Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

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, K(t,s)=aκ(t−s) within convolution integrals, where a,κ∈R. 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 O(Nt) for Nt iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of O(Ntlog⁡Nt) and O(Nt(log⁡Nt))2. 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.