doi: 10.3934/mfc.2021039
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Iterative Bernstein splines technique applied to fractional order differential equations

Department of Mathematics and Informatics, University of Oradea, Str. Universităţii no.1, Oradea, 410087, Romania

Received  August 2021 Revised  November 2021 Early access December 2021

In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.

Citation: Zoltan Satmari. Iterative Bernstein splines technique applied to fractional order differential equations. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021039
References:
[1]

R. P. AgarwalM. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033.  doi: 10.1007/s10440-008-9356-6.

[2]

M. BachherN. Sarkar and A. Lahiri, Fractional order thermoelastic interactions in an infinite porous material due to distributed timedependent heat sources, Meccanica, 50 (2015), 2167-2178.  doi: 10.1007/s11012-015-0152-x.

[3]

R. L. Bagley and R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Contr. Dynam., 14 (1991), 304-311.  doi: 10.2514/6.1989-1213.

[4]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods (2nd Edition), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

[5]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2004, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2.

[6]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 378 (1997), 223–276.

[7]

A. GoswamiJ. SushilaJ. Singh and D. Kumar, Numerical computation of fractional Kersten-Krasil'shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math., 5 (2020), 2346-2368.  doi: 10.3934/math.2020155.

[8]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.

[9]

M. A. Matlob and Y. Jamali, The concepts and applications of fractional order differential calculus in modelling of viscoelastic systems: A primer, Critical Reviews in Biomedical Engineering, (2019), 249-276. 

[10]

S. Micula, An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math., 339 (2018), 124-133.  doi: 10.1016/j.cam.2017.12.006.

[11]

M. Mohammad and A. Trounev, Implicit Riesz wavelets based-method for solving singular fractional integrodifferential equations with applications to hematopoietic stem cell modeling, Chaos Solitons Fractals, 138 (2020), 11pp.  doi: 10.1016/j.chaos.2020.109991.

[12]

A. PedasE. Tamme and M. Vikerpuur, Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems, Journal of Computational and Applied Mathematics, 317 (2017), 1-16.  doi: 10.1016/j.cam.2016.11.022.

[13]

P. RahimkhaniY. Ordokhani and E. Babolian, A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations, Numer Algor, 74 (2017), 223-245.  doi: 10.1007/s11075-016-0146-3.

show all references

References:
[1]

R. P. AgarwalM. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033.  doi: 10.1007/s10440-008-9356-6.

[2]

M. BachherN. Sarkar and A. Lahiri, Fractional order thermoelastic interactions in an infinite porous material due to distributed timedependent heat sources, Meccanica, 50 (2015), 2167-2178.  doi: 10.1007/s11012-015-0152-x.

[3]

R. L. Bagley and R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Contr. Dynam., 14 (1991), 304-311.  doi: 10.2514/6.1989-1213.

[4]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods (2nd Edition), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

[5]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2004, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2.

[6]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 378 (1997), 223–276.

[7]

A. GoswamiJ. SushilaJ. Singh and D. Kumar, Numerical computation of fractional Kersten-Krasil'shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math., 5 (2020), 2346-2368.  doi: 10.3934/math.2020155.

[8]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.

[9]

M. A. Matlob and Y. Jamali, The concepts and applications of fractional order differential calculus in modelling of viscoelastic systems: A primer, Critical Reviews in Biomedical Engineering, (2019), 249-276. 

[10]

S. Micula, An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math., 339 (2018), 124-133.  doi: 10.1016/j.cam.2017.12.006.

[11]

M. Mohammad and A. Trounev, Implicit Riesz wavelets based-method for solving singular fractional integrodifferential equations with applications to hematopoietic stem cell modeling, Chaos Solitons Fractals, 138 (2020), 11pp.  doi: 10.1016/j.chaos.2020.109991.

[12]

A. PedasE. Tamme and M. Vikerpuur, Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems, Journal of Computational and Applied Mathematics, 317 (2017), 1-16.  doi: 10.1016/j.cam.2016.11.022.

[13]

P. RahimkhaniY. Ordokhani and E. Babolian, A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations, Numer Algor, 74 (2017), 223-245.  doi: 10.1007/s11075-016-0146-3.

Table 1.  The numerical results for $q = 1$
$t_{i}$ $n = 10$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $3, 31E-02$ $1, 06E-02$ $2, 88E-03$ $1, 11E-03$
$0, 2$ $1, 74E-02$ $6, 48E-03$ $1, 84E-03$ $7, 17E-04$
$0, 3$ $1, 35E-02$ $5, 14E-03$ $1, 47E-03$ $5, 77E-04$
$0, 4$ $1, 18E-02$ $4, 55E-03$ $1, 31E-03$ $5, 14E-04$
$0, 5$ $1, 12E-02$ $4, 32E-03$ $1, 25E-03$ $4, 91E-04$
$0, 6$ $1, 12E-02$ $4, 35E-03$ $1, 26E-03$ $4, 96E-04$
$0, 7$ $1, 19E-02$ $4, 63E-03$ $1, 34E-03$ $5, 28E-04$
$0, 8$ $1, 34E-02$ $5, 21E-03$ $1, 51E-03$ $5, 96E-04$
$0, 9$ $1, 62E-02$ $6, 27E-03$ $1, 82E-03$ $7, 18E-04$
$1, 0$ $2, 11E-02$ $8, 18E-03$ $2, 37E-03$ $9, 37E-04$
$t_{i}$ $n = 10$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $3, 31E-02$ $1, 06E-02$ $2, 88E-03$ $1, 11E-03$
$0, 2$ $1, 74E-02$ $6, 48E-03$ $1, 84E-03$ $7, 17E-04$
$0, 3$ $1, 35E-02$ $5, 14E-03$ $1, 47E-03$ $5, 77E-04$
$0, 4$ $1, 18E-02$ $4, 55E-03$ $1, 31E-03$ $5, 14E-04$
$0, 5$ $1, 12E-02$ $4, 32E-03$ $1, 25E-03$ $4, 91E-04$
$0, 6$ $1, 12E-02$ $4, 35E-03$ $1, 26E-03$ $4, 96E-04$
$0, 7$ $1, 19E-02$ $4, 63E-03$ $1, 34E-03$ $5, 28E-04$
$0, 8$ $1, 34E-02$ $5, 21E-03$ $1, 51E-03$ $5, 96E-04$
$0, 9$ $1, 62E-02$ $6, 27E-03$ $1, 82E-03$ $7, 18E-04$
$1, 0$ $2, 11E-02$ $8, 18E-03$ $2, 37E-03$ $9, 37E-04$
Table 2.  The numerical results for $q = 4$
$t_{i}$ $n = 10$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $8, 75E-03$ $3, 24E-03$ $9, 16E-04$ $3, 58E-04$
$0, 2$ $5, 35E-03$ $2, 05E-03$ $5, 91E-04$ $2, 32E-04$
$0, 3$ $4, 24E-03$ $1, 64E-03$ $4, 75E-04$ $1, 87E-04$
$0, 4$ $3, 76E-03$ $1, 46E-03$ $4, 24E-04$ $1, 67E-04$
$0, 5$ $3, 57E-03$ $1, 39E-03$ $4, 04E-04$ $1, 59E-04$
$0, 6$ $3, 60E-03$ $1, 40E-03$ $4, 08E-04$ $1, 61E-04$
$0, 7$ $3, 83E-03$ $1, 49E-03$ $4, 35E-04$ $1, 72E-04$
$0, 8$ $4, 31E-03$ $1, 68E-03$ $4, 91E-04$ $1, 94E-04$
$0, 9$ $5, 19E-03$ $2, 03E-03$ $5, 91E-04$ $2, 33E-04$
$1, 0$ $6, 78E-03$ $2, 65E-03$ $7, 71E-04$ $3, 05E-04$
$t_{i}$ $n = 10$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $8, 75E-03$ $3, 24E-03$ $9, 16E-04$ $3, 58E-04$
$0, 2$ $5, 35E-03$ $2, 05E-03$ $5, 91E-04$ $2, 32E-04$
$0, 3$ $4, 24E-03$ $1, 64E-03$ $4, 75E-04$ $1, 87E-04$
$0, 4$ $3, 76E-03$ $1, 46E-03$ $4, 24E-04$ $1, 67E-04$
$0, 5$ $3, 57E-03$ $1, 39E-03$ $4, 04E-04$ $1, 59E-04$
$0, 6$ $3, 60E-03$ $1, 40E-03$ $4, 08E-04$ $1, 61E-04$
$0, 7$ $3, 83E-03$ $1, 49E-03$ $4, 35E-04$ $1, 72E-04$
$0, 8$ $4, 31E-03$ $1, 68E-03$ $4, 91E-04$ $1, 94E-04$
$0, 9$ $5, 19E-03$ $2, 03E-03$ $5, 91E-04$ $2, 33E-04$
$1, 0$ $6, 78E-03$ $2, 65E-03$ $7, 71E-04$ $3, 05E-04$
Table 3.  The numerical results for $q = 1$
$t_{i}$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $2, 16E-03$ $6, 69E-04$ $2, 65E-04$
$0, 2$ $3, 28E-03$ $9, 65E-04$ $3, 75E-04$
$0, 3$ $4, 03E-03$ $1, 17E-03$ $4, 50E-04$
$0, 4$ $4, 58E-03$ $1, 31E-03$ $5, 04E-04$
$0, 5$ $5, 00E-03$ $1, 42E-03$ $5, 45E-04$
$0, 6$ $5, 32E-03$ $1, 51E-03$ $5, 76E-04$
$0, 7$ $5, 57E-03$ $1, 57E-03$ $6, 01E-04$
$0, 8$ $5, 79E-03$ $1, 63E-03$ $6, 22E-04$
$0, 9$ $5, 99E-03$ $1, 68E-03$ $6, 41E-04$
$1, 0$ $6, 20E-03$ $1, 74E-03$ $6, 61E-04$
$t_{i}$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $2, 16E-03$ $6, 69E-04$ $2, 65E-04$
$0, 2$ $3, 28E-03$ $9, 65E-04$ $3, 75E-04$
$0, 3$ $4, 03E-03$ $1, 17E-03$ $4, 50E-04$
$0, 4$ $4, 58E-03$ $1, 31E-03$ $5, 04E-04$
$0, 5$ $5, 00E-03$ $1, 42E-03$ $5, 45E-04$
$0, 6$ $5, 32E-03$ $1, 51E-03$ $5, 76E-04$
$0, 7$ $5, 57E-03$ $1, 57E-03$ $6, 01E-04$
$0, 8$ $5, 79E-03$ $1, 63E-03$ $6, 22E-04$
$0, 9$ $5, 99E-03$ $1, 68E-03$ $6, 41E-04$
$1, 0$ $6, 20E-03$ $1, 74E-03$ $6, 61E-04$
Table 4.  The numerical results for $q = 4$
$t_{i}$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $7, 29E-04$ $2, 17E-04$ $8, 48E-05$
$0, 2$ $1, 07E-03$ $3, 09E-04$ $1, 19E-04$
$0, 3$ $1, 30E-03$ $3, 71E-04$ $1, 43E-04$
$0, 4$ $1, 47E-03$ $4, 17E-04$ $1, 60E-04$
$0, 5$ $1, 60E-03$ $4, 51E-04$ $1, 72E-04$
$0, 6$ $1, 69E-03$ $4, 77E-04$ $1, 82E-04$
$0, 7$ $1, 77E-03$ $4, 98E-04$ $1, 90E-04$
$0, 8$ $1, 84E-03$ $5, 15E-04$ $1, 96E-04$
$0, 9$ $1, 90E-03$ $5, 31E-04$ $2, 02E-04$
$1, 0$ $1, 96E-03$ $5, 48E-04$ $2, 09E-04$
$t_{i}$ $n = 20$ $n = 50$ $n = 100$
$0, 0$ $0, 00$ $0, 00$ $0, 00$
$0, 1$ $7, 29E-04$ $2, 17E-04$ $8, 48E-05$
$0, 2$ $1, 07E-03$ $3, 09E-04$ $1, 19E-04$
$0, 3$ $1, 30E-03$ $3, 71E-04$ $1, 43E-04$
$0, 4$ $1, 47E-03$ $4, 17E-04$ $1, 60E-04$
$0, 5$ $1, 60E-03$ $4, 51E-04$ $1, 72E-04$
$0, 6$ $1, 69E-03$ $4, 77E-04$ $1, 82E-04$
$0, 7$ $1, 77E-03$ $4, 98E-04$ $1, 90E-04$
$0, 8$ $1, 84E-03$ $5, 15E-04$ $1, 96E-04$
$0, 9$ $1, 90E-03$ $5, 31E-04$ $2, 02E-04$
$1, 0$ $1, 96E-03$ $5, 48E-04$ $2, 09E-04$
Table 5.  The numerical results for $q = 1$
$t_{i}$ $n = 10$ $n = 100$
$0, 0$ $0, 00$ $0, 00$
$0, 3$ $4, 44E-016$ $4, 16E-016$
$0, 6$ $1, 33E-015$ $1, 22E-015$
$0, 8$ $3, 02E-014$ $2, 33E-015$
$1, 0$ $7, 24E-09$ $9, 07E-011$
$t_{i}$ $n = 10$ $n = 100$
$0, 0$ $0, 00$ $0, 00$
$0, 3$ $4, 44E-016$ $4, 16E-016$
$0, 6$ $1, 33E-015$ $1, 22E-015$
$0, 8$ $3, 02E-014$ $2, 33E-015$
$1, 0$ $7, 24E-09$ $9, 07E-011$
Table 6.  The numerical results for $q = 4$
$t_{i}$ $n = 10$ $n = 100$
$0, 0$ $0, 00$ $0, 00$
$0, 3$ $4, 16E-016$ $3, 61E-016$
$0, 6$ $1, 22E-015$ $1, 22E-015$
$0, 8$ $4, 66E-015$ $2, 33E-015$
$1, 0$ $1, 01E-09$ $8, 16E-011$
$t_{i}$ $n = 10$ $n = 100$
$0, 0$ $0, 00$ $0, 00$
$0, 3$ $4, 16E-016$ $3, 61E-016$
$0, 6$ $1, 22E-015$ $1, 22E-015$
$0, 8$ $4, 66E-015$ $2, 33E-015$
$1, 0$ $1, 01E-09$ $8, 16E-011$
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