Fast numerical solution of weakly singular Volterra integral equations
The authors present an algorithm for the solution of nonlinear, weakly singular Volterra integral equations of the form \[ y(t)=f(t)+1/\sqrt{\pi}\int^{t}_{0}(t-s)^{-1/2}k(t-s)g(s,y(s))ds, \] whose solution is assumed to be a smooth function of \(\sqrt{t}\). This algorithm is based on the fractional backward differentiation formula of order four introduced by the second author [Math. Comput. 45, 463-469 (1985; Zbl 0584.65090)]. The resulting discrete equations are solved by Newton's method; FFT techniques are employed in the generation of the quadrature weights and the computation of the convolution terms. Two numerical illustrations and the listing of the FORTRAN subroutine ABELFT conclude the paper.
nonlinear, weakly singular Volterra integral equations
fractional backward differentiation formula
scientific article; zbMATH DE number 4068227
scientific article