Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering", Springer Publishing Company , (2007).

[15] Sallam, S. and Karaballi, A. A. "A quartic spline collection method for solving second-order initial value problems", Computational and Applied Mathematics, Vol. 75, No. 2, pp. 295-304, (1996).

[16] Sallam, S. and Anwar, N. M. "One parameter quadratic spline collocation method for solving first order ordinary initial value problems", Applied Mathematical Modelling, Vol. 23, No.2, pp. 153-160. (1999).

[17] Sallam, S. and Anwar, N. M. "Quintic spline integration methods for solving second-order ordinary initial value problems", Journal of Computational and Applied Mathematics, Vol. 115, No. (1-2), pp. 495-502. (2000).

[18] 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).

[19] Svante, W. "Spline function in data analysis", Technimetrics, American Statistical Assonciation, Vol. 16, No. 1, pp. 1-11. (1974).

[20] Varma, A. K. and Howell, G. "Best error bounds for derivatives in two point Birkhoff interpolation problems", J. Approx. Theory, Vol. 38, No. 3, pp. 258-268. (1983).