# On some difference analogues of PDEs with a delay

## Abstract

- 1.
No difference scheme for equation (2) seems to be definitely convergent

- 2.
In our experiments the schemes of type A and B and mixed A-B behave less stable than C or D, especially close to the boundary errors for A and B are relatively big

- 3.
The schemes C and D are comparable and the error expressions are reasonable

- 4.
Our scheme of type D for equation (2) is easily adaptable to a parallel computing process. It is enough to divide the set

*E = [0, T] × [−b, b]*into rectangles*E = [0, T] ×*[*b*_{ j },*b*_{ j },*b*^{ j+1 }] (*j*= 0,..., m − 1) where*−b = b*_{0}< ... <*b*_{ m }*= b*, and then we can minimize functional*F*_{ j }defined on*E*like*F*on*E*. Using many processors one can achieve better results for meshes with smaller steps*h*_{ t }and*h*_{ x }. The exchange of data between neighbouring regions is minimal. - 5.
A proper division of the set

*E*and the question of well-posedness of initialor initial-boundary-value problem for equation (3) is still open and demands some more extensive investigations with the use of multi-processor mashines.

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