A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations
This paper investigates a numerical method for solving two-dimensional nonlinear Volterra-Fredholm integral equations of the following form: \[ u(x,t)-\lambda_1\int_{0}^{t}\int_{0}^{x} k_1(x,t,y,z) u^r(y,z)dydz- \] \[ -\lambda_2\int_{0}^{T_2}\int_{0}^{T_1} k_2(x,t,y,z) u^s(y,z)dydz=f(x,t), \quad r,s\in \mathbb{Z}^{+}, \] for the unknown function \(u(x,t)\) given on some region \(\Omega:=[0,T_1]\times[0,T_2]\). A matrix based method is considered for this equation. The method is based on the operational tau method. Some theoretical results are given to simplify and reduce the computation costs. In addition, \(L^2\)-convergence is proved when the given data are sufficiently smooth. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.
two-dimensional nonlinear Volterra-Fredholm integral equation
scientific article; zbMATH DE number 6418188
scientific article