Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations
Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations
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DOI：
10.1016/j.apnum.2019.05.024
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发表时间：
2019-05
影响因子：
2.8
通讯作者：
S. Nemati;P. Lima;S. Sedaghat
中科院分区：
文献类型：
--
作者：
S. Nemati;P. Lima;S. Sedaghat
本文将Legendre小波与Gauss-Jacobi求积公式相结合，提出了一种求解一类分数阶时滞型积分-微分方程组的配置方法。该问题用初值或边值条件来考虑，分数导数用卡普托意义来描述。首先，利用勒让德小波基函数对未知解进行了逼近。然后，我们将这个近似及其导数代入所考虑的方程。未知函数的卡普托导数用Gauss-Jacobi求积公式近似。通过在著名的平移切比雪夫点处配置得到的残差，我们得到了一个非线性代数方程组。为了获得连续解，在所得到的系统中加入了一些条件。给出了任意函数的Legendre小波逼近的误差界。最后给出了几个算例，说明了该方法的有效性和准确性。
In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss–Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss–Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.