Computational experiment for one class of evolution mathematical models in quasi-Sobolev spaces Текст научной статьи по специальности «Математика»

КРАТКИЕ СООБЩЕНИЯ

MSC 35G05 DOI: 10.14529/mmpl60413

COMPUTATIONAL EXPERIMENT FOR ONE CLASS OF EVOLUTION MATHEMATICAL MODELS IN QUASI-SOBOLEV SPACES

J.K.T. Al-Isawi, South Ural State University, Chelyabinsk, Russian Federation, [email protected],

A.A. Zamyshlyaeva, South Ural State University, Chelyabinsk, Russian Federation, [email protected]

In the article the mathematical model representing one class of evolution equations in quasi-Banach spaces is studied. A theorem on the unique solvability of the Cauchy problem is stated. The conditions for the phase space existence are presented. We also give the conditions for exponential dichotomies of solutions. Based on the theoretical results there was developed an algorithm for the numerical solution of the problem. The algorithm is implemented in Maple. The article includes description of the algorithm which is illustrated by variety of model examples showing the work of the developed program and represent the main properties of solutions.

Keywords: evolution equation; quasi-Banach spaces; numerical solution.

Introduction

Let [Xk} С R+ be a monotone sequence such that lim Xk = The quasi-Banach

space

С = {u=[uk}: £ Х \uk|)"< +

( oo ( mm \q \1/q

with a quasi-norm m ||u|| = (5^k=1 [X^ \uk\ , m E R is called a quasi-Sobolev space [1]. Obviously, for q E [1, the spaces are Banach spaces; 10 =£q, and there is a

q cxi i-j c«ij. сял^ jlj. о • q — q ^

dense and continuous embedding ^tо Р^ for n > m and q E R+.

Example 1. Let U = ; F = l™] Qn(X), Rs(X) ^e polynomials of powers n and s

(n < s) respectively with real coefficients, without common roots. Consider an operator L = Qn(A)u = [Qn(Xk)uk}, where [uk} E U. It is easy to see that the operator L E L(U;F) Construct an operator M = Rs(A)u = [Rs(Xk)uk}. It is easy to show that M E Cl(U; F), domM = , the ^^^rum aL(M) of operator M consists of points = Rs(Xk)(Qn(Xk))_1, k E N Xk is not the root of the polynomial Qn (X). Further we consider polynomials with the coefficients at the highest powers having opposite signs.

Definition 1. Vector-function u E C 1(R+; U), satisfying

Lu=Mu. (1)

pointwise is called a classical solution of this equation. The solution u = u(t) of (1) is called a solution to the weakened Cauchy problem (in sense of S.G. Krein), if in addition for u0 E U

lim u(t) = u0. (2)

holds.

If ker L = {0} then (1) is called a Sobolev type equation. Interest in Sobolev type equations has recently increased significantly [2-5], moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated by the desire to fill up the theory as well as by the aspiration to comprehend non-classical models of mathematical physics in quasi-Banach spaces [6].

Since the Cauchy problem for the Sobolev type equation is not solvable for arbitrary initial data it is necessary to construct the phase space of equation as the set of admissible initial values containing all solutions of equation [2]. The phase spaces of evolution and dynamical Sobolev type equations were constructed earlier in Banach spaces [2]. These ideas were used to study one class of evolution Sobolev type equations in quasi-Banach spaces of sequences [6]. There was held an analytical investigation of the considered problem. A theorem of existence of unique solution was proved. Our gual is to develop an algorithm for the numerical solution of the problem and carry out computational experiments.

1. Analytical Study of the Mathematical Model of One Class of Evolution Equations in Quasi-Banach Spaces

Lemma 1. [6] Operator M defined in example 1 is strongly L-seetorial.

Theorem 1. [6] Let operators M and L be defined as in example 1. Then

(i) operators L and M generate on spaces U and F degenerate holomorphie semigroups {U* : t E R+} and {F* : t E R+} respectively given by

U*=2- j RL (M) ef*df E L (U) F*— J LLM)e^dj E L(F) (3)

for t E where the contour r C pL(M) is such that \argj\ ^ 9 npu j ^ j E r.

(ii) there exist semigroup's units which are the projectors P E L(U) and Q E L(F) given by

!I, if Xk is not the root of Qn(X) for all k E N;

I — < -,ek > ek, if there exist I E N : X^ is the root of Qn(X),

km:k=t,

(the projector Q has the same form), splitting the quasi-Banach spaces U and F into direct sums

U=U0 © U1, F=F° © F1-

Definition 2. The set P C U is called a phase space of equation (1), if

(i) any solution u = u(t) of (1) lies in P pointwise, i.e. u(t) E P for all t E R+;

(ii) for all u0 E P there exists a unique solution to (1), (2).

ML

U1 is a phase space of (1), for arbitrary u0 E U1 there exists a unique solution to (1), (2).

Definition 3. We say that solutions of (1) have exponential dichotomy, if

(i) the phase space of (1) can be represented as P = J1 © J2, where J1(2) are invariant spaces of equation (1);

(ii) for arbitrary u0 E J1 (u0 E J2) solution u = u(t) of (1), (2) is such that ^ ||u(t)|| < C1(u0)e-at Hu(t)H > C2(u0)eat) for some a> 0 and all t E R+.

Theorem 3. [6] Let operators L,M E L(U; F) be defined as in example 1 and condition

aL(M) n iR = 0 and there exists jk E aL(M) with Rejk > 0 hold. Then solutions of (1) have exponential dichotomy.

2. Numerical Solution Algorithm

Based on the theoretical results there was developed an algorithm for numerical solution of problem (1). (2). implemented in a software environment Maple 15.0. The program uses a phase space method [2].

Fig. 1. A block diagram of algorithm

A numerical solution algorithm is shown in a block diagram in Fig. 1. The developed program allows you to:

1. Enter the polynomials of the Laplace quasi-operator and consider one class of evolution equations in quasi-Sobolev spaces.

2. Take into account degeneracy of equation and apply the phase space method.

3. Find the necessary for the accuracy e number of nonzero members of approximate solution.

4. Find and print the approximate solution of the problem.

5. Get a graphical image of the components of received solution over time.

A detailed description of the algorithm (each block of the algorithm corresponds to one step):

Step 1. After the start of the program the user enters the powers of the polynomials s, n, the time period T : t E [0, T] and the accuracy of an approximate solution e.

s

Step 2. In a cycle the polynomial Rs(x) = dkxk is constructed.

k=l

n

Step 3. In a cycle the polynomial Qn(x) = Y1 °kxk is constructed.

k=l

Step 4- The monotone increasing sequence [Xk} is entered by user. Step 5. The initial sequence u0 is set.

Step 6. Input of parameters of a quasi-norm of a quasi-Sobolev space.

Step 1. Calculation of a quasi-norm of the initial sequence.

Step 8. Checking if the quasi-norm of the initial sequence is infinite.

If the eighth step is true go to Step 5.

If the eighth step is false:

Step 9. Verification of number N of nontrivial components of approximate solution

T

needed to take into account degeneracy and to obtain the accuracy e so that J m\\u{t) —

0

UN (t)\\dt.

Step 10. In a cycle check if Qn(Xk) = 0, i.e. equation is degenerate. If the tenth step is true:

u0

If the eleventh step is true: Step 12. End of cycle by k. If the eleventh step is false:

Step 13. Output the message:"There is no solution". If the tenth step is false:

Step 14■ The k-th equation is a differential one. Add it to the system of differential equations to solve.

k

Step 16. Solve the system of differential equations to find the nontrivial components of an approximate solution.

Step 17. The resulting approximate solution is put on the screen and displayed as graphs of the components of an approximate solution. Step 18. End of program.

3. Numerical Experiment

Let U = imm+2ni F = I™- Consider the Cauchy problem

u(0) = u0,t E [0,T], u0 E U, (4)

Qn(A)U = Rs(A)u. (5)

Example 2. It is required to find a numerical solution of problem (4) (5) where Qi(x) = 2 - x, R2(x) = x2, Xk = k2, uok = q = 0, 5, m =1, T = 0, 5 e = 0, 01.

The program checks the condition ™||u0|| = Since it holds, there is no solution to the problem. The program gives output: "There is 110 solution".

Example 3. It is required to find a numerical solution of problem (4) (5) where Q1(x) = 2 — x, R2(x) = x2, Xk = k2, u0k = £2 > q = 0, 5, m =1, T = 0, 5. Depending on accuracy e we received the following results: For e1 = 0,1,

ui(t)=^,0,25 e-8t, 0,1 1 e-11'57t, 0,06 e-18,29^04 e-27'17t, 0, 0, ..., 0,...). e2 = 0, 01,

{¡¡2(t)=(e^25 e-8t,0,l 1 e-11'57t, 0,06 e-18'29^04 e-27'17^03 e-38'12^02 e-51t, 0,02 e-66^01 e-83^01 e-102^^)08 e-123^^)07 e-146t, 0,006 e-171^^)05 e-198t, 0,004 0. ...).

;

227t

, 0, 0,

The graph of the solution is presented in Fig. 2.

Fig. 2. Solution from example 3

Fig. 3. Solution from example 5

Example 4. It is required to find a numerical solution of problem (4) (5) where Q1(x) = 4 — x, R2(x) = x2, Xk = k2, u0k = kj, , q = 0, 5, m =1, T = 0, 5 e1 = 0,1. Equation (5) in this case is degenerate. Since the initial data do not belong to the phase space of equation the program gives output: "There is 110 solution".

Example 5. It is required to find the numerical solution of problem (4) (5) where Q1(x) = 4 — x, R2(x) = x2,Xk = k2,

Jf , if M 2;

u0k = <

\0 iffc = 2.

Equation is degenerate (Qi(A2) = 0) and initial data belong to the phase space of equation (5). For e = 0, 01 we have

u(t)=0, 0,1 e-11'57t, 0,06 e-18'29t, 0,04 e-27'17, 0,03 e-38'12t, 0,02 e-51t, 0,02 e-66t, 0,01 e-83, 0,01 e-102t, 0,008 e-123t, 0,007 e-146t, 0,006 e-171t, 0,005 e-198t, 0,004 e-227t, 0,004 e-258t, 0,003 e-291t, 0,003 e-326t, 0,003 e-363t, 0,003 e-402t, 0,002 e-443t, 0,002 e-486t, 0, (),..., (),...).

The graph of the solution is presented in Fig. 3.

Example 6. It is required to find a numerical solution of problem (4) (5) where Q2(x) = 7 + x2, R3(x) = — 5 + x — 5x2 — x3, Xk = k4, u0k = k?, q = 0.5, m =1, T = 0.5.

For e1 = 0,1,

U1(t)=(e-1'25t, 0,13 e-20'39^04 e-85'9t, 0, 0,.., 0,...). For £2 = 0, 01,

U2(t)=(e-1'25^l3 e-20'39^04 e-85'9t, 0,02 e-260t, 0, 0,..., 0,...). The graph of the solution is presented in Fig. 4.

Fig. 4. Solution from example 6

References

1. Al-Delfi J.K. Quasi-Sobolev Spaces i™. Bulletin of the South Ural State University. Series: Mathematics, Mechanics, Physics, 2013, vol. o, no. 1, pp. 107-109. (in Russian)

2. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003.

3. Zamyshlyaeva A.A. Linear Sobolev Type Equations of High Order. Chelyabinsk, Publishing Center of SUSU, 2012. (in Russian)

4. Manakova N.A. Problems of Optimal Control for the Semilinear Sobolev Type Equations. Chelyabinsk, Publishing Center of SUSU, 2012. (in Russian)

o. Sagadeeva M.A. Dihotomies of Solutions to Linear Sobolev Type Equations. Chelyabinsk, Publishing Center of SUSU, 2012. (in Russian)

6. Zamyshlyeva A.A., Al-Isawi J.K.T. On Some Properties of Solutions to One Class of Evolution Sobolev Type Mathematical Models in Quasi-Sobolev Spaces. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 4, pp. 113-119. DOI: 10.14529/mmpl50410 (in Russian)

Received, September 5, 2016

УДК 517.9 DOI: 10.14529/mmpl60413

ВЫЧИСЛИТЕЛЬНЫЙ ЭКСПЕРИМЕНТ ДЛЯ ОДНОГО КЛАССА ЭВОЛЮЦИОННЫХ МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ В КВАЗИСОБОЛЕВЫХ ПРОСТРАНСТВАХ

Д.К.Т. Аль Исави, A.A. Замышляева

В статье изучается математическая модель, представляющая один класс эволюционных уравнений в квази банаховых пространствах последовательностей. Представлена теорема об однозначной разрешимости в виде условий существования фазового пространства уравнения и приведены условия существования экспоненциальных дихотомий решений. На основе теоретических результатов разработан алгоритм численного решения задачи. Алгоритм реализован в среде Мар1е. Статья содержит описание алгоритма и различные примеры иллюстрации работы программы на его основе, демонстрирующие различные свойства решений.

Ключевые слова: эволюционное уравнение; квазибанаховы пространства; численное решение.

Литература

1. Аль-Делфи Дж.К. Квазисоболевы пространства Im / Дж.К. Аль-Делфи // Вестник ЮУрГУ. Серия: Математика, механика, физика. - 2013. - Т. 5, № 1. - С. 107-109.

2. Sviridyuk G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht, Boston: VSP, 2003.

3. Замышляева A.A. Линейные уравнения соболевского типа выского порядка /A.A. Замышляева. - Челябинск: Издат. центр ЮУрГУ, 2012.

4. Манакова H.A. Задачи оптимального управления для полулинейных уравнений соболевского типа / H.A. Манакова. - Челябинск: Издат. центр ЮУрГУ, 2012.

5. Сагадеева М.А. Дихотомии решений линейных уравнений соболевского типа / М.А. Са-гадеева. - Челябинск: Издат. центр ЮУрГУ, 2012.

6. Замышляева A.A. О некоторых свойствах решений одного класса эволюционных математических моделей соболевского типа в квазисоболевых пространствах / A.A. Замышляева, Д.К.Т. Аль Исави // Вестник ЮУрГУ. Серия: Математическое моделирование и прораммирование. - 2015. - Т. 8, № 4. - С. 113-119.

Джавад Кадим Тахир Аль Исави, аспирант, кафедра «Прикладная математика и программирование:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].

Алена Александровна Замышляева, доктор физико-математических наук, доцент, кафедра «Прикладная математика и программирование», ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), /.ашунЫтсуааа "нини. га.

Поступила в редакцию 5 сентября 2016 г.