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Iterative Bernstein splines technique applied to fractional order differential equations

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  • 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.

    Mathematics Subject Classification: Primary: 65R20.

    Citation:

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

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

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

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

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

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

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

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

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

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

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