Weakly singular Volterra integral equation with combined logarithmic-power-law kernel: Analytical and computational consideration

Abstract

In this article, we consider weakly singular Volterra integral equation (WSVIE) with the kernel combining the power-law and logarithmic singularitiesf(ζ)=Au(ζ)+∫0ζlog(ζ−η)u(η)(ζ−η)βdη,ζ∈[0,1],β∈[0,1), where, f(ζ) is a smooth function. Equations of this kind, introduced by V. Volterra himself within the framework of the theory of functional compositions, are also directly related to the modern problems of fractional dynamics. Our approach is based on employing the operational matrix technique with the shifted Legendre polynomials (SLP) as a basis. This approach reduces WSVIE to a system of linear algebraic equations with coefficients, which contain closed-form analytical expressions. This allows the use of computer algebra systems for obtaining approximate solutions based on the finite terms truncation of series. We provided a number of examples highlighting such a workflow in addition to establishing the error bound, convergence analysis and stability analysis.