Product integration methods for solving a system of nonlinear Volterra integral equations
In their earlier paper [J. Math. Anal. Appl. 190, No. 3, 806-820 (1995; Zbl 0822.35076)] the authors showed that the heat equation with nonlinear and nonlocal boundary conditions, \[ u_t(x,t)=u_{xx} (x,t)\;(0<x<1,\;t>0);\;u(x,0)=1; \] \[ u_x(0,t)=0;\;u_x(1,t)={Em\over 1+L} \biggl\{L \gamma (t)- \bigl(1-\gamma(t)\bigr) u(1,t) \biggr\}; \] \[ m\gamma (t)+ \int^1_0 u(x,t)dx = 1 \] (with given positive constants \(E\), \(L\), \(m)\) is equivalent to a system of two nonlinear Volterra integral equations of the second kind with weakly singular kernel. Here, they derive and analyze the product Euler methods and a product trapezoidal method for solving this system of integral equations numerically. The known singularity of the exact solution at \(t=0\) permits the use of singularity subtraction techniques and the derivation of (classical) optimal order convergence results. A numerical example illustrates these results.
nonlinear and nonlocal boundary conditions
nonlinear Volterra integral equations of the second kind
scientific article; zbMATH DE number 908417
scientific article