Solution of irregular systems of partial differential equations using skeleton decomposition of linear operators Текст научной статьи по специальности «Математика»

MSC 35G15, 35R25

DOI: 10.14529/mmp170205

SOLUTION OF IRREGULAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS USING SKELETON DECOMPOSITION OF LINEAR OPERATORS

D.N. Sidorov1'2, N.A. Sidorov3

1Melentiev Energy Systems Institute SB RAS, Irkutsk, Russian Federation

2

3

E-mail: dsidorov@isem.irk.ru, sidorovisu@gmail.com

The linear system of partial differential equations is considered. It is assumed that there is an irreversible linear operator in the main part of the system. The operator is assumed to enjoy the skeletal decomposition. The differential operators of such system are assumed to have sufficiently smooth coefficients. In the concrete situations the domains of such differential operators are linear manifolds of smooth enough functions with values in Banach space. Such functions are assumed to satisfy additional boundary conditions. The concept of a skeleton chain of linear operator is introduced. It is assumed that the operator generates a skeleton chain of the finite length. In this case, the problem of solution of a given system is reduced to a regular split system of equations. The system is resolved with respect to the highest differential expressions taking into account certain initial and boundary conditions. The proposed approach can be generalized and applied to the boundary value problems in the nonlinear case. Presented results develop the theory of degenerate differential equations summarized in the monographs MR 87a:58036, Zbl 1027.47001.

Keywords: ill-posed problems; Cauchy problems; irreversible operator; skeleton decomposition; skeleton chain; boundary value problems.

Introduction

Consider linear equation

BL( ¿)u = Li( m)u +'«

where linear bounded operator B acting from linear space E to E has no inverse operator. Differential operators

3n

d\ an Qko+^+kr,

+ > ak0 ...km {X)-

I dx) _ dtn + ^

4 ' br,-!-£*-, -I-----\-b

ko+kiH-----+km<n-1

dtko dxk1 ...xm

( д \ v-^ д ko+- + km

L1 ( 7Г" I _ > bko km (x)--, n1 < n.

M dx ) ^ k0-km\ ) dtkodxki xkm 1

ko+ki+-+km<ni dt dx1 ■■■xm

Here coefficients ako..,kmj. Q С Rm+1 ^ R1, bko..,km : Q С Rm+1 ^ R1 are sufficiently smooth and defined in Q, 0 G Q. ^te domains of definition of operators L, L1 consist of linear manifolds Ed of sufficiently smooth functions in Q with their values in E, which satisfy certain system of homogeneous boundary conditions. An abstract function f : Q С Rm+1 ^ E of argument x _ (t,x1, xm) is assumed to be given and the problem is to find the solution u : Q С Rm+1 ^ Ed.

The operator B is assumed to be independent of x. If the operator B has an inverse bounded operator then equation (1) is called regular and otherwise it is called irregular equation. If E = RN and det B = 0, then (1) is the system of linear partial differential equations (PDE) of Kovalevskaya type, and we have a well known regular problem of the PDE theory. The foundation of many branches of modern general theory of PDE systems was constructed by I.G. Petrovskii [1]. In regular case the initial conditions for (1) can be defined as follows

3%u

= Pi(xi,... ,xm), i = 0,1,... ,n - 1. (2)

t=0

dt

Here functions ^ are analytical functions in Q. If f is an analytic function of t, xi,... ,xm in Q, then the Cauchy problem (1), (2) is not only solvable but also well-posed in class of analytic functions.

The well-posedness of the Cauchy problem is a challenging issue even for linear PDE systems in spaces of non-analytic functions. They are usually solved in a class of functions satisfying certain estimates [1].

Irregular models enable study of systems behavior in critical situations [2]. At present, the basis of relevant theory is constructed for certain classes of equations. For example, the theory and numerical methods for differential-algebraic equations has been constructed.

The intensive studies of more complex theory of irregular PDE and abstract irregular differential operator equations are conducted but there are still a lot of unexplored problems.

If B is a normally solvable operator, x E R1 then the approach in the theory of equations of the form (1) can be based on the splitting of the Banach space into a direct

B

from the theory of semigroups with kernels [6]. These approaches are applied to various problems [7,8].

In this field analytical methods were proposed for constructing classical and generalized solutions of the Cauchy problem for ordinary operator-differential equations with x E R1 in Banach spaces with irreversible operator in the main part.

The theory of irregular operator-differential PDEs in Banach spaces in the multidimensional case with x E Rra, n > 2, is to be constructed. There are only initial results in this field published in preprints. Therefore, the construction of the general theory of

B

important for the state of the art mathematical models of compex systems [2,9,10].

It is to be outlined that classical initial Cauchy conditions (2) for equation (1) play

B

is characteristic and functions ^ can not be arbitrary selected in the initial conditions (2)! Then there appears a question of reasonable formulation and methods of solution of

B.

objective of the present work is to solve this problem. In Sec. 2 and Sec. 3 this problem is solved for irreversible operator B which enjoy skeleton decomposition B = A1A2, where A i E L(E ^ Ei), A2 E L(Ei ^ E), and Ei is a normed space.

The remainder of the paper is organized as follows. Sec. 1 presents an introduction concerning the skeleton chains of linear operators using results [4]. The concept of regular

B

nilpotent in case of singular skeleton chain. In Sec. 2 it is assumed that the noninvertible

operator B generates a skeleton chain of linear operators of finite length p and it is demonstrated that irregular equation (1) can be reduced to the recurrent sequence of p +1 equations. It is to be noted that each equation of this sequence is regular under the natural restrictions on differential operators L, Li and certain initial-boundary conditions.

Bp equation (1) can be reduced to a regular system from the (p + 1)-th equation.

The proposed approach can be employed for wide range of concrete problems (1) due

B.

The formulas connecting the solution of (1) with the solution of reduced regular system are derived. This result allows us in Sec. 1 to set new well-posed non-classic boundary conditions for (1) for which the equation enjoys a unique solution what is demonstrated in Sec. 2. For applications, it is important that this solution can be found by solving the sequence of regular problems proposed in this paper. Corresponding results and examples are given in Sec. 3 and Sec. 4.

1. Skeleton Chains of Linear Operator

Let B £ L(E — E), and B = AiA2, where A2 £ L(E — E1), Ai £ L(E1 — E), and Ei,E are linear normed spaces. The following definitions can be introduced.

A decomposition B = A1A2 is called a skeleton decomposition of the operator B. Introduce a linear operator B1 = A2A1. Obviously B1 £ L(E1 — E1). If the operator B1 has a bounded inverse or it is null operator acting from E1 to E1, then B generates a skeleton chain {B1} of length 1. Then the operator B1 can be called a skeleton-attached operator to the operator B. This chain is railed singular if B1 = 0 and regular if B1 = 0. If B1

B 1 = A3A4, where A4 £ L(E1 — E2), A3 £ L(E2 — E1), and E2 is new linear normed space. Obviously in this case A2A1 = A3A4 and an operator B2 = A4A3 £ L(E2 — E2) can be introduced. If it turns out that B2 has a bounded inverse or B2 = 0, then B has a skeleton chain {Bb B2} of length 2. The chain {Bb B2} is singular if B2 = 0 and regular

B2

Then chain length is greater than 2 and one should continue chain's construction.

Indeed, this process can be continued for a number of linear operators by introduction of the normed linear spaces Ei, i = 1,... ,p and by bounded operators construction A2i £ L(Ei-1 — Ei), A2i-1 £ L(Ei — Ei-1), which satisfy the following equalities

A2i A2i-1 = A2i+1A2i+2, i =1, 2,...,p - 1. (3)

Equation (3) defines a sequence of linear operators {Bb ..., Bp} as follows

Bi = A2iA2i-1, i = 1, 2,...,p. (4)

Obviously Bi £ L(Ei — Ei). Here the operator Bp either has a bounded inverse or Bp is a null operator acting from Ep to Ep. This process can be formalized as the following definition.

Definition 1. Let B = A1A2 and operators {Ai}2= 1 satisfy (3). Let operators {B1;..., Bp} be defined by (4), operators {B1;..., Bp-1} be noninvertible, and operator Bp have a bounded inverse or be a null operator acting from Ep to Ep. Then the operator

B generates a skeleton chain of linear operators {Bi,..., Bp} of length p. If Bp = 0 then the chain is regular, if Bp = 0 then the chain is called singular. Operators {B1;..., Bp}

B.

The most important linear operators generating skeleton chains of the finite lenghts are given below:

1. Let E = Rm, then a square matrix B : Rm — Rm with det B = 0 obviously has skeleton chain {B1;..., Bp} of decreasing dimentions. The final matrix Bp will be regular or null matrix, det Bi = 0, i = 1,... ,p — 1.

n

2. Let E be an infinite dimentional normed space, then a finite operator B = Y1 (',Yi)zi,

i=1

where {zi} E E, y E E* has a skeleton chain consisting of finite number of matrices {B1;..., Bp} of decreasing dimentions. Here B1 = ll(zi,Yj)llnj=1 is the first element of this chain, det Bi = 0, i = 1,... ,p — 1. Bp is a null matrix or det Bp = 0.

Here according to Definition 1 the length of the chain p = 1 if det[(zi,7j-)]nj=1 = 0 or (zi, Yj) = 0, i, j = 1, 2,... ,n. In general case the chain always consists of finite number of matrices.

Using (3), (4) and Definition 1 the following result can be formulated.

Bp

Bn = A1A3 . . . A2n-1Bn-1A2n-2A2n-4 ... A2, n = 1, . . . ,p +1, (5)

where B1; B2,... Bp are elements of the skeleton chain of the operator B. From Lemma 1 it follows

B p, B

p + 1.

To proof the Corollary it is sufficient to put n = p + 1 and demonstrate that Bp+1 is Bp

singular skeleton chain.

2. Reduction of Abstract Irregular Equation to the Sequence of Regular Equations

The operator B and linear operators {Ai}2;= 1 from a skeleton chain of the operator B are assumed to be independent of x and commutative with linear operators L and L1. In this paragraph for sake of clarity it is assumed that operators L and L1 can be different from the introduced above differential operators L(d), L1( d) and equation can be considered in abstract form

BLu = L1U + f. (6)

Equation (1) can be considered as a special case of equation (6). Obviously, the introduced

Bx

introduced differential operators L( dx), L1(dx).

p + 1

conditions imposed on operators L, Li. Let us start with the simple case when p = 1. Introduce a system of two equations

BiLui = Liui + Af, (7)

Liu = —f + AiLui. (8)

where u G E and ui G Ei. The decomposed system (7) - (8) can be obtained by formal multiplication of (6) by the operator A2 from the skeleton decomposition of the operator B and making notation ui = A2u.

B1

B1L - L1, L1

unique solution can be constructed. Of course without additional conditions there remains a question: Does a constructed solution u(x) satisfy (6) ?

Let us introduce two lemmas establishing the link between (6) and system (7), (8) to answer that question.

Lemma 2. Let u* satisfy (6) and operator Li have left inverse. Then a pair u* = A2u*,u* satisfies (7), (8).

Proof. Based on conditions of the Lemma the following equality is satisfied

AiA2Lu* = Liu* + f. (9)

From (9) because of commutativity condition the following equality is valid

AiLA2u* = Liu* + f (10)

and

A2AiLA2 u* = LiA2u* + A2f. (11)

u* = A2u* u*

u

right hand side

Liu = -f + AiA2Lu*.

Here the operators commutativity property is employed. The solution exists for such

L1u* .

u* u* = A2u* L1.

Therefore, Liu = Liu*. Since the operator Li has left inverse, then u* is a unique solution u1 = A2u*

□

( u* , u* ) L1

right inverse L-1 . Then element u* satisfies (6).

Proof. The element —f + AiLu* belongs to the range of operator Li, because a pair u*,u* satisfies (7), (8). Hence u* = L-i(—f + AiLu*) because —f + AiLui G ^(Li), where ^(Li) is the range of the operator Li. It is to be demonstrated that the constructed

element u* satisfies (6) because by hypothesis u1 satisfies (7). Indeed, substitution of the u* B = A1A2,

A1A2LL-1 — f + A1Lu1) = A1LU*.

Taking into account operators commutativity, the following equality is valid

A1L{A2L-1(—f + A1Lu1) — u*} = 0. (12)

A1L

brackets is zero. Since u* E E1 satisfies (7), where B1 = A2A1; then the following equality is valid

A2(A1Lu1 — f ) = L1u*. (13)

Hence A2(A1Lu1 — f) E ^(L1^d u* = L-1A2(A1Lu* — f), where L-1 is the right inverse to the operator L1. Since A2L1 = L1A2 then A2 = L-1A2L1. This yields A2L-1 = L-1A2L1L-1. From equality L1L-1 = I it follows that the right inverse L-1 is also commutative with the operator A2. Then (13) can be represented as

A2L-1(A1Lu1 — f) — u* = 0. Thus we have shown that expression in brackets in (12) is zero. Lemma 3 is proved.

□

p.

assume operator B to have a skeleton chain {B1;..., Bp}, p > 1, and linear operators {Ai}f1= 1 represent decomposition of Bi which skeleton is attached to B. Introduce

i

u = 11 A2j u,i =1,...,p, (14)

j=1

where ui E Ei, ui = A2iui-1 ,u0 := u.

If u0 satisfies (6) then for p E N by Definition 1 we get equalities

p

BpLup = Lu + JJ A2j f, (15)

j=1

i

Lu = — JJ A2j f + A2i+1Lui+1, (16)

j=1

L1 u = —f + A1Lu1. (17)

For p > 2 there is a connection between the solution of (6) and system (15) - (17). In particular the following two lemmas can be formulated.

Lemma 4. Let u* satisfy (6) and operator L1 have left inverse. Then elements u* = nj=1 A2ju*, i = p,p — 1,..., 1 satisfy (15), (16), and u* satisfies (17).

Lemma 5. Let elements u*,u*-1,..., u1, u* satisfy (15) - (17) and operator L1 have right

u*

to (6).

Proof of Lemma 4 and Lemma 5 for any natural number p can be reduced to employment of the skeleton chain via operators {Aj}2= 1 and repeats of stages of the proofs of Lemmas 2 and 3 for the case of p = 1. Based on Lemmas 1-5 the following main result can be formulated.

Main Theorem. Let irreversible bounded operator B have a skeleton chain {B1,..., Bp}. Let opera tor BpL — L1 have a bounded inverse with a domain in Ep. Let operator L1 be defined on do mains Ei, i = 1,... ,p and E. Let opera tor L1 have an inverse bounded operator on Ei, i = 1,... ,p or E. Then system (15) - (17) enjoys a unique solution {u*, u*- ]_,... ,u\,u*}, where

p

u*p = (BpL — L1)-1H A2j f,

u* = LrH- J] A2j f + A2j+1L<+1}, i = p - 1,..., 1, j=i

u* = L-1{-f + Ai Lui}.

Moreover, element u* satisfies (6) and u* = nj=i A2ju*,i = 1,... ,p.

By setting the initial-boundary conditions to ensure the reversibility of the operators L^d BPL — Li with specific differential operators ^d Li and using the Main Theorem the existence and uniqueness theorems can be derived. Moreover, the formula obtained in Theorem can effectively build the desired classical solution of (1) with sufficient smoothness of f : n C Rm+i ^ E and the coefficients of the differential operators L and Li. Such applications of the theory are discussed further in section 3.

3. The Existence and Methods of Constructing Solutions of Nonclassic BVP with Partial Derivatives

Consider the system

_ dkl+k2u(x,t) ^ dkl+k2u(x,t)

^ dtkl dxk2 = 2^ Cfc1fc2 dtkl dxk2 + f ^V. ^

ki+k2<n ki+k2<m

Here m < n, B is a const ant N x N matrix, det B = 0, aklk2, cklk2 are numbe rs, an0=0, a0n = 0, c0m = 0, cm0 = 0. The vector-functions u(x,t) = (u1(x,t),... ,uN(x,t))T, f (x,t) = (f1(x, t),..., fN(x, t))T are supposed to be defined and analytical for < x,t < x>.

Let rankB = r < N. Then based on [11] B = A1A2, where A1 is an N x r matrix, A2 is an r x N matrix. Let us introduce an r x r matrix B1 = A2A1 and assume that det B1 = 0.

the successive solution of (7), (8), which are in this case as follows:

дkl+k2 u1(x,t) v-^ дkl+k2 u1(x,t)

\—^ д U\(X, t) ^—"V д U\(X,b) г/ \ f \

Bl akk Qtk,dxk2 = Qtklдхк2 + A2/(x,t), (19)

ki+k2<n ki +k2<m

u(x,t) „ . A дkl+k2 ui(x,t)

= -f (x,t) + Al , (20)

ki+k2<m ki+k2<n

where detB1 = 0, u1(x,t) = (u11 (x,t),... ,u1r(x,t))T,r < N,u1 = A2u. By hypothesis an0 = 0, c0m = 0, therefore for system (18) one may introduce the initial conditions

d*u(x, t)

A-

dx% d*u(x, t)

0, i = 0,1,...,m- 1,

x=0

dt%

0, i = 0,1,...,n — 1.

(21) (22)

t=0

The vector-function u1(x, t) based on Kovalevskaya theorem can be defined as a unique solution to system (19) with initial conditions

d*u1(x,t)

dt*

0, i = 0,1,...,n — 1.

t=0

u1 (x, t) u(x, t)

can be found as a unique solution to the Cauchy problem (20), (21). Consider system

B

dnu(x, t) dxn

(- — a2 dL ^

\ dt dx2 )

u(x, t) + f (x, t), n > 3.

(23)

As in system (18), B is a singular N x N matrix with rank B = r < N, B = A1A2, B1 = A2A1, det B1 = 0. Let f (x,t) = (f1(x,t),..., fN(x,t))T be a vector-function defined for 0 < x < 1, 0 < t < x>, continuous with respect to x and analytical by t, u = (u1,..., uN)T. The objective is to construct a solution of (23) in^ = {0 < x < 1, 0 < t < to}. Based

u1 = A2u

Bi

dnu1(x,t) f d

dxn

(

d 2 \

dt - a2dx2)ui(x't) + A2f(x't)'

.dt- a

with initial-boundary conditions

(- - a2 \ dt dx2 )

u(x, t) = —f (x, t) + A1

dnu1(x, t) dxn

diu1(x, t)

dx*

0, i = 0,1,..., n — 1,

(24)

(25)

(26)

x=0

u(x, t) |t=o = 0, (27)

u(x,t)|x=o = 0, u(x,t)lx=1 = 0. (28)

det B1 = 0 u1 (x, t)

u1 (x, t)

right hand side of (25). A unique solution of the first boundary value problem (25), (27), (28) is constructed for the heat equation using known formula (here readers may refer to

u(x, t)

unique solution of (23) in domain Q = {0 < x < 1, 0 < t < to}. This solution satisfies

the initial conditions . ,

diu(x, t)

A2

dx*

0, i = 1,

n1

x=0

and (27), (28).

4. Skeleton Decomposition in the Theory of Irregular ODE in a Banach Space

Consider the simplest irregular ODE

B dut1 = u(t) + f (t), (29)

f (t) : [0, to) ^ E, B G L(E ^ E). Let {B1,..., Bp} be a skeleton chain of the operator B.

Theorem 1. Let {B1,..., Bp} be a regular skeleton chain, the function f (t) be (p — 1)-

times dijferentiable. Then equation (29) with initial condition

p

UA2J u(t)lt =0 = co, co G Ep (30)

j=1

enjoys a unique classic solution u0(t,c0). Here

uo(t,co) = —f (t) + A1 d1, (31)

u1

Let us outline the scheme for construction of the function u1(t,c0) in solution (31) to problem (29), (30):

1. If p = 1 then u1(t,c0) satisfies the regular Cauchy problem

{

Bi -f = U + A2f (t), ui(0) = co.

2. If p > 2 then the function u1(t,c0) can be constructed by the following recursion

A P

B/-UP = up ^ Ajf (t), j=i

Up (0) = co.

Ui(t, co) = А2г+1 - П A2j f (t), i = p - l,p - 2,..., 1.

j=i

Theorem 2. Let {B1,..., Bp_1, 0} be a singular chain of length p > 1, 0 is a null operator-acting from Ep to Ep. Then B is a nilpotent operator and the homogeneous equation B-u = u has only trivial solution. In this case, if the function f (t) is p-times dijferentiable then the unique classic solution of (29) can be constructed as follows

d

Un(t) = -f (t) + B-un_1 (t), Uo(t) = -f (t), n =1, 2,...,p. dt

Here the function up(t) is a unique classic solution of (29).

Conclusion

This paper reports on the novel method of skeleton chains initiated in [4] for the linear operators in order to produce new non-classical boundary value problems for systems of differential and integral-differential equations with partial derivatives arising in modern mathematical modelling.

Acknowledgements. This work is fuilfilled within International Science and Technology Cooperation Program (No. 2015DFA70850) of China & Russia; NSFC Grant No. 61673398. The first author's work is partly supported by RSF Grant No. 14-19-00054-

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О ПОСТРОЕНИИ РЕШЕНИЙ НЕРЕГУЛЯРНЫХ СИСТЕМ С ЧАСТНЫМИ ПРОИЗВОДНЫМИ НА ОСНОВЕ ТЕОРИИ СКЕЛЕТНЫХ РАЗЛОЖЕНИЙ ЛИНЕЙНЫХ ОПЕРАТОРОВ

Д.Н. Сидоров1'2, Н.А. Сидоров3

1Институт систем энергетики имени Л.А. Мелентьева СО РАН, г. Иркутск 2

3Иркутский государственный университет, г. Иркутск

Рассматриваются линейные системы уравнений с частными производными. В главной части систем стоит линейный необратимый оператор, допускающий скелетное разложение. Входящие в систему дифференциальные операторы имеют достаточно гладкие коэффициенты. Области определения дифференциальных операторов в конкретных ситуациях, рассмотренных в работе, состоят из линейных многообразий достаточно гладких функций со значениями в банаховом пространстве подчиненных дополнительным граничным условиям. Вводится понятие скелетной цепочки линейного оператора, стоящего в главной части системы. Предполагается, что этот оператор порождает скелетную цепочку конечной длины. В этом случае решение исходной системы сводится к регулярной расщепленной системе уравнений, разрешенных относительно старших дифференциальных выражений с определенными начально-краевыми условиями. Указаны возможные обобщения предложенного подхода и рассмотрено его приложение к постановке граничных задач в нелинейном случае. Результаты дополняют элементы теории дифференциальных уравнений с вырождениями, заложенные в монографиях МИ 87а:58036, ЪЬ\ 1027.47001.

Ключевые слова: некорректная задача; задача Коши; необратимый оператор; скелетное разложение; скелетные цепочки, граничные задачи.

Денис Николаевич Сидоров, доктор физико-математических наук, профессор, Институт систем энергетики имени Л.А. Мелентьева СО РАН, Иркутский национальный исследовательский технический университет (г. Иркутск, Российская Федерация), dsidorov@isem.irk.ru.

Николай Александрович Сидоров, доктор физико-математических наук, профессор, Иркутский государственный университет (г. Иркутск, Российская Федерация), sidorovisu@gmail.com.

Received 28 December, 2016

УДК 517.9

DOI: 10 Л4529/ттр170205

Поступила в редакцию 28 декабря 2016 г.