Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 18, 2005, 151–175 \begin{matrix}c \\ ◯ \end{matrix} Ye.A. Utkina

Ye.A. Utkina

ON A PARTIAL DIFFERENTIAL EQUATION IN 4-DIMENSIONAL EUCLIDEAN SPACE

(submitted by M. M. Arslanov)

Abstract. The objective of this article is to construct in a hyperrectangular region of the 4-dimensional Euclidean space a solution of the Goursat problem for the following equation:

L (u) = ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} a_{i_{1}i_{2}i_{3}i_{4}} (x_{1}, x_{2}, x_{3}, x_{4}) \frac{∂^{i_{1}+i_{2}+i_{3}+i_{4}}u}{∂x_{1}^{i_{1}}∂x_{2}^{i_{2}}∂x_{3}^{i_{3}}∂x_{4}^{i_{4}}}

= F (x_{1}, x_{2}, x_{3}, x_{4}) .

Introduction. Let D = {x_{10} < x_{1} < x_{11}, x_{20} < x_{2} < x_{21}, x_{30} < x_{3} < x_{31}, x_{40} < x_{4} < x_{41}}, where X_{1}, X_{2}, X_{3}, X_{4} are the faces of D for x_{i} = x_{i0} (i=1,2,3,4), respectively. The objective of this article is to construct in D a solution of the Goursat problem for the following equation:

L (u) = ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} a_{i_{1}i_{2}i_{3}i_{4}} (x_{1}, x_{2}, x_{3}, x_{4}) \frac{∂^{i_{1}+i_{2}+i_{3}+i_{4}}u}{∂x_{1}^{i_{1}}∂x_{2}^{i_{2}}∂x_{3}^{i_{3}}∂x_{4}^{i_{4}}}

= F (x_{1}, x_{2}, x_{3}, x_{4}) , (1)

where a_{m_{1}m_{2}m_{3}m_{4}} ≡ 1, while the smoothness of remaining coefficients is defined by the inclusions

a_{i_{1}i_{2}i_{3}i_{4}} ∈ C^{i_{1}+i_{2}+i_{3}+i_{4}} (\overline{D}) , F ∈ C^{0+0+0+0} (\overline{D}) .  (2)

Here C^{α_{1}+α_{2}+α_{3}+α_{4}} stands for a class of functions continuous in \overline{D} together with their partial derivatives

∂^{r_{1}+r_{2}+r_{3}+r_{4}}/∂x_{1}^{r_{1}}∂x_{2}^{r_{2}}∂x_{3}^{r_{3}}∂x_{4}^{r_{4}} (r_{1} = 0, ..., α_{1}, r_{2} = 0, ..., α_{2}, r_{3} = 0, ..., α_{3}, r_{4} = 0, ..., α_{4}).

The Goursat problem: Determine in D a regular solution of the equation (1), which satisfies the following conditions:

\begin{matrix}\frac{∂^{i_{1}}u}{∂x_{1}^{i_{1}}} (x_{10}, x_{2}, x_{3}, x_{4}) = φ_{1i_{1}} (x_{2}, x_{3}, x_{4}) , (i_{1} = \overline{0, m_{1} − 1}) , \\ \frac{∂^{i_{2}}u}{∂x_{2}^{i_{2}}} (x_{1}, x_{20}, x_{3}, x_{4}) = φ_{2i_{2}} (x_{1}, x_{3}, x_{4}) , (i_{2} = \overline{0, m_{2} − 1}) , \\ \frac{∂^{i_{3}}u}{∂x_{3}^{i_{3}}} (x_{1}, x_{2}, x_{30}, x_{4}) = φ_{3i_{3}} (x_{1}, x_{2}, x_{4}) , (i_{3} = \overline{0, m_{3} − 1}) , \\ \frac{∂^{i_{4}}u}{∂x_{4}^{i_{4}}} (x_{1}, x_{2}, x_{3}, x_{40}) = φ_{4i_{4}} (x_{1}, x_{2}, x_{3}) , (i_{4} = \overline{0, m_{4} − 1}) , \end{matrix} (3)

where

\begin{matrix}φ_{1i_{1}} ∈ C^{m_{2}+m_{3}+m_{4}} (\overline{X_{1}}) , φ_{2i_{2}} ∈ C^{m_{1}+m_{3}+m_{4}} (\overline{X_{2}}) , \\ φ_{3i_{3}} ∈ C^{m_{1}+m_{2}+m_{4}} (\overline{X_{3}}) , φ_{4i_{4}} ∈ C^{m_{1}+m_{2}+m_{3}} (\overline{X_{4}}) , \end{matrix}

and the boundary values in (3) are coordinated on the edges of D as follows:

φ_{10} (x_{20}, x_{3}, x_{4}) = φ_{20} (x_{10}, x_{3}, x_{4}) , φ_{10} (x_{2}, x_{30}, x_{4}) = φ_{30} (x_{10}, x_{2}, x_{4}) ,

φ_{10} (x_{2}, x_{3}, x_{40}) = φ_{40} (x_{10}, x_{2}, x_{3}) ;

φ_{20} (x_{1}, x_{30}, x_{4}) = φ_{30} (x_{1}, x_{20}, x_{4}) , φ_{20} (x_{1}, x_{3}, x_{40}) = φ_{40} (x_{1}, x_{20}, x_{3}) , φ_{30} (x_{1}, x_{2}, x_{40}) = φ_{40} (x_{1}, x_{2}, x_{30}) ,

and the coordinated values are supposed to be differentiable.

Equation (1) can be treated as an analog of the two-dimensional pseudoparabolic equation in [1]. However, a version of the Riemann’s method to be applied here differs essentially from that used in [1]. One can note that equations considered in this paper have modifications frequently used in applications. One can consider, for example, the Boussinesq-Love equation in the theory of oscillations (see [1], formula (20)), the Aller equation in [2], p.261, applied in the mathematical simulation of moisture absorption by roots of plants.

1. First of all we consider a special case, where the derivatives with respect to the variables x_{2}, x_{3}, x_{4} are not present:

\frac{∂^{m}u}{∂x_{1}^{m}} + ∑_{i=0}^{m−1} a_{i} (x_{1}, y) \frac{∂^{i}u}{∂x_{1}^{i}} = 0.  (4)

Here and on in this section we denote by y the set of coordinates x_{2}, x_{3}, x_{4}. One item of the investigation of (4) will be also used in the general case (see Section 2). In addition, it seems to be of interest to compare the investigation of (4) with the study of the following ordinary differential equation:

\frac{d^{m}u}{dx^{m}} + ∑_{i=0}^{m−1} a_{i} (x) \frac{d^{i}u}{dx^{i}} = 0.  (5)

Obviously, (4) and (5) possess a similar structure; however, the general theory of the second case (see [3], p.145) depends essentially on the theorem stating that the Wronskian m of its solutions either does not vanish or is identically equal to zero. However, this theorem cannot be generalized to equation (1): one can easily see, for example, that the equation u_{xx} − 3yu_{x} + 2y^{2}u = 0 possesses the solutions exp (xy), exp (2xy). At the same time, its Wronskian w = y · exp (3xy) vanishes for y = 0 and differs from zero in any other point of a bounded domain. Therefore there arises a problem of construction of a theory for the solutions of (4), which will not use the above property of the Wronskian. In what follows we develop a version of the Riemann method in [4], [5].

The Goursat problem can be transformed to the following well-known Cauchy problem: find in D a solution of equation (1) in the class C^{m+0}, which will satisfy the following conditions:

\frac{∂^{i}u}{∂x_{1}^{i}} |_{x_{1}=x_{10}} = φ_{i} (y) (0 ≤ i ≤ m − 1)  (6)
φ_{i} ∈ C (p) , p = [y_{0}, y_{1}] .

We assume that the coefficients a_{i} are continuous in \overline{D} with respect to the variables x_{2}, x_{3}, x_{4} and belong to the class C^{i} (\overline{D}) on x_{1} . We define the Riemann function V as a solution of the following Volterra integral equation:

V (x_{1}, y) + ∑_{i=0}^{m−1} (−1)^{m−i} ∫_{ξ}^{x_{1}} \frac{(x_{1} − t)^{m−i−1}}{(m − i − 1)!} a_{i} (t, y) V (t, y) dt = 1, (7)

which is known to exist and be unique. In addition, from (7) it follows that V is a solution of the equation conjugate to (4):

L^{∗} (V) = ∑_{i=0}^{m} (−1)^{m−i} \frac{∂^{i} (a_{i}V)}{∂x_{1}^{i}} ≡ 0, a_{m} = 1.  (8)

To indicate that V depends not only on x_{1}, y but also on ξ, we will write this function as R (x_{1}, y, ξ). The next identities (which will be of use later) easily follow from (7):

∑_{α=i}^{m} (−1)^{m−α} \frac{∂^{α−i} (a_{α}R)}{∂x_{1}^{α−i}} (x_{1}, y, x_{1}) ≡ 0, (i ≤ m − 1)  (9)
R (x_{1}, y, x_{1}) ≡ 1.

The following identity takes place for any function u from the class C^{m+0} (D):

\frac{∂^{m} [uR]}{∂x_{1}^{m}} ≡ RL (u) + ∑_{i=1}^{m−1} (−1)^{m−i+1} \frac{∂^{i}}{∂x_{1}^{i}} [u ∑_{α=i}^{m} (−1)^{m−α} \frac{∂^{α−i} [a_{α}R]}{∂x_{1}^{α−i}} ]

+ ∑_{i=0}^{m} ∑_{b=0}^{i} K_{ib} \frac{∂^{b}u}{∂x_{1}^{b}} · \frac{∂^{i−b} [a_{i}R]}{∂x_{1}^{i−b}}, (10)

where a_{i} depend on (x_{1}, y), while R and its derivatives depend on

(x_{1}, y, ξ). In this situation, K_{ib} = ∑_{α=b}^{i} (−1)^{i−α} C_{α}^{b} − M_{ib}, M_{ib} = 1 if

b = i, and M_{ib} = 0 otherwise. To prove (10) rewrite this identity as follows:

\frac{∂^{m} [uR]}{∂x_{1}^{m}} = R·∑_{i=0}^{m} a_{i}· \frac{∂^{i}u}{∂x_{1}^{i}} +∑_{i=1}^{m−1} (−1)^{i−1} \frac{∂^{i}}{∂x_{1}^{i}} [u · ∑_{α=i}^{m} (−1)^{α} \frac{∂^{α−i} [a_{α}R]}{∂x_{1}^{α−i}}]

+ ∑_{i=0}^{m} ∑_{b=0}^{i} (−1)^{i−b} \frac{∂^{b}}{∂x_{1}^{b}} [u · \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}}] − ∑_{i=0}^{m} \frac{∂^{i}u}{∂x_{1}^{i}} · a_{i}R .

In the right-hand side the first and last addends cancel each other. Transfer the addend from the left-hand side to the right-hand side and add to the second term:

∑_{i=1}^{m} ∑_{α=i}^{m} (−1)^{i−α+1} \frac{∂^{i}}{∂x_{1}^{i}} [u · \frac{∂^{α−i} [a_{α}R]}{∂x_{1}^{α−i}}]

+ ∑_{i=0}^{m} ∑_{b=0}^{i} (−1)^{i−b} \frac{∂^{b}}{∂x_{1}^{b}} [u · \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}}] = 0.

By using the equality

∑_{b=0}^{i} ∑_{α=b}^{i} C_{α}^{b} \frac{∂^{b}u}{∂x_{1}^{b}} · \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}} = ∑_{α=0}^{i} ∑_{b=0}^{α} C_{b}^{α} \frac{∂^{b}u}{∂x_{1}^{b}} · \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}},

we see that identity (10) holds.

To construct the solution of the problem we represent (10) in another way:

\frac{∂^{m} [uR]}{∂x_{1}^{m}} ≡ RL (u) + ∑_{i=1}^{m−1} (−1)^{m−i+1} \frac{∂^{i}}{∂x_{1}^{i}} [u ∑_{α=i}^{m} (−1)^{m−α} \frac{∂^{α−i} [a_{α}R]}{∂x_{1}^{α−i}} ]

+ ∑_{i=2}^{m} ∑_{b=0}^{i} \frac{∂^{b+1}u}{∂x_{1}^{b+1}} \{C_{i−1−b}^{i−2−2b} \frac{∂^{i−2−2b} (u)}{∂x_{1}^{i−2−2b}} · \frac{∂^{b+1} (a_{i}R)}{∂x_{1}^{b+1}}\} . (11)

In order to verify that (11) is valid, one can use the following equality:

∑_{b=0}^{i} [∑_{α=b}^{i} (−1)^{i−α} C_{α}^{b} − M_{ib}] \frac{∂^{b}u}{∂x_{1}^{b}} · \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}}

= ∑_{b=0}^{i} [∑_{k=0}^{b/2} C_{\frac{i}{2} −(int(\frac{b}{2})−k)}^{2k+1} · C_{\frac{i}{2} +(int(\frac{b}{2})−k)}^{b−(2k+1)} − M_{ib}] \frac{∂^{b}u}{∂x_{1}^{b}} · \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}}. (12)

The latter takes place if b is odd and i is even. If both i and b are odd, then the right-hand side of (12) can be replaced with the following expression

∑_{b=0}^{i} [ ∑_{k=0}^{int(b/2)} C_{\frac{i+1}{2} −(int(\frac{b}{2})−k)}^{2k+1} · C_{\frac{i−1}{2} +(int(\frac{b}{2})−k)}^{b−(2k+1)} − M_{ib}] \frac{∂^{b}u}{∂x_{1}^{b}}

· \frac{∂^{i−b} (a_{i}R)}{∂x_{1}^{i−b}} . (13)

For b even, instead of 2k +1 one must take 2k. Relations (12) and (13) can be verified straightforwardly. By changing the summation order in (12), we obtain the last addend in (11). Now, by taking in (11) x_{1} equal to ξ and denoting x_{1} by ξ, we calculate the integral in x_{10} < ξ < x_{11} and take into account identities (9). We then arrive at the following:

\frac{∂^{m−1}u}{∂x_{1}^{m−1}} (x, y) = ∫_{x_{10}}^{x_{1}} (R · L (u)) (ξ, y) dξ−

− ∑_{i=1}^{m} (−1)^{i−1} ∑_{α=i}^{m} (−1)^{α} (\frac{∂^{i−1}u}{∂x_{1}^{i−1}} · \frac{∂^{α−i} (a_{α}R)}{∂x_{1}^{α−i}}) (x_{10}, y, x_{1}) . (14)

Obtain the required solution by integrating (14) m−1 times with respect to x_{1} between the limits x_{10} to x_{1} and taking into account (6):

u (x_{1}, y) = φ_{0} (y) + ∑_{i=1}^{m−2} φ_{i} (y) ∫_{x_{10}}^{x_{1}} \frac{(x_{1} − t)^{i−1}}{(i − 1)!} dt − ∑_{i=1}^{m} (−1)^{i−1} φ_{i−1} (y)

· ∑_{α=i}^{m} (−1)^{α} · ∫_{x_{10}}^{x_{1}} \frac{(x_{1} − t)^{m−2}}{(m − 2)!} · \frac{∂^{α−i} (a_{α}R (x_{10}, y, t))}{∂x_{1}^{α−i}}dt . (15)

As in [6], p. 66, we can assume that for arbitrary φ_{i} (y) the formula (15) supplies the general presentation of the solutions of equation (4).

With u = u (x) and a_{i} = a_{i} (x) equation (4) turns into an ordinary differential equation (5). Obviously, formula (15) represents a solution for (5), moreover, here φ_{i} (y) are constants and both the coefficients a_{i} and the Riemann function do not depend on y. Thus, formula (15) contains the classical result; moreover, it is written there in a closed form which means that the classical theory indicates a method for solving, while such a solution is represented here by a formula.

Now assume a_{i} = a_{i} (y), i.e., here we deal with a certain generalization of ordinary differential equations with constant coefficients. We will consider only a particular case with m = 2. The formula (15) then takes

PARTIAL DIFFERENTIAL EQUATION 157

the form

u (x, y) = φ_{0} (y) · {1 + ∫_{x_{0}}^{x} [R_{ξ} (x_{0}, y, ξ) − a_{1} (y) R (x_{0}, y, ξ)] dξ}

+ φ_{1} (y) ∫_{x_{0}}^{x} R (x_{0}, y, ξ) dξ

= φ_{0} (y) · {1 − ∫_{x_{0}}^{x} ∫_{x_{0}}^{t} a_{0} (y) R (x_{0}, y, t_{1}) dt_{1}dt}

+ φ_{1} (y) ∫_{x_{0}}^{x} R (x_{0}, y, ξ) dξ . (16)

Find the Riemann function whose equation in this case is as follows:

V (x, y) = ∑_{k=0}^{∞} V_{k} (x, y) ,

where

V_{0} ≡ 1, V_{1} (x, y) = ∫_{x_{0}}^{x} H (x, t, y) dt, V_{2} (x, y) = ∫_{x_{0}}^{x} H (x, t, y) V_{1} (t, y) dt, …,

V_{k} (x, y) = ∫_{x_{0}}^{x} H (x, t, y) V_{k−1} (t, y) dt, …,

and besides H (x, t, y) = a_{1} (y) − (x − t) a_{0} (y). We thus have

R (x, y, x_{0}) = V (x, y) = ∑_{n=0}^{∞} ∑_{k=0}^{n} (−1)^{k} C_{n}^{k}a_{1}^{n−k} (y) a_{0}^{k} (y) \frac{(x − x_{0})^{n+k}}{(n + k)!} .

Put a_{1} (y) = λ_{1} (y) + λ_{2} (y), a_{0} (y) = λ_{1} (y) · λ_{2} (y). For the case with λ_{1} (y) ≠ λ_{2} (y), we obtain

R (x, y, x_{0}) = \frac{λ_{2} (y) e^{λ_{2}(y)(x−x_{0})} − λ_{1} (y) e^{λ_{1}(y)(x−x_{0})}}{λ_{2} (y) − λ_{1} (y)} ,

where λ_{1;2} (y) = ^{\frac{a_{1}(y)∓\sqrt{a_{1}^{2}(y)−4a_{0}(y)}}{2}}. By interchanging x and x_{0}, we obtain

R (x_{0}, y, x) = \frac{−λ_{2} (y) e^{−λ_{2}(y)(x−x_{0})} + λ_{1} (y) e^{−λ_{1}(y)(x−x_{0})}}{−λ_{2} (y) + λ_{1} (y)} .

Substitute the result into (16). One can easily see that

u (x, y) = C_{1} (y) · e^{−λ_{1}(y)x} + C_{2} (y) · e^{−λ_{2}(y)x},

where

C_{1} (y) = \frac{−φ_{0} (y) λ_{2} (y) − φ_{1} (y)}{(λ_{1} (y) − λ_{2} (y)) e^{−λ_{1}(y)x_{0}}} , C_{2} (y) = \frac{−φ_{0} (y) λ_{1} (y) − φ_{1} (y)}{(λ_{2} (y) − λ_{1} (y)) e^{−λ_{2}(y)x_{0}}} .

Now set λ_{1} (y) = λ_{2} (y). Then the Riemann function is

R (x, y, x_{0}) = e^{λ_{1}(y)(x−x_{0})} [1 + (x − x_{0}) λ_{1} (y)] .

We again interchange x and x_{0} and calculate the respective terms. We thus obtain

u (x, y) = C_{1} (y) · e^{−λ_{1}(y)x} + C_{2} (y) · x · e^{−λ_{1}(y)x},

where

C_{1} (y) = (φ_{0} (y) [−x_{0} · λ_{1} (y) + 1] − φ_{1} (y) x_{0}) e^{λ_{1}x_{0}},
C_{2} (y) = (φ_{0} (y) λ_{1} (y) + φ_{1} (y)) e^{λ_{1}x_{0}}.

Let us write the solution in the case of conjugate complex roots λ_{1} (y), λ_{2} (y) of the characteristic equation. The roots can be represented as follows

λ_{1;2} = \frac{α ± β · i}{2},

where β · i = \sqrt{a_{1}^{2} (y) − 4a_{0} (y)}, α = a_{1} (y).

In this case, by the straightforward calculation yields:

u (x, y) = φ_{0} (y) · e^{− \frac{α}{2} ·(x−x_{0})} [\frac{α sin (\frac{β}{2} (x − x_{0})) + β cos (\frac{β}{2} (x − x_{0}))}{−β}]

+φ_{1} (y) · e^{− \frac{α}{2} ·(x−x_{0})} \frac{sin (\frac{β}{2} (x − x_{0}))}{β}

= e^{− \frac{α}{2} ·(x−x_{0})} · [− sin (\frac{β}{2} (x − x_{0})) \{φ_{0} (y) \frac{α}{β} − φ_{1} (y) · \frac{1}{β} \}

+ cos (\frac{β}{2} (x − x_{0})) · α · φ_{0} (y)] .

Consequently, in contrast to the classical technique for solving ordinary differential equations with constant coefficients, the formula obtained contains simultaneously both the case of a non-multiple root of the characteristic equation and that of a multiple root, as well as the case of conjugate roots.

Consider also a non-homogeneous equation

\frac{∂^{m}u}{∂x^{m}} + ∑_{i=0}^{m−1} a_{i} (x, y) \frac{∂^{i}u}{∂x^{i}} = f (x, y) .

Its solution has the form

u (x, y) = φ_{0} (y) + ∑_{i=1}^{m−2} φ_{i} (y) ∫_{x_{0}}^{x} \frac{(x − t)^{i−1}}{(i − 1)!} dt

− ∑_{i=1}^{m} (−1)^{i−1} φ_{i−1} (y) · ∑_{α=i}^{m} (−1)^{α} ∫_{x_{0}}^{x} \frac{(x − t)^{m−2}}{(m − 2)!} · \frac{∂^{α−i} (a_{α}R (x_{0}, y, t))}{∂x^{α−i}}dt

+ ∫_{x_{0}}^{x} \frac{(x − t)^{m−1}}{(m − 1)!} R (t, y, x) dt .

Denote by K (t, y, x) \frac{(x−t)^{m−1}}{(m−1)!} R (t, y, x). One can then easily verify that we get a particular solution obtained in ([7], p. 121), by the Cauchy method. Since K (t, y, t) = K′_{x} (t, y, t) = ... = K_{x}^{(m−2)} (t, y, t) = 0,

K_{x}^{(m−1)} (t, y, t) = 1 and u_{∗} (x, y) = ∫_{x_{0}}^{x} K (t, y, x) f (t, y) dt is a particular

solution of the equation, satisfying the zero initial conditions φ_{0} (y) ≡ φ_{1} (y) ≡ ... ≡ φ_{n−1} (y) ≡ 0. In addition, K (t, y, x) is written in the explicit form.

2. In this section us consider a general case. For the sake of simplicity we use following notation introduced in [1]: D_{t}^{k}φ ≡ ∂^{k}φ/∂t^{k} for

k = 1, 2, .. and D_{t_{0}}^{k}φ ≡ ∫_{t_{0}}^{t} \frac{(t−τ)^{−k−1}φ(τ)}{(−k−1)!}dτ if k = −1, −2, .. and D_{t}^{0} being

operator of identical transform.

By the Riemann function R (x_{1}, x_{2}, x_{3}, x_{4}; ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) we call any solution of the integral equation

∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} (−1)^{∑_{α=1}^{4}^{(m_{α}−i_{α})}} ∏_{α=1}^{4} D_{x_{α}}^{i_{α}−m_{α}} (a_{i_{1}i_{2}i_{3}i_{4}}R) = 1,  (17)

which both exists and is unique (see [8], p.180). From (17) it follows that, if we consider the first four arguments of the equation, then R is a solution of the following equation conjugate to (1):

L^{∗} (V) = ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} (−1)^{∑_{α=1}^{4}^{(m_{α}−i_{α})}} ∏_{α=1}^{4} D_{x_{α}}^{i_{α}} (a_{i_{1}i_{2}i_{3}i_{4}}V) = 0. (18)

In what follows we need the next proposition.

For any function u in C^{∑_{α=1}^{4} m_{α}} (D), the following identity holds:

L (u) + \begin{matrix}∑_{i_{1}=0}^{m_{1}} & ∑_{i_{2}=0}^{m_{2}} & ∑_{i_{3}=0}^{m_{3}} & ∑_{i_{4}=0}^{m_{4}} \\ i_{1}+i_{2}+i_{3}+i_{4}>0 \end{matrix}

(−1)^{∑_{k=1}^{4} i_{k}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}} [u ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}}

(−1)^{∑_{k=1}^{4} α_{k}−1} · ∏_{k=1}^{4} D_{x_{k}}^{α_{k}−i_{k}} (a_{α_{1}α_{2}α_{3}α_{4}}R)]

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{1}=0}^{i_{1}} ∑_{b_{2}=0}^{i_{2}} ∑_{b_{3}=0}^{i_{3}} ∑_{b_{4}=0}^{i_{4}} K_{i_{1}i_{2}i_{3}i_{4}b_{1}b_{2}b_{3}b_{4}}

· ∏_{k=1}^{4} D_{x_{k}}^{b_{k}} (u) ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−b_{k}} (a_{α_{1}α_{2}α_{3}α_{4}}R) ≡ 0 , (19)

where

K_{i_{1}i_{2}i_{3}i_{4}b_{1}b_{2}b_{3}b_{4}} =

∑_{α_{1}=b_{1}}^{i_{1}} (−1)^{i_{1}−α_{1}} C_{α_{1}}^{b_{1}} ∑_{α_{2}=b_{2}}^{i_{2}} (−1)^{i_{2}−α_{2}} C_{α_{2}}^{b_{2}} ∑_{α_{3}=b_{3}}^{i_{3}} (−1)^{i_{3}−α_{3}} C_{α_{3}}^{b_{3}}

∑_{α_{4}=b_{4}}^{i_{4}} (−1)^{i_{4}−α_{4}} C_{α_{4}}^{b_{4}} − M_{b_{1}b_{2}b_{3}b_{4}},

and M_{b_{1}b_{2}b_{3}b_{4}} = 1 if b_{1} = i_{1}, b_{2} = i_{2}, b_{3} = i_{3}, b_{4} = i_{4}, otherwise M_{b_{1}b_{2}b_{3}b_{4}} = 0. Here a_{i_{1}i_{2}i_{3}i_{4}} depend on (x_{1}, x_{2}, x_{3}, x_{4}), while R and its derivatives on (x_{1}, x_{2}, x_{3}, x_{4}, ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}). In order to prove this rewrite the last addend in (19) and note that R is a solution of equation (18). Next, present this addend as follows:

D_{x_{1}}^{b_{1}+1} \{C_{i_{1}−1−b_{1}}^{i_{1}−2−2b_{1}}D_{x_{1}}^{i_{1}−2−2b_{1}} ∏_{α=2}^{4} D_{x_{α}}^{i_{α}}u · D_{x_{1}}^{b_{1}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)\}^{∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1}} (20)

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1}

D_{x_{2}}^{b_{2}+1} {C_{i_{2}−1−b_{2}}^{i_{2}−2−2b_{2}}D_{x_{2}}^{i_{2}−2−2b_{2}} ∏_{α=1,α≠2}^{4} D_{x_{α}}^{i_{α}}u · D_{x_{2}}^{b_{2}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1}

D_{x_{3}}^{b_{3}+1} {C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}D_{x_{3}}^{i_{3}−2−2b_{3}} ∏_{α=1,α≠3}^{4} D_{x_{α}}^{i_{α}}u · D_{x_{3}}^{b_{3}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1}

D_{x_{4}}^{b_{4}+1} {C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{4}}^{i_{4}−2−2b_{4}} ∏_{α=1}^{3} D_{x_{α}}^{i_{α}}u · D_{x_{4}}^{b_{4}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{2}}^{b_{2}+1}

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{3}}^{b_{3}+1}

· {C_{i_{1}−1−b_{1}}^{i_{1}−2−2b_{1}}C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}D_{x_{1}}^{i_{1}−2−2b_{1}}D_{x_{2}}^{i_{2}}D_{x_{3}}^{i_{3}−2−2b_{3}}D_{x_{4}}^{i_{4}}u · D_{x_{1}}^{b_{1}+1}D_{x_{3}}^{b_{3}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{4}}^{b_{4}+1}

· {C_{i_{1}−1−b_{1}}^{i_{1}−2−2b_{1}}C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{1}}^{i_{1}−2−2b_{1}}D_{x_{2}}^{i_{2}}D_{x_{3}}^{i_{3}}D_{x_{4}}^{i_{4}−2−2b_{4}}u

· D_{x_{1}}^{b_{1}+1}D_{x_{4}}^{b_{4}+1} (a_{i_{1}i_{2}i_{3}nnnni_{4}}R)}

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} D_{x_{2}}^{b_{2}+1}D_{x_{3}}^{b_{3}+1}

· {C_{i_{2}−1−b_{2}}^{i_{2}−2−2b_{2}}C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}D_{x_{1}}^{i_{1}}D_{x_{2}}^{i_{2}−2−2b_{2}}D_{x_{3}}^{i_{3}−2−2b_{3}}D_{x_{4}}^{i_{4}}u · D_{x_{2}}^{b_{2}+1}D_{x_{3}}^{b_{3}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{2}}^{b_{2}+1}D_{x_{4}}^{b_{4}+1}

· {C_{i_{2}−1−b_{2}}^{i_{2}−2−2b_{2}}C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{1}}^{i_{1}}D_{x_{2}}^{i_{2}−2−2b_{2}}D_{x_{3}}^{i_{3}}D_{x_{4}}^{i_{4}−2−2b_{4}}u · D_{x_{2}}^{b_{2}+1}D_{x_{4}}^{b_{4}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{3}}^{b_{3}+1}D_{x_{4}}^{b_{4}+1}

· {C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{1}}^{i_{1}}D_{x_{2}}^{i_{2}}D_{x_{3}}^{i_{3}−2−2b_{3}}D_{x_{4}}^{i_{4}−2−2b_{4}}u · D_{x_{3}}^{b_{3}+1}D_{x_{4}}^{b_{4}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R)}

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=0}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{2}}^{b_{2}+1}D_{x_{3}}^{b_{3}+1}

· {C_{i_{1}−1−b_{1}}^{i_{1}−2−2b_{1}}C_{i_{2}−1−b_{2}}^{i_{2}−2−2b_{2}}C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}D_{x_{1}}^{i_{1}−2−2b_{1}}D_{x_{2}}^{i_{2}−2−2b_{2}}D_{x_{3}}^{i_{3}−2−2b_{3}}D_{x_{4}}^{i_{4}}u

· ∏_{k=1}^{3} D_{x_{k}}^{b_{k}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R) }

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=0}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{2}}^{b_{2}+1}D_{x_{4}}^{b_{4}+1}

· {C_{i_{1}−1−b_{1}}^{i_{1}−2−2b_{1}}C_{i_{2}−1−b_{2}}^{i_{2}−2−2b_{2}}C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{1}}^{i_{1}−2−2b_{1}}D_{x_{2}}^{i_{2}−2−2b_{2}}D_{x_{3}}^{i_{3}}D_{x_{4}}^{i_{4}−2−2b_{4}}u

· ∏_{k=1,k≠3}^{4} D_{x_{k}}^{b_{k}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R) }

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=0}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{3}}^{b_{3}+1}D_{x_{4}}^{b_{4}+1}

· {C_{i_{1}−1−b_{1}}^{i_{1}−2−2b_{1}}C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{1}}^{i_{1}−2−2b_{1}}D_{x_{2}}^{i_{2}}D_{x_{3}}^{i_{3}−2−2b_{3}}D_{x_{4}}^{i_{4}−2−2b_{4}}u

· ∏_{k=1,k≠2}^{4} D_{x_{k}}^{b_{k}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R) }

+ ∑_{i_{1}=0}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{2}}^{b_{2}+1}D_{x_{3}}^{b_{3}+1}D_{x_{4}}^{b_{4}+1}

· {C_{i_{2}−1−b_{2}}^{i_{2}−2−2b_{2}}C_{i_{3}−1−b_{3}}^{i_{3}−2−2b_{3}}C_{i_{4}−1−b_{4}}^{i_{4}−2−2b_{4}}D_{x_{1}}^{i_{1}}D_{x_{2}}^{i_{2}−2−2b_{2}}D_{x_{3}}^{i_{3}−2−2b_{3}}D_{x_{4}}^{i_{4}−2−2b_{4}}u

· ∏_{k=2}^{4} D_{x_{k}}^{b_{k}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R) }

+ ∑_{i_{1}=2}^{m_{1}} ∑_{i_{2}=2}^{m_{2}} ∑_{i_{3}=2}^{m_{3}} ∑_{i_{4}=2}^{m_{4}} ∑_{b_{1}=0}^{\frac{i_{1}}{2} −1} ∑_{b_{2}=0}^{\frac{i_{2}}{2} −1} ∑_{b_{3}=0}^{\frac{i_{3}}{2} −1} ∑_{b_{4}=0}^{\frac{i_{4}}{2} −1} D_{x_{1}}^{b_{1}+1}D_{x_{2}}^{b_{2}+1}D_{x_{3}}^{b_{3}+1}D_{x_{4}}^{b_{4}+1}

· { ∏_{k=1}^{4} C_{i_{k}−1−b_{k}}^{i_{k}−2−2b_{k}}D_{x_{k}}^{i_{k}−2−2b_{k}}u · ∏_{k=1}^{4} D_{x_{k}}^{b_{k}+1} (a_{i_{1}i_{2}i_{3}i_{4}}R) }.

To prove the above representation one can use formula (11). Add to and

subtract from each factor of the form ∑_{α_{1}=b_{1}}^{i_{1}} (−1)^{i_{1}−α_{1}} C_{α_{1}}^{b_{1}} the correspond-

ing addend M_{i_{1}b_{1}}. Then we have

K_{i_{1}i_{2}i_{3}i_{4}b_{1}b_{2}b_{3}b_{4}} = ∑_{α_{1}=b_{1}}^{i_{1}} (−1)^{i_{1}−α_{1}} C_{α_{1}}^{b_{1}} ∑_{α_{2}=b_{2}}^{i_{2}} (−1)^{i_{2}−α_{2}} C_{α_{2}}^{b_{2}}

∑_{α_{3}=b_{3}}^{i_{3}} (−1)^{i_{3}−α_{3}} C_{α_{3}}^{b_{3}} ∑_{α_{4}=β_{4}}^{i_{4}} (−1)^{i_{4}−α_{4}} C_{α_{4}}^{b_{4}} − M_{b_{1}b_{2}b_{3}b_{4}} =

= [ ∑_{α_{1}=b_{1}}^{i_{1}} (−1)^{i_{1}−α_{1}} C_{α_{1}}^{b_{1}} ± M_{i_{1}b_{1}}] · [ ∑_{α_{2}=b_{2}}^{i_{2}} (−1)^{i_{2}−α_{2}} C_{α_{2}}^{b_{2}} ± M_{i_{2}b_{2}}] ·

· [ ∑_{α_{3}=b_{3}}^{i_{3}} (−1)^{i_{3}−α_{3}} C_{α_{3}}^{b_{3}} ± M_{i_{3}b_{3}}] · [ ∑_{α_{4}=b_{4}}^{i_{4}} (−1)^{i_{4}−α_{4}} C_{α_{4}}^{b_{4}} ± M_{i_{4}b_{4}}]

− M_{b_{1}b_{2}b_{3}b_{4}}.

Remove the parenthesis and apply formula (11).

Moreover, from (17) one can see the next formulas:

∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{∑_{k=1}^{4}^{(m_{k}−α_{k})}} D_{x_{1}}^{α_{1}−i_{1}}

∏_{k=2}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}}R) (x_{1}, ξ_{2}, ξ_{3}, ξ_{4}) ≡ 0.

Let us integrate (19) with regard for (20) and the above identities. To this end in (19) we set x_{1} = ξ_{1}, x_{2} = ξ_{2}, x_{3} = ξ_{3},x_{4} = ξ_{4} and calculate fourfold integrals of the left- and right-hand sides of the identity within the limits x_{10} < ξ_{1} < x_{1}, x_{20} < ξ_{2} < x_{2}, x_{30} < ξ_{3} < x_{3}, x_{40} < ξ_{4} < x_{4}:

∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} (u) (x_{1}, x_{2}, x_{3}, x_{4})

= ∏_{k=1}^{4} D_{x_{k0}}^{−1} (RL (u)) + ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{2}, x_{3}, x_{4})  (21)

+ ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{20}, x_{3}, x_{4}) + ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{2}, x_{30}, x_{4})

164 YE.A. UTKINA

+ ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{2}, x_{3}, x_{40}) − ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{20}, x_{3}, x_{4})

− ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{2}, x_{30}, x_{4}) − ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{2}, x_{3}, x_{40})

− ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{20}, x_{30}, x_{4}) − ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{20}, x_{3}, x_{40})

− ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{2}, x_{30}, x_{40}) + ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{20}, x_{30}, x_{4})

+ ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{20}, x_{3}, x_{40}) + ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{2}, x_{30}, x_{40})

+ ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{1}, x_{20}, x_{30}, x_{40}) − ∏_{k=1}^{4} D_{x_{k}}^{m_{k}−1} [u] · R (x_{10}, x_{20}, x_{30}, x_{40})

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{2}, x_{3}, x_{4}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, x_{3}, x_{4})

·R (x_{10}, x_{2}, x_{3}, x_{4}))

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{20}, x_{3}, x_{4}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, x_{3}, x_{4})

·R (x_{1}, x_{2}, x_{3}, x_{4})) −

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{2}, x_{30}, x_{4})) ·

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{2}, x_{30}, x_{4})

·R (x_{1}, x_{2}, x_{30}, x_{4}))

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{2}, x_{3}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{2}, x_{3}, x_{40})

·R (x_{1}, x_{2}, x_{3}, x_{40}))

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{20}, x_{3}, x_{4}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, x_{3}, x_{4})

·R (x_{10}, x_{20}, x_{3}, x_{4}))

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{2}, x_{30}, x_{4}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, x_{30}, x_{4})

·R (x_{10}, x_{2}, x_{30}, x_{4}))

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{2}, x_{3}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, x_{3}, x_{40})

·R (x_{10}, x_{2}, x_{3}, x_{40}))

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{20}, x_{30}, x_{4}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, x_{30}, x_{4})

·R (x_{1}, x_{20}, x_{30}, x_{4}))

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{20}, x_{3}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, x_{3}, x_{40})

·R (x_{1}, x_{20}, x_{3}, x_{40}))

166 YE.A. UTKINA

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{2}, x_{30}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{2}, x_{30}, x_{40})

·R (x_{1}, x_{2}, x_{30}, x_{40}))

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{20}, x_{30}, x_{4}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, x_{30}, x_{4})

·R (x_{10}, x_{20}, x_{30}, x_{4}))

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{20}, x_{3}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, x_{3}, x_{40})

·R (x_{10}, x_{20}, x_{3}, x_{40}))

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{2}, x_{30}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, x_{30}, x_{40})

·R (x_{10}, x_{2}, x_{30}, x_{40}))

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{1}, x_{20}, x_{30}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, x_{30}, x_{40})

·R (x_{1}, x_{20}, x_{30}, x_{40}))

+ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{3}+i_{4}−1} ∏_{k=1}^{4} D_{x_{k}}^{i_{k}−1} (u (x_{10}, x_{20}, x_{30}, x_{40}))

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} ∏_{k=1}^{4} D_{x_{k}}^{α_{k}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, x_{30}, x_{40})

·R (x_{10}, x_{20}, x_{30}, x_{40}))

− ∫_{x_{10}}^{x_{1}} {∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{2}+i_{3}+i_{4}−1} { D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{20}, x_{3}, x_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, x_{3}, x_{4}) ·R (ξ_{1}, x_{20}, x_{3}, x_{4}))

+ D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{2}, x_{30}, x_{4})

∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{2}, x_{30}, x_{4}) ·R (ξ_{1}, x_{2}, x_{30}, x_{4}))

+ D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{2}, x_{3}, x_{40})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{2}, x_{3}, x_{40})

· R (ξ_{1}, x_{2}, x_{3}, x_{40})) − (−1)^{i_{2}+i_{3}+i_{4}−1} D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{20}, x_{30}, x_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, x_{30}, x_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{2}, x_{30}, x_{40}) ·R (ξ_{1}, x_{2}, x_{30}, x_{40}))

+ D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{20}, x_{30}, x_{40})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}} D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}}

(a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, x_{30}, x_{40}) · R (ξ_{1}, x_{20}, x_{30}, x_{40}))} dξ_{1}} −

− ∫_{x_{20}}^{x_{2}} {∑_{i_{1}=1}^{m_{1}} ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{3}+i_{4}−1} { D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, ξ_{2}, x_{3}, x_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, x_{3}, x_{4})

·R (x_{10}, ξ_{2}, x_{3}, x_{4}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, ξ_{2}, x_{30}, x_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, ξ_{2}, x_{30}, x_{4})·R (x_{1}, ξ_{2}, x_{30}, x_{4})) + D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, ξ_{2}, x_{3}, x_{40})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, ξ_{2}, x_{3}, x_{40}) · R (x_{1}, ξ_{2}, x_{3}, x_{40}))

− D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, ξ_{2}, x_{30}, x_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, x_{30}, x_{4}) · R (x_{10}, ξ_{2}, x_{30}, x_{4}))

− D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, ξ_{2}, x_{3}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, x_{3}, x_{40}) · R (x_{10}, ξ_{2}, x_{3}, x_{40}))

− D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, ξ_{2}, x_{30}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, ξ_{2}, x_{30}, x_{40})·R (x_{1}, ξ_{2}, x_{30}, x_{40}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, ξ_{2}, x_{30}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, x_{30}, x_{40})·R (x_{10}, ξ_{2}, x_{30}, x_{40}))} dξ_{2}}

− ∫_{x_{30}}^{x_{3}} {∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{2}+i_{4}−1} { D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, x_{2}, ξ_{3}, x_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, ξ_{3}, x_{4})·R (x_{10}, x_{2}, ξ_{3}, x_{4}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, x_{20}, ξ_{3}, x_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, ξ_{3}, x_{4}) · ·R (x_{1}, x_{20}, ξ_{3}, x_{4})) + D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, x_{2}, ξ_{3}, x_{40}) ·

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{2}, ξ_{3}, x_{40}) · ·R (x_{1}, x_{2}, ξ_{3}, x_{40}))

− D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, x_{20}, ξ_{3}, x_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, ξ_{3}, x_{4}) ·R (x_{10}, x_{20}, ξ_{3}, x_{4}))

− D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, x_{2}, ξ_{3}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, ξ_{3}, x_{40}) ·R (x_{10}, x_{2}, ξ_{3}, x_{40}))

− D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, x_{20}, ξ_{3}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, ξ_{3}, x_{40}) · R (x_{1}, x_{20}, ξ_{3}, x_{40}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}} − u (x_{10}, x_{20}, ξ_{3}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, ξ_{3}, x_{40})·R (x_{10}, x_{20}, ξ_{3}, x_{40}))} dξ_{3}}

− ∫_{x_{40}}^{x_{4}} {∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} (−1)^{i_{1}+i_{2}+i_{3}−1} { D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{10}, x_{2}, x_{3}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, x_{3}, ξ_{4}) ·R (x_{10}, x_{2}, x_{3}, ξ_{4}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{1}, x_{20}, x_{3}, ξ_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, x_{3}, ξ_{4})·R (x_{1}, x_{20}, x_{3}, ξ_{4}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{1}, x_{2}, x_{30}, ξ_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{2}, x_{30}, ξ_{4})·R (x_{1}, x_{2}, x_{30}, ξ_{4}))

− D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{10}, x_{20}, x_{3}, ξ_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, x_{3}, ξ_{4}) · R (x_{10}, x_{20}, x_{3}, ξ_{4}))

− D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{10}, x_{2}, x_{30}, ξ_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, x_{30}, ξ_{4}) · R (x_{10}, x_{2}, x_{30}, ξ_{4}))

− D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{1}, x_{20}, x_{30}, ξ_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, x_{30}, ξ_{4}) · ·R (x_{1}, x_{20}, x_{30}, ξ_{4}))

+D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (x_{10}, x_{20}, x_{30}, ξ_{4}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, x_{30}, ξ_{4}) · ·R (x_{10}, x_{20}, x_{30}, ξ_{4}))} dξ_{4}}

− ∫_{x_{10}}^{x_{1}} ∫_{x_{20}}^{x_{2}} { ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{3}+i_{4}−1} { D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, ξ_{2}, x_{30}, x_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, ξ_{2}, x_{30}, x_{4}) · ·R (ξ_{1}, ξ_{2}, x_{30}, x_{4}))

+ D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, ξ_{2}, x_{3}, x_{40}) · ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, ξ_{2}, x_{3}, x_{40})·R (ξ_{1}, ξ_{2}, x_{3}, x_{40}))

− D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, ξ_{2}, x_{30}, x_{40}) · ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, ξ_{2}, x_{30}, x_{40})·R (ξ_{1}, ξ_{2}, x_{30}, x_{40}))} dξ_{2}dξ_{1}}

− ∫_{x_{10}}^{x_{1}} ∫_{x_{30}}^{x_{3}} { ∑_{i_{2}=1}^{m_{2}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{2}+i_{4}−1} { D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{20}, ξ_{3}, x_{4}) ·

∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, ξ_{3}, x_{4})·R (ξ_{1}, x_{20}, ξ_{3}, x_{4})) + D_{x_{2}}^{i_{2}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, x_{2}, ξ_{3}, x_{40})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{2}, ξ_{3}, x_{40}) · ·R (ξ_{1}, x_{2}, ξ_{3}, x_{40}))

− ∫_{x_{10}}^{x_{1}} ∫_{x_{20}}^{x_{2}} { ∑_{i_{3}=1}^{m_{3}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{3}+i_{4}−1} { D_{x_{3}}^{i_{3}−1}D_{x_{4}}^{i_{4}−1}u (ξ_{1}, ξ_{2}, x_{30}, x_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, ξ_{2}, x_{30}, x_{4})

− ∫_{x_{10}}^{x_{1}} ∫_{x_{40}}^{x_{4}} { ∑_{i_{2}=1}^{m_{2}} ∑_{i_{3}=1}^{m_{3}} (−1)^{i_{2}+i_{3}−1} { D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (ξ_{1}, x_{20}, x_{3}, ξ_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, x_{3}, ξ_{4}) · ·R (ξ_{1}, x_{20}, x_{3}, ξ_{4}))

+ D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (ξ_{1}, x_{2}, x_{30}, ξ_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{2}, x_{30}, ξ_{4}) ·

·R (ξ_{1}, x_{2}, x_{30}, ξ_{4})) − D_{x_{2}}^{i_{2}−1}D_{x_{3}}^{i_{3}−1}u (ξ_{1}, x_{20}, x_{30}, ξ_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, x_{30}, ξ_{4})

·R (ξ_{1}, x_{20}, x_{30}, ξ_{4}))} dξ_{4}dξ_{1}}

− ∫_{x_{20}}^{x_{2}} ∫_{x_{30}}^{x_{3}} { ∑_{i_{1}=1}^{m_{1}} ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{1}+i_{4}−1} { D_{x_{1}}^{i_{1}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, ξ_{2}, ξ_{3}, x_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, ξ_{3}, x_{4}) ·R (x_{10}, ξ_{2}, ξ_{3}, x_{4}))

+ D_{x_{1}}^{i_{1}−1}D_{x_{4}}^{i_{4}−1}u (x_{1}, ξ_{2}, ξ_{3}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, ξ_{2}, ξ_{3}, x_{40}) ·R (x_{1}, ξ_{2}, ξ_{3}, x_{40}))

− D_{x_{1}}^{i_{1}−1}D_{x_{4}}^{i_{4}−1}u (x_{10}, ξ_{2}, ξ_{3}, x_{40}) · ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

172 YE.A. UTKINA D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, ξ_{3}, x_{40})

·R (x_{10}, ξ_{2}, ξ_{3}, x_{40}))} dξ_{3}dξ_{2}}

− ∫_{x_{20}}^{x_{2}} ∫_{x_{40}}^{x_{4}} { ∑_{i_{1}=1}^{m_{1}} ∑_{i_{3}=1}^{m_{3}} (−1)^{i_{1}+i_{3}−1} { D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}u (x_{10}, ξ_{2}, x_{3}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, x_{3}, ξ_{4})·R (x_{10}, ξ_{2}, x_{3}, ξ_{4})) + D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}u (x_{1}, ξ_{2}, x_{30}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, ξ_{2}, x_{30}, ξ_{4})·R (x_{1}, ξ_{2}, x_{30}, ξ_{4})) − D_{x_{1}}^{i_{1}−1}D_{x_{3}}^{i_{3}−1}u (x_{10}, ξ_{2}, x_{30}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, x_{30}, ξ_{4})·R (x_{10}, ξ_{2}, x_{30}, ξ_{4}))} dξ_{4}dξ_{2}}

− ∫_{x_{30}}^{x_{3}} ∫_{x_{40}}^{x_{4}} { ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} (−1)^{i_{1}+i_{2}−1} { D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}u (x_{10}, x_{2}, ξ_{3}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{2}, ξ_{3}, ξ_{4})·R (x_{10}, x_{2}, ξ_{3}, ξ_{4})) + D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}u (x_{1}, x_{20}, ξ_{3}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{1}, x_{20}, ξ_{3}, ξ_{4})·R (x_{1}, x_{20}, ξ_{3}, ξ_{4})) − D_{x_{1}}^{i_{1}−1}D_{x_{2}}^{i_{2}−1}u (x_{10}, x_{20}, ξ_{3}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, x_{20}, ξ_{3}, ξ_{4})·R (x_{10}, x_{20}, ξ_{3}, ξ_{4}))} dξ_{4}dξ_{3}}

− ∫_{x_{10}}^{x_{1}} ∫_{x_{20}}^{x_{2}} ∫_{x_{30}}^{x_{3}} { ∑_{i_{4}=1}^{m_{4}} (−1)^{i_{4}−1} { D_{x_{4}}^{i_{4}−1}u (ξ_{1}, ξ_{2}, ξ_{3}, x_{40})

PARTIAL DIFFERENTIAL EQUATION 173

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=i_{4}}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}−i_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, ξ_{2}, ξ_{3}, x_{40})·R (ξ_{1}, ξ_{2}, ξ_{3}, x_{40}))} dξ_{3}dξ_{2}dξ_{1} }

− ∫_{x_{10}}^{x_{1}} ∫_{x_{20}}^{x_{2}} ∫_{x_{40}}^{x_{4}} { ∑_{i_{3}=1}^{m_{3}} (−1)^{i_{3}−1} { D_{x_{3}}^{i_{3}−1}u (ξ_{1}, ξ_{2}, x_{30}, ξ_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=i_{3}}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}−i_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, ξ_{2}, x_{30}, ξ_{4}) ·R (ξ_{1}, ξ_{2}, x_{30}, ξ_{4}))} dξ_{4}dξ_{2}dξ_{1} }

− ∫_{x_{10}}^{x_{1}} ∫_{x_{30}}^{x_{3}} ∫_{x_{40}}^{x_{4}} { ∑_{i_{2}=1}^{m_{2}} (−1)^{i_{2}−1} { D_{x_{2}}^{i_{2}−1}u (ξ_{1}, x_{20}, ξ_{3}, ξ_{4})

· ∑_{α_{1}=0}^{m_{1}} ∑_{α_{2}=i_{2}}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}}D_{x_{2}}^{α_{2}−i_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (ξ_{1}, x_{20}, ξ_{3}, ξ_{4}) ·R (ξ_{1}, x_{20}, ξ_{3}, ξ_{4}))} dξ_{4}dξ_{3}dξ_{1} }

− ∫_{x_{20}}^{x_{2}} ∫_{x_{30}}^{x_{3}} ∫_{x_{40}}^{x_{4}} { ∑_{i_{1}=1}^{m_{1}} (−1)^{i_{1}−1} { D_{x_{1}}^{i_{1}−1}u (x_{10}, ξ_{2}, ξ_{3}, ξ_{4})

· ∑_{α_{1}=i_{1}}^{m_{1}} ∑_{α_{2}=0}^{m_{2}} ∑_{α_{3}=0}^{m_{3}} ∑_{α_{4}=0}^{m_{4}} (−1)^{α_{1}+α_{2}+α_{3}+α_{4}}

D_{x_{1}}^{α_{1}−i_{1}}D_{x_{2}}^{α_{2}}D_{x_{3}}^{α_{3}}D_{x_{4}}^{α_{4}} (a_{α_{1}α_{2}α_{3}α_{4}} (x_{10}, ξ_{2}, ξ_{3}, ξ_{4}) ·R (x_{10}, ξ_{2}, ξ_{3}, ξ_{4}))} dξ_{4}dξ_{3}dξ_{2} }

Now it remains to integrate (21) with respect to x_{1},x_{2}, x_{3}, and x_{4} m_{1} − 1, m_{2} − 1, m_{3} − 1, and m_{4} − 1 times, respectively. Denote by H (x_{1}, x_{2}, x_{3}, x_{4}, x_{1}, x_{2}, x_{3}, x_{4}) the right-hand side of (21) without the first addend. Then

u (x_{1}, x_{2}, x_{3}, x_{4}) = ∏_{k=1}^{4} D_{x_{k}}^{−m_{k}} [RF] + ∏_{k=1}^{n} D_{x_{k}}^{−m_{k}+1} [H] + F_{1},  (22)

where

\begin{matrix}F_{1} = ∑_{i_{1}=0}^{m_{1}−2} D_{x_{1}}^{−i_{1}} (φ_{i_{1}} (x_{2}, x_{3}, x_{4})) + ∑_{i_{2}=0}^{m_{2}−2} D_{x_{2}}^{−i_{2}} (φ_{i_{2}} (x_{1}, x_{3}, x_{4})) \\ + ∑_{i_{3}=0}^{m_{3}−2} D_{x_{3}}^{−i_{3}} (φ_{i_{3}} (x_{1}, x_{2}, x_{4})) + ∑_{i_{4}=0}^{m_{4}−2} D_{x_{4}}^{−i_{4}} (φ_{i_{4}} (x_{1}, x_{2}, x_{3})) \end{matrix}

174 YE.A. UTKINA

− ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{2}=0}^{m_{2}−2} D_{x_{1}}^{−i_{1}}D_{x_{2}}^{−i_{2}} (D_{x_{2}}^{i_{2}}φ_{i_{1}} (x_{20}, x_{3}, x_{4}))

− ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{3}=0}^{m_{3}−2} D_{x_{1}}^{−i_{1}}D_{x_{3}}^{−i_{3}} (D_{x_{3}}^{i_{3}}φ_{i_{1}} (x_{2}, x_{30}, x_{4}))

− ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{1}}^{−i_{1}}D_{x_{4}}^{−i_{4}} (D_{x_{4}}^{i_{4}}φ_{i_{1}} (x_{2}, x_{3}, x_{40}))

− ∑_{i_{2}=0}^{m_{2}−2} ∑_{i_{3}=0}^{m_{3}−2} D_{x_{2}}^{−i_{2}}D_{x_{3}}^{−i_{3}} (D_{x_{3}}^{i_{3}}φ_{i_{2}} (x_{1}, x_{30}, x_{4}))

− ∑_{i_{2}=0}^{m_{2}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{2}}^{−i_{2}}D_{x_{4}}^{−i_{4}} (D_{x_{4}}^{i_{4}}φ_{i_{2}} (x_{1}, x_{3}, x_{40})) −

− ∑_{i_{3}=0}^{m_{3}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{3}}^{−i_{3}}D_{x_{4}}^{−i_{4}} (D_{x_{4}}^{i_{4}}φ_{i_{3}} (x_{1}, x_{2}, x_{40}))

+ ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{2}=0}^{m_{2}−2} ∑_{i_{3}=0}^{m_{3}−2} D_{x_{1}}^{−i_{1}}D_{x_{2}}^{−i_{2}}D_{x_{3}}^{−i_{3}} (D_{x_{2}}^{i_{2}}D_{x_{3}}^{i_{3}}φ_{i_{1}} (x_{20}, x_{30}, x_{4}))

+ ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{2}=0}^{m_{2}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{1}}^{−i_{1}}D_{x_{2}}^{−i_{2}}D_{x_{4}}^{−i_{4}} (D_{x_{2}}^{i_{2}}D_{x_{4}}^{i_{4}}φ_{i_{1}} (x_{20}, x_{3}, x_{40}))

+ ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{3}=0}^{m_{3}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{1}}^{−i_{1}}D_{x_{3}}^{−i_{3}}D_{x_{4}}^{−i_{4}} (D_{x_{3}}^{i_{3}}D_{x_{4}}^{i_{4}}φ_{i_{1}} (x_{2}, x_{30}, x_{40}))

+ ∑_{i_{2}=0}^{m_{2}−2} ∑_{i_{3}=0}^{m_{3}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{2}}^{−i_{2}}D_{x_{3}}^{−i_{3}}D_{x_{4}}^{−i_{4}} (D_{x_{3}}^{i_{3}}D_{x_{4}}^{i_{4}}φ_{i_{2}} (x_{1}, x_{30}, x_{40}))

− ∑_{i_{1}=0}^{m_{1}−2} ∑_{i_{2}=0}^{m_{2}−2} ∑_{i_{3}=0}^{m_{3}−2} ∑_{i_{4}=0}^{m_{4}−2} D_{x_{1}}^{−i_{1}}D_{x_{2}}^{−i_{2}}D_{x_{3}}^{−i_{3}}D_{x_{4}}^{−i_{4}} (D_{x_{2}}^{i_{2}}D_{x_{3}}^{i_{3}}D_{x_{4}}^{i_{4}}φ_{i_{1}} (x_{20}, x_{30}, x_{40})) .

If one assumes that the functions φ_{pi_{p}} (i_{p} = \overline{0, m_{p}} , p = 1, 2, 3, 4) are arbitrary, then (22) can be treated as the general representation of the solution of equation (1), as this was done in [6], p.66.

References

[1] Soldatov A.P. and Schkhanukov M.H., Boundary value problems with the general non-local A.A.Samarskii’s condition for higher order pseudo parabolic equations, DAN USSR, v.297, 1987, pp.547-552.

[2] Nakhushev A.M., Equations of Mathematical Biology, Moscow: High school publ.comp., 1995.

PARTIAL DIFFERENTIAL EQUATION 175

[3] Pontryagin L.S., Ordinary Differential Equations. Moscow: Nauka publ.comp., 1961.

[4] Zhegalov V.I. and Utkina E.A., Goursat problem for a three-dimensional equation with the main derivation, Izvestiya Vuzov, Math., no. 11, 2001, P.77-81.

[5] Zhegalov V.I. and Utkina E.A., On a fourth order partial differential equation with three independent variables // Differential equations, 2002, v.38, p.93-97.

[6] Bitsadze A.V., Some Classes of Partial Differential Equations. Moscow: Nauka publ. comp.,1981.

[7] El’sgol’ts L.E., Differential Equations and Calculus of Variations. Moscow: Science publ. comp.,1969.

[8] Muentz H. Integral Equations, Vol. I. Moscow: GTTI publ. comp.,1934.

Kazan State Pedagogical University, Kazan, RUSSIA E-mail address: eutkina1@yandex.ru

Received December 23, 2004; revised version 2005, April 27