Evaluating Analytical Approximations and Numerical Solutions for Volterra’s Population Growth Equation

Abstract

This paper presents a comprehensive study to solve the nonlinear Volterra integro-differential equation (NLVIDE) i.e. Volterra’s population growth , with initial condition , is examined using three distinct approaches: a power series solution, a novel hybrid Laplace transform-series method, and numerical integration via MATLAB’s ode45. We derive analytical approximations, implement a MATLAB script to compute and visualize solutions, and compare the computational efficiency and accuracy of each method. The results indicate that the series and hybrid methods yield accurate approximations for small time intervals, whereas the numerical method provides robust solutions over larger domains. This study elucidates the strengths and limitations of each approach, offering insights into their applicability in nonlinear integro-differential systems.