ON THE PROBLEM OF RECOVERING BOUNDARY CONDITIONS IN THE THIRD BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATION Текст научной статьи по специальности «Математика»

МАТЕМАТИКА

MATHEMATICS

УДК 517.9

doi:10.21685/2072-3040-2021-2-1

On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation

I.V. Boykov1, V.A. Ryazantsev2

1,2Penza State University, Penza, Russia [email protected], [email protected]

Abstract. Background. In recent decades the theory for solution of inverse and ill-posed problems has become one of the most important and fast-growing branch of modern mathematics. A relevancy of this theory is due to not only significant growth in the number of applications of inverse and ill-posed problems in different fields of physical and technical sciences but also rapid development of computer technology. It is known that most of inverse problems of mathematical physics belong to the class if ill-posed problems, and the most important property of these problems is their instability to small perturbations of the initial data of the problem. This property induces the need for the development of regulari-zation methods of special type. The boundary value problems constitute one of the most significant classes of inverse problems. An inverse problem is termed boundary if it is required to recover the function that is the part of a boundary value. Such problems arise when direct measuring of heat field characteristics at a domain boundary is difficult or impossible at all. Constructing numerical methods for solution of these problems is very crucial due to a vast number of their physical and technical applications. Materials and methods. In this paper we propose a numerical method for simultaneous recovering of boundary value coefficients in the third boundary problem for a heat equation. At the core of the method there is continuous method for solution of operator equations in Ba-nach spaces. The main idea of the method is composing and solving the auxiliary system of differential equations of special type relative to the unknown coefficients of the basic problem. This system is to be solved numerically with the help of some method for approximate solution of differential equations. Simultaneous recovering of the coefficients by means of the proposed method additionally requires knowledge of values of a solution of the basic parabolic equation at two different points. Results. A numerical method for solution of the problem of recovering boundary value coefficients in the third boundary value problem for one-dimensional heat equation is constructed. We show applicability of continuous operator method to solution of inverse boundary value problems for parabolic equations. Convergence of the method is proven with the help of the theory of stability of ordinary differential equations. Successful solution of the model example shows effectiveness of the proposed method. Conclusions. An efficient method for solution of the problem of recovering boundary value coefficients in the third boundary value problem for linear one-dimensional parabolic equation is described in the paper. The key advantages of the method are its simplicity, flexibility and also stability to perturbations of initial data of the problem. It is of significant theoretical and practical interest to extend the proposed method to a wider class of boundary problems and also to multidimensional and nonlinear parabolic equations.

© Бойков И. В., Рязанцев В. А., 2021. Контент доступен по лицензии Creative Commons Attribution 4.0 License / This work is licensed under a Creative Commons Attribution 4.0 License.

Keywords: inverse problems, parabolic equations, third boundary value problem, stability theory, continuous operator method

For citation: Boykov I.V., Ryazantsev V.A. On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2021;2:3-13. doi:10.21685/ 2072-3040-2021-2-1

К вопросу о восстановлении граничных условий в третьей краевой задаче для параболического уравнения

И. В. Бойков1, В. А. Рязанцев2

1,2Пензенский государственный университет, Пенза, Россия 1 [email protected], [email protected]

Аннотация. Актуальность и цели. В последние десятилетия теория решения обратных и некорректных задач стала одним из важнейших и наиболее быстро развивающихся направлений современной математической науки. Актуальность этого направления обусловлена как существенно возросшим числом приложений обратных и некорректных задач в различных областях физики и техники, так и бурным развитием вычислительной техники. Известно, что большинство обратных задач математической физики принадлежит к классу некорректных задач, наиболее важным свойством которых является их неустойчивость относительно малых возмущений исходных данных задачи. Это свойство некорректных задач вызывает к жизни необходимость в разработке для их решения специальных методов регуляризации. Одним из важнейших классов обратных задач являются граничные обратные задачи. Обратная задача называется граничной, если требуется восстановить функцию, входящую в граничное условие задачи. Такие задачи возникают в случае, если непосредственные измерения характеристик теплового поля на границе области или ее части затруднены либо же вовсе невозможны. Построение численных методов решения таких задач вследствие большого числа их приложений в физике и технике является актуальной задачей. Материалы и методы. Предложен численный метод одновременного восстановления коэффициентов граничных условий в третьей краевой задаче для уравнения теплопроводности. В основе метода лежит непрерывный метод решения операторных уравнений в банаховых пространствах. Сущность метода состоит в составлении и решении относительно неизвестных коэффициентов исходной задачи вспомогательной системы дифференциальных уравнений специального вида. Эту систему затем предлагается численно решать при помощи одного из приближенных методов решения дифференциальных уравнений. Одновременное восстановление коэффициентов при помощи предлагаемого метода дополнительно требует знания значений решения исходного параболического уравнения в двух различных точках. Результаты. Построен численный метод решения задачи восстановления коэффициентов граничных условий в третьей краевой задаче для одномерного уравнения теплопроводности. Показана применимость непрерывного операторного метода к решению граничных обратных задач для параболических уравнений. Сходимость метода обосновывается с помощью теории устойчивости решений обыкновенных дифференциальных уравнений. Успешное решение модельного примера продемонстрировало эффективность предложенного метода. Выводы. Описан эффективный численный метод решения задачи восстановления коэффициентов граничных условий в третьей краевой задаче для линейного одномерного параболического уравнения. Ключевыми преимуществами метода являются его простота, универсальность, а также устойчивость к возмущениям исходных данных задачи. Представляет значительный теорети-

ческий и практический интерес обобщение предложенного метода на более широкий класс граничных задач, а также на случай многомерных и нелинейных параболических уравнений.

Ключевые слова: обратные задачи, параболические уравнения, третья краевая задача, теория устойчивости, непрерывный операторный метод

Для цитирования: Boykov I. V., Ryazantsev V. A. On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation // Известия высших учебных заведений. Поволжский регион. Физико-математические науки. 2021. № 2. С. 3-13. doi:10.21685/2072-3040-2021-2-1

Introduction

Not with standing the fact that the theory for solution of inverse and ill-posed problems is relatively new research direction, at present time it represents a powerful instrument of modern mathematical science that incorporated a lot of results and allows to solve many difficult applied problems effectively, for example, in composite materials [1], in geophysics [2], in the theory of nuclear reactors [3], in quantum theory, electric and seismic prospecting, potential theory, computational tomography, astronomy etc. A range of methods is used for solving inverse problems is quite wide and includes particularly the method of functional analyses, iteration methods, the finite elements method and also regularization methods.

Now many important results in constructing method for solution of inverse problems of mathematical physics have been obtained. Some of the results are presented in [4-7]. Between the methods mentioned above we should highlight a group of iteration methods and particularly gradient methods. These methods by reason of their high efficiency are very promising research direction for the theory and practice of solving inverse problems. One of such methods is proposed in this paper.

We consider a particular case of inverse boundary problem for the heat equation. Specifically, it is required to recover the coefficients ki > 0, k2 > 0 which are the parts of boundary conditions in the third boundary problem for linear one-dimensional heat equation:

d- = 0 < X < A t > 0, (1)

dt Эх2

u (0, х) = ф( х), (2)

du (t, х)

dx du(t, x)

— k\u + k^u

= Vi(t ), (3)

x=0

= V2(t )• (4)

x=f.

dx

It is supposed that we additionally know a pair of values u (, x* ),

t , x ) of the solution of the problem (1)-(4) at two different points

x[0,f] and (,x**)e K+x[0,f]

\t*,x* |e

Before describing the proposed method we give a brief description of the continuous method for solution of operator equations and the definition of logarithmic norm.

1. A description of the continuous operator method

The continuous operator method is described in accordance with the paper [8]. Consider the operator equation

T(x) = f. (5)

Here x e B and the Banach space B is mapped by the operator T onto itself.

We introduce a Cauchy problem for the differential equation

^ = T (x(c))- f, (6)

d a

X(0) = x, (7)

where a is nonnegative independent variable. The following theorem occurs.

Theorem 1. Consider that the equation (5) has the solution x = x*, and on every differentiable curve g (t) that is placed in a ball 5(0, r) of radius r the following conditions hold.

1) At every a> 0 the following inequality is valid:

a

JЛ(Т(g(s)))ds ^ 0;

0

2) The following inequality takes place: a

lim - JЛ(Т'((s)))ds ^ -< rr_a »

v v— -/, - a, a>0.

O 0 v v "

Then a solution of the problem (6)-(7) converges to the solution x* of the equation (5).

In the paper [8] it is also shown that the assertion of this theorem holds for every initial condition x belonging to the ball B(x*, r).

Here T' should be understood as Frechet derivative of the operator By A(T) we denote logarithmic norm of the operator T defined by the formula [9]

, „ IIl - h®1|-1

A(<5') = lim J-1—,

V ' hi0 h

where I denotes an identity operator and the symbol X denotes that h tends to zero, decreasing.

General description of the proposed method

Let us proceed with the description of the method for recovering both boundary value coefficients in the third boundary value problem (1)-(4).

In is known [10] that a general solution of the problem (1)-(4) is defined by the formula

where

i(t, x) = J ç(Ç)G (x, Ç, t; kh k2 )d Ç-y-J Vi(s)G (x,0; t - 5, kb )ds

0 0 t

+Y-J ¥2(s)G (x, I, t - s; ki, k2 )ds,

» 1

G ( x,Ç,t; ki, k2 ) = ^ 77—) • yn( x) • yn (Ç) • exp ( ), n=1 7 (yn )

k

yn = cos (lnx ) +-sin (lnx )

In

+

7 (yn ) = l2+ki2 + _A

+ —br + —

i k 2 ^ k1

1+lr

V 1 n J

(8)

2^ ^ +kf 2

and are positive roots of the transcendental equation

tg(^) = h + k2 ^ i2 - hh

Substituting (t, x) = (t , x*) and (t, x) = (t , x* ) in (8), we rewrite this equation in the following way:

J 9(Ç)G (, Ç, t*, k1, k2 )d Ç-y J ^1(s)G ( ,0, t*- s; kb k2 )ds

+

+Y- J ¥2(s)G (x*, I, t* - s; kb k2 )ds - u (t *, x*) = 0.

(9)

-C I

J 9(Ç)g( x**, Ç, t**, k1,k2 )dÇ - y J V1(s)G (x** ,0, t** - s; k1, k2 )ds

+

+Y- J ¥2(s)G(x**,I,t**-s;k1,k2 )ds -u((,x**) = 0.

(10)

*

In accordance with the continuous operator method we introduce auxiliary functions ki(a), ¿¿(a), a^0, such as

lim ki (a) = ki, lim ¿2 (a) = k2.

a^» a^»

The system of nonlinear equations (9)-(10) we associate with a system of ordinary differential equations in unknown functions ki (a), ¿2 (a):

= Vl ■ \J ' ^ ' ^ ki(0)' F2(o) )-

l

J Vi(s)G (x* ,0, t *- 5; ki(a), k2(o) ) +

J9(^)G(x*,Ç/;ki(a),^2(0))-u(t*,x*)

(ll)

d^2 (0) d о

= V2 ■ \J 9(^)G (x**, 1t** ; ki(o), ^2(0) )-

I

J Vi(5)G(x**,0,t** - 5;ki(o),k2(o)) +

l

+Y- J (x**,t**;ki(o),k2(o)) -u(t**,x**)

(i2)

where the parameters Vj =±1, v 2 =±1 are fixed in such as way that all conditions of the theorem 1 be fulfilled.

In order to provide the numerical method we add to the system (11)-(12) the following initial conditions:

ki(0) = Xi, MO) =X2,

(i3)

where the values x, %2 are chosen in a neighborhood of supposed solution of the problem. As the problem (11)-(12) is nonlinear, choice of initial conditions is sophisticated.

The problem (11)-(13) can be solved by arbitrary numeric method for solution of differential equations. In particular, Euler method is quite efficient. Let 0 be the step of Euler method and L is the number of iterations of Euler method. Let us denote

k1,r = k1 (cr ) k2 r = k2 (°r ) ar = r0.

*

Then numeric solution of the problem (11)-(13) is implemented through successive calculations by the following formulae (r = 0, L -1).

k1,r+1 = k1,r + 0-v1 • <

J x , t*; klrr, k2,r )d

I

-Y• JVi(s)G(x*,0,t* -s;\r,k2>r )

+

+Y- J¥2(s)G(x*,f,t* -s;ki r,k2 r )ds -u(t*,x* j

(14)

k2,r+1 = k2,r + 0V2 •<

L

J (x**, t**; k1,r, k2,r )d£ -

-y- J ^1(s)G(x**,0,t** - s;k1r,k2 r )ds +

+Y• J s)G(x**,f,t** - s;k\r,k2 r )ds - u (t**, x** j

0

(15)

The result of the method is the approximate values of parameters kj, k2, which are fixed in accordance with the formula

k1 ~ k1,L, k2 ~ k2,L.

(16)

It should be noted that the integrals in (14)-(15) are to be computed approximately with the aid of one of the quadrature formulae.

2. Solving the model problem

Let it be required to recover the values of the parameters k2 for the problem (1)-(4) with f = 1. Let be known that

e~t e~t

9( x) = cos( x), V1(t) =—, V2(t) = — (3cos(1) - 2sin(1)), (17) and the following values are additionally known:

u ((*, *)=u (4,3 )=exp (- 1 )cos (f

*

/****\ ( 3 i ) ( 3 ) ( i

Т 'x )=u[4,4J =exp[-4Jcos[4

The exact solution of this problem at the specified conditions is defined by the following formulae

k = k2 = |, (18)

while u (t, x) = e— cos( x).

The integrals on the right hand sides of the formulae (14)-(15) are computed

—1 —1

using the trapezoidal rule with the steps t = 10 , h = 10 in variable t and variable x respectively. When calculating G(x,t;^ ) function values only first

—2

five terms of the sum are used. The step 0 of Euler method equals to 10 , and the

number L of iterations for Euler method equals to 3 104. The initial guess values are defined by the formula x = %2 = 2.

The result of numerical computations is given by the formula

k1 - k1L = 0.49545, k2 = k2L = 1.37563.

It is easily seen that these approximate values of k1, k2 are close to the exact values of k1, k2 which are defined by the formulae (17).

It should be noted that in case of an unsuccessful choice of the points

¡^ \ I ^^ ¡^^ \

t ,x ), It ,x ) the problem of simultaneous restoration of both unknown coefficients k1, k2 may have non unique solution. As an example we give the results of solution of the considered problem for functions q>( x), ^ (t ), y 2 (t ) defined by the formulae (17) with £ = 1. For approximate computing of the integrals in (14)-

—2

(15) we use trapezoidal rule with steps T = h = 10 . While calculating G(x,t;k1,k2 ) we use first five terms of the sum. For Euler method we fix the following parameters: L = 500, 0 = 10—1. The exact solution of the problem is defined by the formulae (18) while u (t, x) = e— cos( x).

In the first series of experiments we fix (( *, x*) = ^ j, = I —,— I, so that we have the following additional information:

/ ** ** \ ( i 2 ) (t 'x ) = )3'?J

u((/) = u\ i-i J = exP\-3)cosI 3 I,

/ ** ** \ i i 2 ) ( i i(( ,x )=u[3,3J=exp[-3

COS^

The results of computations are listed in the following table 1.

Table 1

Initial quess for k1 Initial quess for k2 Computed value of £1 Computed value of £2 Absolute value of difference of exact and approximate value of u(t*,x") Absolute value of difference of exact and approximate value of u(t",x")

0.1 0.1 0.48953 1.49241 8.58-10-4 2.83-10-3

1.0 1.0 0.48808 1.49780 8.22-10-4 2.5M0-3

2.0 2.0 0.48345 1.50305 6.82-10-4 2.26-10-3

3.0 3.0 0.48345 1.51221 5.27-10-4 1.77-10-3

2.0 1.0 0.49929 1.46157 1.4M0-3 4.50-10-3

1.0 2.0 0.47210 1.55026 7.53-10-5 2.65-10-4

It is easily seen that for all foregoing initial guesses for £1, £2 the iteration process converges to the exact solution defined by the formulae (18).

In the second series of experiments we fix It*, x*

For the same parameters of the method the results of computations are presented below.

Table 2

Initial quess for £ Initial quess for k2 Computed value of £1 Computed value of £2 Absolute value of difference of exact and approximate value of u(f,x*) Absolute value of difference of exact and approximate value of u(t",x")

0.1 0.1 0.69208 1.11506 1.09-10-3 1.98-10-3

1.0 1.0 0.91740 0.77624 4.3M0-4 5.48^10-4

2.0 2.0 1.02894 0.64677 4.12-10-4 4.77-10-4

3.0 3.0 1.09851 0.57593 1.1110-3 1.10^10-3

2.0 1.0 1.28986 0.40932 3.32-10-3 3.24^10-3

1.0 2.0 0.63216 1.22807 1.02-10-3 2.04^10-3

It can be easily seen that the iteration process converges to different solutions depending on the choice of initial guess.

Therefore it turns out that in order to provide uniqueness of solution of the

problem of finding two coefficients £1, £2 we must choose the points (, x*),

(,x**) in a proper way. Unfortunately to the authors' knowledge there is no known technique for choosing these points with aim of providing the unique pair of

values ki, k2 which solves the stated problem. The question of devising such technique calls for further investigations.

Conclusion

An iterative method for solution of the problem of simultaneous recovery of coefficients kj, k2 in the third boundary problem for linear one-dimensional parabolic equation is proposed. The method is based on the continuous operator method for solution of nonlinear operator equations in Banach spaces. Among basic advantages of the method we should mention its simplicity and versatility in the sense of its applicability to a wider range of problems. For successful recovery of the coefficients kj, k2 we must additionally know the values of u (t, x) at only two different points (t*,x*), ((**,x**). The main disadvantage of the method lies in the

fact that there is no sharply defined procedure for choosing ((*,x*) and ((**,x**)

to provide uniqueness of the pair kj, k2. Solving several model problems shows high efficiency of the proposed method.

References

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Список литературы

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Информация об авторах / Information about the authors

Илья Владимирович Бойков Il'ya V. Boykov

доктор физико-математических наук, Doctor of physical and mathematical

профессор, заведующий кафедрой sciences, professor, head of

высшей и прикладной математики, the sub-department of higher and applied

Пензенский государственный mathematics, Penza State University

университет (Россия, г. Пенза, (40 Krasnaya street, Penza, Russia)

ул. Красная, 40)

E-mail: [email protected]

Владимир Андреевич Рязанцев Vladimir A. Ryazantsev

кандидат технических наук, доцент, Candidate of engineering sciences,

кафедра высшей и прикладной associate professor, sub-department

математики, Пензенский of higher and applied mathematics,

государственный университет (Россия, Penza State University (40 Krasnaya

г. Пенза, ул. Красная, 40) street, Penza, Russia)

E-mail: [email protected]

Поступила в редакцию / Received 3i.03.202i

Поступила после рецензирования и доработки / Revised i0.04.202i Принята к публикации / Accepted i5.04.202i