Numerical methods for singular nonlinear integrodifferential equations
The equation which is studied here is the integrodifferential equation \[ y'(t)=f(t,y(t))+(1/\Gamma (\alpha))\int^{t}_{0}(t-s)^{\alpha - 1}K(t,s,y'(s))ds,\quad t>0, \] where \(y(0)=y_ 0\). The functions f and K are known and assumed to be smooth; \(0<\alpha <1\). It is known that the solution may not be smooth due to the presence of the weak singularity. \textit{C. Lubich} [Math. Comp. 45, 463-469 (1985; Zbl 0584.65090)] proposed a ``fractional linear multistep'' method for the numerical solution of an integral equation with a weakly singular kernel. The present authors show how this method can be adapted to deal with their equation. They prove convergence and present a stability analysis with diagrams of stability domains. The paper concludes with an examination of Basset's integrodifferential equation which describes the behaviour of a particle in a turbulent fluid. The equation is solved numerically and an associated function, the diffusion coefficient, is graphed.
singular nonlinear integrodifferential equations
scientific article; zbMATH DE number 4026778
scientific article