The method of finding coefficients of Taylor series is proposed for the solution of partial

differential equations (PDEs) with initial conditions. It is used to find interval solution

of Gursa problem

Uxy + a(x, y)Ux + b(x, y)Uy + c(x, y)U = f(x, y) (1)

(a, b, c - are analytical functions in the given field) with following conditions

U(x0, y) = ϕ1(y), y0 ≤ y ≤ b,

U(x, y0) = ϕ2(x), x0 ≤ x ≤ a, (2)

and besides ϕ1(y0) = ϕ2(x0).

Functions ϕ1(y) and ϕ2(x) have continuous derivatives of the first order.

The same method of finding coefficients of Taylor series is developed for the solution

of homogeneous differential equations system (HDE) [1].

The Taylor series function f(x, y) in the point (x0, y0) is the following:

f(x, y) =

X∞

i=0

X∞

j=0

(f)ij(x − x0)

i

(y − y0)

j

,

where

(f)ij :=

1

i!j!

∂

i+j

f

∂x

i∂y

j

.

The following formulas are used to find coefficients of Taylor series

(fx)ij = (i + 1)(f)i+1,j; (fxy)ij = (i + 1)(j + 1)(f)i+1,j+1;

(fxx)ij = (i + 1)(i + 2)(f)i+2,j; (f ± g)ij = (f)ij + (g)ij;

(fg)ij =

X

i

k=0

X

j

l=0

(f)kl(g)i−k,j−l

.

The recurrence formula is as follows:

(Uxy)ij = −(aUx)ij − (bUy)ij − (cU)ij + (f)ij. (3)

The remaining member is calculated according to the formula:

Rn+1 =

1

(n + 1)!(

∂

∂x

h1 +

∂

∂y

h2)

n+1U(τ1, τ2)

where

x0 ≤ τ1 ≤ x0 + h1,

y0 ≤ τ2 ≤ y0 + h2.

The polynomial to the n power is found and the remaining member of the (n + 1)

order is evaluated. Using interval methods get guaranteed a posteriori error estimation

including error calculation for the approximate problem solution (1)(2).

Reference

[1] Moore R.E. Interval analysis.-Englewood Cliffs.N.J.: Prentice-Hall.1966.