On the nonlinear inverse problem for a partial difference equation of the higher order Текст научной статьи по специальности «Математика»

Решетнеескцие чтения. 2015

UDC 517.95

ON THE NONLINEAR INVERSE PROBLEM FOR A PARTIAL DIFFERENCE EQUATION

OF THE HIGHER ORDER

T. K. Yuldashev

Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: tursunbay@rambler.ru

The paper proposes a method to study the inverse problem for a nonlinear partial difference equation with difference hyperbolic operator of the arbitrary natural power. In solving the inverse problem with respect to the restore function the nonlinear summary equation of the first kind is obtained, the equation is reduced by the aid of special nonclassical summary transform into nonlinear summary equation of the second kind. Further the method of successive approximations combined it with the method of compressing maps is used.

Keywords: inverse problem, difference equation, nonlinear right-hand side, one-value solvability, the method of successive approximation.

О НЕЛИНЕЙНОЙ ОБРАТНОЙ ЗАДАЧЕ ДЛЯ УРАВНЕНИЯ В ЧАСТНЫХ РАЗНОСТЯХ ВЫСШЕГО ПОРЯДКА

Т. К. Юлдашев

Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

Е-mail: tursunbay@rambler.ru

Предложен метод изучения однозначной разрешимости обратной задачи для нелинейного разностного уравнения с гиперболическим разностным оператором произвольной натуральной степени. При решении обратной задачи относительно функции восстановления получено нелинейное суммарное уравнение первого рода, которое с помощью неклассического суммарного преобразования сведено к нелинейному суммарному уравнению второго рода. Далее применен метод последовательных приближений в сочетании с методом сжимающих отображений.

Ключевые слова: обратная задача, разностное уравнение, нелинейная правая часть, однозначная разрешимость, метод последовательных приближений.

Mathematical modeling of many processes occurring in On the set D we consider a nonlinear difference

the real world leads to the study of inverse problems for equation of the following form equations of mathematical physics. Inverse problems are rapidly developing area of modern theory of differential

equations. Intensive study of inverse problems is largely with end-point and additional conditions

due to the need to develop mathematical methods for u (n,m)|n=N = << (m),

solving a broad class of important applied problems related . _ (2)

to the processing and interpretation of observations. д nu (n, m) | n=N = < i+1

(m), i =1,2k-1,

We note, that difference equations as analog of differ- u (n m ) = ^ (n) (3)

ential equations are important in numerical solving the 0

differential equations (see, for examples [1-5]). where u (n, m) is unknown functwn, f (n, m, u, 3 ),

In solving the inverse problem in this paper we даИ i = 1,2k are defined for all n > n0, m0 > 0, use the method of characteristics, because with the aid 0 0

of this method we can change the considering problem D = DN x Z, DN n 0 < n < N} , Z is the set of

to equivalent nonlinear summary equation. -

Inverse problem is called nonlinear if the restore func- integer numbers the given fonctions <г (m), i =1,2k

tion is included in this equation is nonlinear [6]. With are defined for all integer numbers, k is arbitrary natural

respect to the restore function it is obtained the nonlinear

r . . . . number, ш (n) and restore function 3 (n) are defined

summary equation of the first kind, which is reduced by

the aid of special nonclassical summary transform into for 311 n > n 0,

n°nlmear smmmrny equation °f the second kind. Then we д 2 u (n, m) = Д nu (n +1, m)-Д nu (n, m) = use the method of successive approximations in combination it with the method of compressing mapping. = u (n +1 m) - 2 u (n, m) + u (n -1, m),

(д 2 — Д m ) u (n, m) = f (n , m , u (n, m) , 3 (n)) (1)

Прикладная математика

A m u (n, m) = A m u (n, m +1) -A m u (n, m) = = u (n, m +1) - 2 u (n, m) + u (n, m -1) , A nu (n, m) = u (n +1, m) - u (n, m) , A mu (n, m) = u (n, m +1) - u (n, m). Definition. As a solution of the inverse problem

(1)-(3) we call a pair of functions { u (n, m), S (n)},

satisfying the given summary equation (1) and conditions

(2), (3).

Left-hand side of equation (1) we rewrite as

(a n-A m )u(n,m) =

= (A n-A m ) (A n +A m ) u (n , m) =

= L

k [ l2 [« ]].

where

i=1

( i -1)!

n-1 n-1 n-1

+(-1)k S S ••• S f >

v 1=и v 2 =v 1 v k =v k-1

x(v k, да, m (v k , m), S(v k )),

(5)

i=1

k N-1 N-1 N-1 (-1) J +k+1(N -v k - 1) J-1

+S S S ••• S

J =1 v1 =n v2 =v1 vk =vk-1

(i -1)!

k

( J -1)!

(m - N + v k +1) +

Pk+j

n-1 n-1

+(-1)2k S S

v1=n v 2=v1

... s f (v 2k , m , u (v 2k , m), S (v 2k)). (6)

v 2 k =v2 k-1

Lemma. Assume that the following conditions will be fulfilled:

2 k ( \ T - n - 1) i-1

1. L, (m) I <Mt , 0 <SMi - , 1— <80 <œ;

i=1

(i -1)!

2. f (n, m, u, S) e Bnd (M 0) n Lip {L|u }, 0 < L = const ;

(n-n -1)2k

3. 0 ^^-—— M0 <8j <œ; p<1,

(2 k )!

p = L

( N - n -1) (2 k )!

2 k

Then the initial value problem (1), (2) for any restore function S (n) has a unique solution in the domain D.

We use the additional condition in (6). Then we have the following nonlinear summary equation of first kind

n-1 n-1

(-1)2k S S •••

v 1 =n v 2 =v 1

n-1

••• S f (v 2 k > m 0> V (v 2k S (v 2k ) )= (n (7)

Zi [ Lk2 [u (A n-A m) Lk2 [u ],

L 2 [ U ] = (A n +A m) U .

Then the equation (1) takes the form

Li [ L2 [u ]] = / (n,m, u (n,m) ,S(n)). (4)

From (4) we derive, that the difference equation (1) has two k multiple characteristics:

1) m + N - n -1 = C1;

2) m - N + n +1 = C2, where C1, C2 are arbitrary

constants. Summing the equation (4) k times along the second characteristics, by virtue of the end-point conditions (2) we obtain

L2. [u ] = f ♦ k „ (m - N + n +1)'-1' M<N - n +1)1-1 +

'2k =v2 k-1

where

g (n) = v (n)-S<P i (m 0 + N - n -1)

(-1) '+1 (N - n -1)i-1

Similarly, summing the equation (5) k times along the first characteristics, by virtue of conditions (2) yields

k (-1) i+1 (n - n -1)i-1 u (n, m) = i (m + N - n -1) -—-—---— +

k n_1 N-1 N-1 (-1)J+k+1 (n - v k -1)j-1

Z Z ... Z -7771T1-^k+j x

j=1 V1 = n V2 =V1 vk =vk-1 (7 1) !

x( m 0 - N + v k +1).

After solving the equation (7), we prove that there exists a unique restore function S (n) on the set DN.

Substituting the solution of the equation (7) into (6) we obtain the desired one-valued function u (n, m).

Theorem. Let be fulfilled all conditions of lemma. Then there exists the unique pair of solutions of the inverse problem (1)-(3): { u (n,m), S(n)} .

References

1. Yuldashev T. K. On a nonlinear system of functional difference equations with nonlinear complicated mixed maxima // Advanced studies in contemporary mathematics. 2006. Vol. 13, no. 1, pp. 1-5.

2. Yuldashev T. K. On a first order quasilinear partial difference equation // Advanced studies in contemporary mathematics. 2013. Vol. 23, no. 4, pp. 677-680.

3. Yuldashev T. K. On a summery equation with weak nonlinear right-hand side // Advanced studies in contemporary mathematics. 2007. Vol. 15, no. 1, pp. 95-98.

4. Yuldashev T. K. On a solvability of nonlinear evolution summary equations with nonlinear deviation // Proc. o/ Jangjeon Math. Society. 2008. Vol. 11, no 1, pp. 83-88.

5. Yuldashev T. K. On a solvability of nonlinear summary equation of the third kind // Proc. O/Jangjeon Math. Society. 2013. Vol. 16, no. 1, pp. 151-155.

6. Yuldashev T. K. A double inverse problem for a partial Fredholm integro-differential equation of fourth order // Proc. o/ Jangjeon Math. Society. 2015. Vol. 18, no. 3, pp. 417-426.

© Yuldashev T. K., 2015

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