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\centerline{Approximation of analytic solutions}

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\centerline{to functional differential equations of the first order}

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\centerline{Antoni Augustynowicz (Gda\'nsk)}

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Consider the problem
$$
u_t(t,x)=f\Bigl( t,x,u(\alpha(t,x)),u_x(\beta(t,x))\Bigr)\ ,\ \
u(0,x)=\phi(x)\ ,
$$
where the functions $f,\phi,\alpha,\beta$ are analytic. Suppose that
there exist majorants of $\alpha$ and $\beta$ being delays with respect
to both variables, then the above problem has the unique analytic
solution on some neighbourhood of $x$-axis. If the majorants are
substantial delays, then using the step-by-step method we can extend the
solution on the maximal area of analyticity of known functions. We study
some numerical method of finding approximate solution on a neighbourhood
of the origin and extention of this solution on larger area.







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