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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
\[ u(x, t)-\int_0^t \int_0^x K(x, t, y, z, u(y, z))\,dydz=f(x, t), \]
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 Maple are given. The authors anticipate that the method will be developed for solving two-dimensional Volterra integro-differential equations and their systems.
Reviewer: Pat Lumb (Chester)

MSC:

65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

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