American Institute of Mathematical Sciences

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