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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)- 0 t 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)= 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)= t 0 t x 0 x u(y,z)v(y,z)dydzandg(x,t)=h(x,t) t 0 t x 0 x u(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.

65R20Integral equations (numerical methods)
45G10Nonsingular nonlinear integral equations