Iterative Bernstein splines technique applied to fractional order differential equations

Zoltan Satmari

doi:10.3934/mfc.2021039

Article Contents

2023, Volume 6, Issue 1: 41-53. Doi: 10.3934/mfc.2021039

This issue Previous Article Analysis of directional higher order jump discontinuities with trigonometric shearlets Next Article Generalized Kantorovich modifications of positive linear operators

Iterative Bernstein splines technique applied to fractional order differential equations

Zoltan Satmari

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

Published: February 2023

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Abstract

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.

Keywords:

Initial value problem,
Bernstein splines,
fractional order differential equations,
iterative numerical method,
Volterra integral equations.

Mathematics Subject Classification: Primary: 65R20.

Citation:

Full Text(HTML)

Table 1. The numerical results for $q = 1$

$t_{i}$	$n = 10$	$n = 20$	$n = 50$	$n = 100$
$0, 0$	$0, 00$	$0, 00$	$0, 00$	$0, 00$
$0, 1$	$3, 31E-02$	$1, 06E-02$	$2, 88E-03$	$1, 11E-03$
$0, 2$	$1, 74E-02$	$6, 48E-03$	$1, 84E-03$	$7, 17E-04$
$0, 3$	$1, 35E-02$	$5, 14E-03$	$1, 47E-03$	$5, 77E-04$
$0, 4$	$1, 18E-02$	$4, 55E-03$	$1, 31E-03$	$5, 14E-04$
$0, 5$	$1, 12E-02$	$4, 32E-03$	$1, 25E-03$	$4, 91E-04$
$0, 6$	$1, 12E-02$	$4, 35E-03$	$1, 26E-03$	$4, 96E-04$
$0, 7$	$1, 19E-02$	$4, 63E-03$	$1, 34E-03$	$5, 28E-04$
$0, 8$	$1, 34E-02$	$5, 21E-03$	$1, 51E-03$	$5, 96E-04$
$0, 9$	$1, 62E-02$	$6, 27E-03$	$1, 82E-03$	$7, 18E-04$
$1, 0$	$2, 11E-02$	$8, 18E-03$	$2, 37E-03$	$9, 37E-04$

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Table 2. The numerical results for $q = 4$

$t_{i}$	$n = 10$	$n = 20$	$n = 50$	$n = 100$
$0, 0$	$0, 00$	$0, 00$	$0, 00$	$0, 00$
$0, 1$	$8, 75E-03$	$3, 24E-03$	$9, 16E-04$	$3, 58E-04$
$0, 2$	$5, 35E-03$	$2, 05E-03$	$5, 91E-04$	$2, 32E-04$
$0, 3$	$4, 24E-03$	$1, 64E-03$	$4, 75E-04$	$1, 87E-04$
$0, 4$	$3, 76E-03$	$1, 46E-03$	$4, 24E-04$	$1, 67E-04$
$0, 5$	$3, 57E-03$	$1, 39E-03$	$4, 04E-04$	$1, 59E-04$
$0, 6$	$3, 60E-03$	$1, 40E-03$	$4, 08E-04$	$1, 61E-04$
$0, 7$	$3, 83E-03$	$1, 49E-03$	$4, 35E-04$	$1, 72E-04$
$0, 8$	$4, 31E-03$	$1, 68E-03$	$4, 91E-04$	$1, 94E-04$
$0, 9$	$5, 19E-03$	$2, 03E-03$	$5, 91E-04$	$2, 33E-04$
$1, 0$	$6, 78E-03$	$2, 65E-03$	$7, 71E-04$	$3, 05E-04$

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Table 3. The numerical results for $q = 1$

$t_{i}$	$n = 20$	$n = 50$	$n = 100$
$0, 0$	$0, 00$	$0, 00$	$0, 00$
$0, 1$	$2, 16E-03$	$6, 69E-04$	$2, 65E-04$
$0, 2$	$3, 28E-03$	$9, 65E-04$	$3, 75E-04$
$0, 3$	$4, 03E-03$	$1, 17E-03$	$4, 50E-04$
$0, 4$	$4, 58E-03$	$1, 31E-03$	$5, 04E-04$
$0, 5$	$5, 00E-03$	$1, 42E-03$	$5, 45E-04$
$0, 6$	$5, 32E-03$	$1, 51E-03$	$5, 76E-04$
$0, 7$	$5, 57E-03$	$1, 57E-03$	$6, 01E-04$
$0, 8$	$5, 79E-03$	$1, 63E-03$	$6, 22E-04$
$0, 9$	$5, 99E-03$	$1, 68E-03$	$6, 41E-04$
$1, 0$	$6, 20E-03$	$1, 74E-03$	$6, 61E-04$

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Table 4. The numerical results for $q = 4$

$t_{i}$	$n = 20$	$n = 50$	$n = 100$
$0, 0$	$0, 00$	$0, 00$	$0, 00$
$0, 1$	$7, 29E-04$	$2, 17E-04$	$8, 48E-05$
$0, 2$	$1, 07E-03$	$3, 09E-04$	$1, 19E-04$
$0, 3$	$1, 30E-03$	$3, 71E-04$	$1, 43E-04$
$0, 4$	$1, 47E-03$	$4, 17E-04$	$1, 60E-04$
$0, 5$	$1, 60E-03$	$4, 51E-04$	$1, 72E-04$
$0, 6$	$1, 69E-03$	$4, 77E-04$	$1, 82E-04$
$0, 7$	$1, 77E-03$	$4, 98E-04$	$1, 90E-04$
$0, 8$	$1, 84E-03$	$5, 15E-04$	$1, 96E-04$
$0, 9$	$1, 90E-03$	$5, 31E-04$	$2, 02E-04$
$1, 0$	$1, 96E-03$	$5, 48E-04$	$2, 09E-04$

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Table 5. The numerical results for $q = 1$

$t_{i}$	$n = 10$	$n = 100$
$0, 0$	$0, 00$	$0, 00$
$0, 3$	$4, 44E-016$	$4, 16E-016$
$0, 6$	$1, 33E-015$	$1, 22E-015$
$0, 8$	$3, 02E-014$	$2, 33E-015$
$1, 0$	$7, 24E-09$	$9, 07E-011$

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Table 6. The numerical results for $q = 4$

$t_{i}$	$n = 10$	$n = 100$
$0, 0$	$0, 00$	$0, 00$
$0, 3$	$4, 16E-016$	$3, 61E-016$
$0, 6$	$1, 22E-015$	$1, 22E-015$
$0, 8$	$4, 66E-015$	$2, 33E-015$
$1, 0$	$1, 01E-09$	$8, 16E-011$

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