A Numerical Solution of Volterra's Population Growth Model Based on Hybrid Function

Abstract

In this paper, a new numerical method for solving Volterra's population growth model is presented. Volterra's population growth model is a nonlinear integro-differential equation. In this method, by introducing the combination of fourth kind of Chebyshev polynomials and Block-pulse functions, approximate solution is presented. To do this, at first the interval of equation is divided into small sub-intervals, then approximate solution is obtained for each sub-interval. In each sub-interval, approximate solution is assumed based on introduced combination function with unknown coefficients. In order to calculate unknown coefficients, we imply collocation method with Gauss-Chebyshev points. Finally, the solution of equation is obtained as the sum of solutions at all sub-intervals. Also, it has been shown that upper bound error of approximate solution is O(m^-r/N^0.5). It means that by increasing m and N, error will decrease. At the end, the comparison of numerical results with some existing ones, shows high accuracy of this method.

Keywords

• Integro-differential equation
• Chebyshev polynomials
• Block-pulse functions
• Gauss-Chebyshev points
• Hybrid function