Ali Ishtiaq
Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia.
Math Biosci Eng. 2021 Mar 22;18(3):2764-2774. doi: 10.3934/mbe.2021140.
Functional differential equations of neutral type are a class of differential equations in which the derivative of the unknown functions depends on the history of the function and its derivative as well. Due to this nature the explicit solutions of these equations are not easy to compute and sometime even not possible. Therefore, one must use some numerical technique to find an approximate solution to these equations. In this paper, we used a spectral collocation method which is based on Bernstein polynomials to find the approximate solution. The disadvantage of using Bernstein polynomials is that they are not orthogonal and therefore one cannot use the properties of orthogonal polynomials for the efficient evaluation of differential equations. In order to avoid this issue and to fully use the properties of orthogonal polynomials, a change of basis transformation from Bernstein to Legendre polynomials is used. An error analysis in infinity norm is provided, followed by several numerical examples to justify the efficiency and accuracy of the proposed scheme.
中立型泛函微分方程是一类微分方程，其中未知函数的导数不仅依赖于函数本身的历史，还依赖于其导数的历史。由于这种性质，这些方程的显式解不易计算，有时甚至不可能求出。因此，必须使用一些数值技术来找到这些方程的近似解。在本文中，我们使用了一种基于伯恩斯坦多项式的谱配置方法来求近似解。使用伯恩斯坦多项式的缺点是它们不正交，因此不能利用正交多项式的性质来高效地求解微分方程。为了避免这个问题并充分利用正交多项式的性质，我们使用了从伯恩斯坦多项式到勒让德多项式的基变换。文中给出了无穷范数下的误差分析，随后通过几个数值例子验证了所提方案的有效性和准确性。
