A Composite Wavelet–Rational Approach for Solving the Volterra’s Population Growth Model Over Semi-Infinite Domain

In problems defined on a semi-infinite domain, rational Chebyshev or Laguerre functions are the generic choices of basis functions in spectral methods. The rationale is that if the solution is oscillatory near the origin, then large number of basis functions may be required to retrieve the spectral accuracy that is not convenient. In this paper, we propose a novel idea that combines Chebyshev wavelets and orthogonal rational functions for solving problems in semi-infinite domain numerically. The semi-infinite domain [0,∞) is divided into two subdomains [0,α) and [α,∞). As the basis functions, Chebyshev wavelets are considered in the subdomain [0,α) and a new class of orthonormal rational functions is derived on the semi-infinite subdomain [α,∞). By constructing the operational matrices of derivative, integral, and product, and implementing the collocation method, the original problem is transcribed to a system of algebraic equations. A key feature of the method is that for a suitably chosen α, it yields good approximations to the solutions that first oscillate and then decay fast as t→∞. Application of the proposed method to the Volterra’s model for population growth of a species in a closed system is explained. Numerical results show that the discrete solution exhibits exponential convergence as a function of suitable α and the number of collocation points.