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Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method. (English) Zbl 1176.65164

The authors consider two-dimensional Volterra integral equations of the form

$u\left(x,t\right)-{\int }_{0}^{t}{\int }_{0}^{x}K\left(x,t,y,z,u\left(y,z\right)\right)\phantom{\rule{0.166667em}{0ex}}dydz=f\left(x,t\right),$

where $K$ takes the degenerate form

$K\left(x,t,y,z\right)=\sum _{i=0}^{p}{v}_{i}\left(x,t\right){w}_{i}\left(y,z,u\left(y,z\right)\right)·$

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

$g\left(x,t\right)={\int }_{{t}_{0}}^{t}{\int }_{{x}_{0}}^{x}u\left(y,z\right)v\left(y,z\right)\phantom{\rule{0.166667em}{0ex}}dydz\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}g\left(x,t\right)=h\left(x,t\right){\int }_{{t}_{0}}^{t}{\int }_{{x}_{0}}^{x}u\left(y,z\right)\phantom{\rule{0.166667em}{0ex}}dydz·$

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 Maple are given. The authors anticipate that the method will be developed for solving two-dimensional Volterra integro-differential equations and their systems.

##### MSC:
 65R20 Integral equations (numerical methods) 45G10 Nonsingular nonlinear integral equations
Maple