Ndi Azese Martin
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA.
Applied Mechanics Laboratory, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.
Heliyon. 2024 Jul 3;10(13):e33737. doi: 10.1016/j.heliyon.2024.e33737. eCollection 2024 Jul 15.
We introduce a novel, time-efficient adaptive Runge-Kutta computational scheme tailored for systematically solving linear and nonlinear Volterra-type Integro-Differential Equations (VTIDEs). This scheme is particularly effective for equations featuring a specific class of convolution memory, within convolution integrals, where . Such equations frequently arise in fields like viscoelasticity, population dynamics, epidemiology, control systems, financial mathematics, and neuroscience, where current states are influenced by historical data, especially in dynamical systems within physics. Our proposed scheme demonstrates remarkable computational efficiency, achieving a cost-effective computation time of for iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of and . The key feature of our scheme lies in its ability to elegantly and efficiently handle convoluted integrals of varying complexity such as those including nonlinearities and derivatives. Specifically, our approach eliminates the need for backstage integration, allowing the main Runge-Kutta scheme to operate efficiently with only a few data points at each time step. This not only saves time but also prevents potential data storage issues while rendering numerical implementation straightforward and tractable. Moreover, this article introduces flexible discretizations for integration and differentiation, accommodating unequally spaced data points. We implement a fast RK45-Fehlberg algorithm (TE-RK45-Fehlberg), which we use to solve three examples of nonlinear VTIDE presented in this study. We further demonstrate the effectiveness of our approach by using an example, where we employ a novel parabolic step size generator function to efficiently generate step sizes for a computationally fast RK4 scheme (Flex-RK4). For all examples tackled using TE-RK45-Fehlberg and Flex-RK4, we generate plots that compare numerical data with exact solutions, providing insight into the accuracy and reliability of our approach. We also show how the scheme provides straightforward ways of transforming it into implicit forms and obtaining stability curves.
我们介绍了一种新颖、省时的自适应龙格 - 库塔计算方案，专门用于系统地求解线性和非线性沃尔泰拉型积分 - 微分方程（VTIDE）。该方案对于在卷积积分中具有特定一类卷积记忆的方程特别有效，其中 。这类方程在粘弹性、种群动态、流行病学、控制系统、金融数学和神经科学等领域经常出现，在这些领域中，当前状态受到历史数据的影响，特别是在物理中的动态系统中。我们提出的方案展示了显著的计算效率，在 次迭代中实现了具有成本效益的计算时间 ，与现有的VTIDE方案相比有显著改进，现有方案通常表现出 和 的复杂度。我们方案的关键特性在于其能够优雅且高效地处理各种复杂程度的卷积积分，例如那些包含非线性和导数的积分。具体而言，我们的方法无需后台积分，允许主龙格 - 库塔方案在每个时间步仅使用少量数据点就能高效运行。这不仅节省了时间，还避免了潜在的数据存储问题，同时使数值实现变得直接且易于处理。此外，本文引入了用于积分和微分的灵活离散化方法，以适应不等间距的数据点。我们实现了一种快速的RK45 - 费尔贝格算法（TE - RK45 - 费尔贝格），用于求解本研究中给出的三个非线性VTIDE示例。我们还通过一个示例进一步证明了我们方法的有效性，在该示例中，我们使用一种新颖的抛物线步长生成函数为计算快速的RK4方案（Flex - RK4）高效生成步长。对于使用TE - RK45 - 费尔贝格和Flex - RK4处理的所有示例，我们生成了将数值数据与精确解进行比较的图表，以深入了解我们方法的准确性和可靠性。我们还展示了该方案如何提供将其转换为隐式形式并获得稳定性曲线的直接方法。
