Analytical Solutions of Volterra Integral Equations Involving Jacobi Polynomial Nonlinearities: A Power Series Method

Abstract

Nonlinear Volterra integral equations model several phenomena in a variety of fields, including epidemiology, population growth theory, diffusion and heat transfer problems. In this paper, the analytical solution of a class of nonlinear Volterra integral equations of the second kind is presented using a power series method based on the generalised Cauchy product. The nonlinear terms $\Phi(U(\xi))$ in the proposed problem are given by normalised Jacobi polynomials $\mathsf{P}_{\ell}^{(\phi, \varphi)}(U(\xi))$ ($\ell=1,2,3,\dots;\phi,\varphi>-1$) in the unknown $U(\xi)$. The proposed method first expresses the nonlinear terms as a power series in the unknown $U$, and then uses the generalised Cauchy product to convert the series into power series in the independent variable $\xi$. Subsequently, recurrence relations for the expansion coefficients of the series solution are obtained in terms of the nonlinear terms and their higher derivatives evaluated at the constant term of the series solution. Specialising the nonlinear terms to Gegenbauer or ultraspherical polynomials, Legendre polynomials, and Chebyshev polynomials, several nonlinear Volterra integral equations and their solutions are introduced. Considering the normalised Jacobi polynomials as spherical functions on rank one symmetric spaces of compact type, a number of novel nonlinear integral equations are presented. To validate the proposed method, some known nonlinear Volterra integral equations are solved and the series solutions converge to the known exact solutions. The convergence of the series solutions shows that the proposed power series method is accurate and reliable for solving such nonlinear Volterra integral problems.