Optimizing linear/non-linear Volterra-type integro-differential equations with Runge–Kutta 2 and 4 for time efficiency

A novel non-adaptative time-efficient (TE) numerical scheme based on the Runge–Kutta (RK) algorithm is devised for solving linear and non-linear Volterra-type integro-differential equations (VTIDE) characterized by a specific class of convolution memory, $K(t,s)=a^{\kappa (t-s)}$, where $a,\kappa \in R$. This kernel is commonly encountered in viscoelasticity and various applied sciences. The integration governing the convolution term is elegantly reformulated, allowing for the implementation of a backstage integration within the RK scheme’s main body. This approach circumvents the need for full integration of the convolution during each iteration, thereby significantly reducing computational time to $O\left(N_t\right)$ for $N_t$ iterations. Furthermore, this formulation facilitates the adoption of an implicit scheme by selecting appropriate methods and data stencils. We demonstrate this concept using an implicit trapezoidal method applied to a linear VTIDE, accompanied by stability analyses. Additionally, a complex VTIDE is constructed featuring nonlinearities both within and outside the convolutions, as well as a derivative-of-dependent-variable integrant. This setup enables the synergy of differentiation, integration, and RK schemes to generate data for intricate VTIDE types. Consequently, both the scheme and the equation exhibit uniqueness denoted as TE-RK and TE-RK-VTIDE. We illustrate how this development simplifies the implementation of implicit schemes and the calculation of stability regions concerning the convolution. The TE-RK scheme is tested on VTIDE using RK2 and RK4, yielding plots that exhibit strong agreement, thus validating our approach. Furthermore, we successfully apply the TE-RK scheme to a well-known nonlinear logistic equation. Plots for various step sizes are generated alongside corresponding error graphs, demonstrating consistent trends in error behavior.