A fast iterative method for discretized Volterra-Fredholm integral equations.

*(English)*Zbl 1092.65118The authors develop a method that reduces the cost of solving discretized versions of nonlinear Volterra-Fredholm integral equations of the form
\[
u(t, x)=f(t,x)+\int_0^t \int_{\Omega} G(t, s, x, \xi, u(s, \xi))\,d\xi \,ds
\]
on a bounded spatial domain. The method introduced uses linear methods for both the Fredholm and for the Volterra parts of the equation. Semi-discretization in space by the Nyström method is followed by discretization in time by a direct quadrature method. The resulting system of nonlinear algebraic equations is (conventionally) solved using the (modified) Newton iteration scheme.

The authors present and prove a theorem relating to the total error at the mesh points after \(\sigma\) Newton iterations. A new iterative method is introduced which reduces the computational complexity of the process. The inner iteration process used involves solving \((M+1)\) decoupled scalar equations, decoupled by a split of the coefficient matrix by a diagonal matrix. Convergence of this iterative linear solver is discussed and analyzed for kernels which are degererate with respect to the spatial variables, and for nondegenerate kernels of Hammerstein type (under the assumption that conditions for the existence and uniqueness of the solution are satisfied). Theorems relating to the convergence of the inner iteration process are stated.

Results of numerical experiments illustrating the performance of the method are presented and discussed. The authors comment that many of the results obtained in the paper can be generalized to the case of an unbounded spatial domain. They intend this case, found in many applications, to be the focus of a later paper.

The authors present and prove a theorem relating to the total error at the mesh points after \(\sigma\) Newton iterations. A new iterative method is introduced which reduces the computational complexity of the process. The inner iteration process used involves solving \((M+1)\) decoupled scalar equations, decoupled by a split of the coefficient matrix by a diagonal matrix. Convergence of this iterative linear solver is discussed and analyzed for kernels which are degererate with respect to the spatial variables, and for nondegenerate kernels of Hammerstein type (under the assumption that conditions for the existence and uniqueness of the solution are satisfied). Theorems relating to the convergence of the inner iteration process are stated.

Results of numerical experiments illustrating the performance of the method are presented and discussed. The authors comment that many of the results obtained in the paper can be generalized to the case of an unbounded spatial domain. They intend this case, found in many applications, to be the focus of a later paper.

Reviewer: Pat Lumb (Chester)

##### MSC:

65R20 | Numerical methods for integral equations |

45G10 | Other nonlinear integral equations |

65H10 | Numerical computation of solutions to systems of equations |

##### Keywords:

nonlinear Volterra-Fredholm integral equations; Nyström methods; iterative methods; direct quadrature methods; degenerate and non-degenerate kernels; semi-discretization; Newton iteration; convergence; kernels of Hammerstein type; numerical experiments
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\textit{A. Cardone} et al., J. Comput. Appl. Math. 189, No. 1--2, 568--579 (2006; Zbl 1092.65118)

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