Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method
The authors consider two-dimensional Volterra integral equations of the form \[ u(x, t)-\int_0^t \int_0^x K(x, t, y, z, u(y, z))\,dydz=f(x, t), \] where \(K\) takes the degenerate form \[ K(x, t, y, z)=\sum_{i=0}^p v_i(x, t)w_i(y, z, u(y, z)). \] In section 2 fundamental properties of the TDDT are summarised in a theorem. This is followed by the main theorem in the paper which states and derives differential transforms for \[ g(x, t)=\int_{t_0}^t \int_{x_0}^x u(y, z)v(y, z)\,dydz\text{ and }g(x, t)=h(x, t)\int_{t_0}^t \int_{x_0}^xu(y, z)\,dydz. \] The method is described in section 3. Section 4 includes illustrative examples (including a non-linear problem) to demonstrate the accuracy of the presented method. In each example a recurrence relation for the differential transform is obtained and results arising from the use of \textsc{Maple} are given. The authors anticipate that the method will be developed for solving two-dimensional Volterra integro-differential equations and their systems.
two-dimensional differential transform method
scientific article; zbMATH DE number 5559030