ABSTRACT

In this study, a fractional Bernstein series solution method has been submitted to solve the fractional-order biological population model with one carrying capacity. The numerical method has been implemented by an effective algorithm written on the computer algebraic system Maple 15. An error-bound analysis is performed by using a process similar to the RK45 method. An error estimation technique relating to residual function is presented. In the numerical application, the variations in the population of prey and predator with respect to time and situations of these two species relative to each other are plotted. The outputs obtained from our method are compared with the homotopy perturbation Sumudu transform method and reproducing kernel Hilbert space method. The approximate solutions gained from the Bernstein series method are consistent with those of other methods. The advantage of our method is that it requires less computational cost compared with methods involving more complex operations.

摘要

在这项研究中，提出了一种分数伯恩斯坦级数解法来求解具有一个承载能力的分数阶生物种群模型。数值方法是通过在计算机代数系统 Maple 15 上编写的有效算法实现的。使用类似于 RK45 方法的过程执行误差界限分析。提出了一种与残差函数相关的误差估计技术。在数值应用中，绘制了猎物和捕食者种群随时间和这两个物种相对情况的变化。从我们的方法获得的输出与同伦微扰 Sumudu 变换方法和再生核希尔伯特空间方法进行比较。Bernstein 级数方法得到的近似解与其他方法的近似解是一致的。我们方法的优点是与涉及更复杂操作的方法相比，它需要更少的计算成本。