Vazquez-Leal Hector, Sandoval-Hernandez Mario Alberto, Filobello-Nino Uriel, Huerta-Chua Jesus
Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, Mexico.
Consejo Veracruzano de Investigación Científica y Desarrollo Tecnológico (COVEICYDET), Av Rafael Murillo Vidal No. 1735, Cuauhtemoc, 91069, Xalapa, Veracruz, Mexico.
Heliyon. 2020 Apr 14;6(4):e03695. doi: 10.1016/j.heliyon.2020.e03695. eCollection 2020 Apr.
This work presents the novel Leal-polynomials (LP) for the approximation of nonlinear differential equations of different kind. The main characteristic of LPs is that they satisfy multiple expansion points and its derivatives as a mechanism to replicate behaviour of the nonlinear problem, giving more accuracy within the region of interest. Therefore, the main contribution of this work is that LP satisfies the successive derivatives in some specific points, resulting more accurate polynomials than Taylor expansion does for the same degree of their respective polynomials. Such characteristic makes of LPs a handy and powerful tool to approximate different kind of differential equations including: singular problems, initial condition and boundary-valued problems, equations with discontinuities, coupled differential equations, high-order equations, among others. Additionally, we show how the process to obtain the polynomials is straightforward and simple to implement; generating a compact, and easy to compute, expression. Even more, we present the process to approximate Gelfand's equation, an equation of an isothermal reaction, a model for chronic myelogenous leukemia, Thomas-Fermi equation, and a high order nonlinear differential equations with discontinuities getting, as result, accurate, fast and compact approximate solutions. In addition, we present the computational convergence and error studies for LPs resulting convergent polynomials and error tendency to zero as the order of LPs increases for all study cases. Finally, a study of CPU time shows that LPs require a few nano-seconds to be evaluated, which makes them suitable for intensive computing applications.
这项工作提出了用于逼近不同类型非线性微分方程的新型利尔多项式（LP）。利尔多项式的主要特点是它们满足多个展开点及其导数，以此作为复制非线性问题行为的一种机制，在感兴趣的区域内提供更高的精度。因此，这项工作的主要贡献在于利尔多项式在某些特定点满足连续导数，对于相同次数的多项式，其得到的多项式比泰勒展开更精确。这种特性使利尔多项式成为逼近不同类型微分方程的便捷而强大的工具，这些方程包括：奇异问题、初值问题和边值问题、具有间断性的方程、耦合微分方程、高阶方程等等。此外，我们展示了获取多项式的过程简单直接且易于实现；生成一个紧凑且易于计算的表达式。甚至，我们给出了逼近盖尔范德方程、等温反应方程、慢性粒细胞白血病模型、托马斯 - 费米方程以及一个具有间断性的高阶非线性微分方程的过程，从而得到准确、快速且紧凑的近似解。另外，我们给出了利尔多项式的计算收敛性和误差研究，结果表明对于所有研究案例，随着利尔多项式阶数的增加，多项式收敛且误差趋于零。最后，对CPU时间的研究表明，评估利尔多项式需要几纳秒，这使得它们适用于密集计算应用。
