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A fast iterative method for discretized Volterra-Fredholm integral equations. (English) Zbl 1092.65118
The 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.
Reviewer: Pat Lumb (Chester)

65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations
65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[1] Atkinson, K.E., The numerical solution of integral equations of the second kind, (1997), Cambridge University Press Cambridge, MA · Zbl 0155.47404
[2] Bownds, J.M.; Wood, B., On numerically solving nonlinear Volterra integral equations with fewer computations, SIAM J. numer. anal., 13, 705-719, (1976) · Zbl 0404.65064
[3] Brunner, H., On the numerical solution of nonlinear volterra – fredholm integral equations by collocation methods, SIAM J. numer. anal., 27, 987-1000, (1990) · Zbl 0702.65104
[4] Brunner, H.; van der Houwen, P.J., The numerical solution of Volterra equations, () · Zbl 0611.65092
[5] Brunner, H.; Messina, E., Time-stepping methods for volterra – fredholm integral equations by collocation methods, Rend. mat., serie VII, 23, 329-342, (2003) · Zbl 1095.65117
[6] Diekmann, O., Thresholds and travelling waves for the geographical spread of infection, J. math. biol., 6, 109-130, (1978) · Zbl 0415.92020
[7] Han, G., Asymptotic error expansion for the Nyström method for a nonlinear volterra – fredholm integral equations, J. comput. appl. math., 59, 49-59, (1995) · Zbl 0834.65137
[8] van der Houwen, P.J.; de Swart, J.J.B., Parallel linear system solvers for runge – kutta methods, Adv. comput. math., 7, 157-181, (1997) · Zbl 0886.65078
[9] Kauthen, J.P., Continuous time collocation methods for volterra – fredholm integral equations, Numer. math., 56, 409-424, (1989) · Zbl 0662.65116
[10] Pachpatte, B.G., On mixed volterra – fredholm type integral equations, Indian J. pure appl. math., 17, 488-496, (1986) · Zbl 0597.45012
[11] Thieme, H.R., A model for spatial spread of an epidemic, J. math. biol., 4, 337-351, (1977) · Zbl 0373.92031
[12] Wolkenfelt, P.H.M., The construction of reducible quadrature rules for Volterra integral and integro-differential equations, IMA J. numer. anal., 2, 131-152, (1982) · Zbl 0481.65084
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