Abstract and Applied Analysis

Volume 2012 (2012), Article ID 603018, 14 pages

doi:10.1155/2012/603018

Research Article

The Difference Problem of Obtaining the Parameter of a Parabolic Equation

Charyyar Ashyralyyev1,2 and Oznur Demirdag³

¹Department of Computer Technology, Turkmen Agricultural University, Gerology Street,74400 Asgabat, Turkmenistan

²Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey

³Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey

Received 4 April 2012; Accepted 23 April 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Charyyar Ashyralyyev and Oznur Demirdag. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The boundary value problem of determining the parameter of a parabolic equation in an arbitrary Banach space with the strongly positive operator is considered. The first order of accuracy stable difference scheme for the approximate solution of this problem is investigated. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.

1. Introduction

The differential equations with parameters play a very important role in many branches of science and engineering. Some examples were given in temperature overspecification by Dehghan [1], chemistry (chromatography) by Kimura and Suzuki [2], physics (optical tomography) by Gryazin et al. [3].

The differential equations with parameters have been studied extensively by many researchers (see, e.g., [4–20] and the references therein). However, such problems were not well investigated in general.

As a result, considerable efforts have been expanded in formulating numerical solution methods that are both accurate and efficient. Methods of numerical solutions of parabolic problems with parameters have been studied by researchers (see, e.g., [21–29] and the references therein).

It is known that various boundary value problems for parabolic equations with parameter can be reduced to the boundary value problem for the differential equation with parameter :

in an arbitrary Banach space with the strongly positive operator . In the present work, the first order of accuracy difference scheme for the approximate solution of boundary value problem (1.1) is studied. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.

2. The Boundary Value Problem for Parabolic Equations

Throughout this work, is a Banach space, is the generator of the analytic semigroup with exponentially decreasing norm, when , that is, the following estimates hold:

From estimate (2.1), it follows that

Here, .

Abstract problem (1.1) was investigated in the paper [4] by applying estimates (2.1) and (2.2). The solvability of problem (1.1) in the space of the continuous -valued functions defined on equipped with the norm

was studied under the necessary and sufficient conditions for the operator . The solution depends continuously on the initial and boundary data. More pricisely, we have the following result.

Theorem 2.1. Assume that is the generator of the analytic semigroup and all points , , do not belong to the spectrum . Let , , and . Then, for the solution of problem (1.1) in , the estimates

hold, where M does not depend on , , and . Here is the space obtained by completion of the space of all smooth -valued functions on in the norm

With the help of , we introduce the fractional space , , consisting of all for which the following norms are finite [6, 30]:

We say is the solution of problem (1.1) in if the following conditions are satisfied:

(i), ,(ii) satisfies the equation and boundary conditions (1.1).

Here, , , is the Hölder space with weight obtained by completion of the space of all smooth E-valued functions on in the norm

In the paper [23], the exact estimates in , and , Hölder spaces for the solution of problem (1.1) were proved. In applications, exact estimates for the solution of the boundary value problems for parabolic equations were obtained.

Now, we consider the application of Theorem 2.1. First, the boundary-value problem on the range , for the -order multidimensional parabolic equation is considered:

where and are given as sufficiently smooth functions. Here, is a sufficiently large positive constant.

It is assumed that the symbol

of the differential operator of the form

acting on functions defined on the space satisfies the inequalities

for .

Problem (2.8) has a unique smooth solution. This allows us to reduce problem (2.8) to problem (1.1) in a Banach space of all continuous bounded functions defined on satisfying a Hölder condition with the indicator .

Theorem 2.2. For the solution of boundary problem (2.8), the following estimates are satisfied:

where is independent of , , and .

The proof of Theorem 2.2 is based on the abstract Theorem 2.1 and on the strongly positivity of the operator defined by formula (2.10) (see, [31–33]).

Second, let be the unit open cube in the -dimensional Euclidean space with boundary . In , we consider the mixed boundary value problem for the multidimensional parabolic equation

where , , , and are given smooth functions and Here, is a sufficiently large positive constant.

We introduce the Banach spaces , , of all continuous functions satisfying a Hölder condition with the indicator , ,, and with weight , , which is equipped with the norm

where is the space of the all continuous functions defined on , equipped with the norm

It is known that the differential expression [34]

defines a positive operator acting on with the domain .

Therefore, we can replace mixed problem (2.13) by the abstract boundary problem (1.1). Using the results of Theorem 2.1, we can obtain the following theorem on stability.

Theorem 2.3. For the solution of mixed boundary value problem (2.18), the following estimates are valid:

where does not depend on , , and .

Third, we consider the mixed boundary value problem for parabolic equation

where , , , and are given sufficiently smooth functions and . Here, is a sufficiently large positive constant.

We introduce the Banach spaces of all continuous functions satisfying a Hölder condition for which the following norms are finite,

where is the space of the all continuous functions defined on with the usual norm

It is known that the differential expression [30]

defines a positive operator acting in with the domain

Therefore, we can replace the mixed problem (2.18) by the abstract boundary value problem (1.1). Using the result of Theorem 2.1, we can obtain the following theorem on stability.

Theorem 2.4. For the solution of mixed problem (2.18), the following estimates are valid:

where is independent of , , and .

3. Rothe Difference Scheme for Parabolic Equations with an Unknown Parameter

In this section, our focus is the well-posedness of the Rothe difference scheme

for approximately solving problem (1.1).

Let , , be the uniform grid space with step size , where is a fixed positive integer.

Throughout the section, denotes the linear space of grid functions with values in the Banach space .

Let be the Banach space of bounded grid functions with the norm

For , let be the Hölder space with the following norm:

Let us start with some lemmas we need in the following.

Lemma 3.1 (see [31]). The following estimates hold:

for some , which are independent of , where is a positive small number and is the resolvent of .

Lemma 3.2. The operator has an inverse and the following estimate is satisfied:

Let us now obtain the formula for the solution of problem (3.1). It is clear that the first order of accuracy difference scheme

has a solution and the following formula holds:

Applying formula (3.7) and the boundary condition

we can write

Since

we have that

Using Lemma 3.2, we get

Using formulas (3.7) and (3.12), we get

Since

we have that

Hence, difference equation (3.1) is uniquely solvable, and, for the solution, formulas (3.12) and (3.15) are valid.

Theorem 3.3. For the solution of problem (3.1) in , the stability estimates

hold, where is independent of , , , and .

Proof. From formulas (3.7) and (3.12), it follows that

Using this formula, the triangle inequality, and estimates (3.4), we obtain

The estimate (3.16) is proved. Using formula (3.15), the triangle inequality, and estimates (3.4), we obtain

for any . From that it follows estimate (3.17). Theorem 3.3 is proved.

Theorem 3.4. For the solution of problem (3.1) in , the almost coercive stability estimates

hold, where does not depend on , , , and .

Proof. Using formula (3.12), the triangle inequality and estimates (3.4), we obtain

Since [31]

we have estimate (3.21). Using formula (3.15), the triangle inequality, and estimates (3.4), (3.24), we obtain

for any . Therefore,

This estimate, triangle inequality, and (1.1) yield estimate (3.22). Theorem 3.4 is proved.

4. Applications

Now, we consider the applications of Theorems 3.3 and 3.4. The boundary value problem (2.18) for the parabolic differential equation is considered. The discretization of problem (2.18) is carried out in two steps. In the first step, we define the grid space

Let us introduce the Banach space of the grid functions defined on , equipped with the norm

To the differential operator generated by problem (2.18), we assign the difference operator by the formula

acting in the space of grid functions satisfying the conditions , . It is wellknown that is a strongly positive operator in . With the help of , we arrive at the boundary value problem

In the second step, we replace (4.4) with the difference scheme (3.1)

Theorem 4.1. The solution pairs of problem (4.5) satisfy the stability estimates

where and do not depend on ,, , and , .

Here, is the grid space of grid functions defined on with norm

The proof of Theorem 4.1 is based on Theorem 3.3 and the positivity property of the operator defined by formula (4.3).

Theorem 4.2. The solution pairs of problem (4.5) satisfy the almost coercive stability estimates

where and are independent of , , and , .

The proof of Theorem 4.2 is based on Theorem 3.4 and the positivity property of the operator defined by formula (4.3) and on the estimate

Note that, in a similar manner, we can construct the difference schemes of the first order of accuracy with respect to one variable for approximate solutions of boundary value problems (2.8) and (2.13). Abstract theorems given from above permit us to obtain the stability, the almost stability estimates for the solutions of these difference schemes.

5. Conclusion

In this work, the first order of accuracy Rothe difference scheme for the approximate solution of the boundary value problem of determining the parameter of a parabolic equation

in arbitrary Banach space with the strongly positive operator is studied. The well-posedness of the difference scheme is established. Some results in this paper in Hilbert space with self adjoint positive definite operator were obtained in the paper [25]. The investigation of this paper in arbitrary Banach space with the strongly positive operator permits us to obtain the stability and almost stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained. Of course, such type results for the solution of difference scheme for the following boundary value problems

in an arbitrary Banach space with positive operator and an unknown parameter hold.

Acknowledgment

The authors are grateful to Professor Allaberen Ashyralyev (Fatih University,Turkey) for his comments and suggestions to improve the quality of the paper.

References

M. Dehghan, “Determination of a control parameter in the two-dimensional diffusion equation,” Applied Numerical Mathematics, vol. 37, no. 4, pp. 489–502, 2001.

T. Kimura and T. Suzuki, “A parabolic inverse problem arising in a mathematical model for chromatography,” SIAM Journal on Applied Mathematics, vol. 53, no. 6, pp. 1747–1761, 1993.

Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography,” Inverse Problems, vol. 2, no. 5, pp. 373–397, 1999.

Y. S. Eidelman, Boundary value problems for di¤erential equations with parameters [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1984.

Y. S. Eidelman, “Conditions for the solvability of inverse problems for evolution equations,” Doklady Akademii Nauk Ukrainskoj SSR Serija A, no. 7, pp. 28–31, 1990 (Russian).

A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Di¤erential Equations, Operator Theory Advances and Applications, Birkhäuser, Boston, Berlin, 2004.

R. P. Agarwal and V. B. Shakhmurov, “Multipoint problems for degenerate abstract differential equations,” Acta Mathematica Hungarica, vol. 123, no. 1-2, pp. 65–89, 2009.

A. O. Ashyralyev and P. E. Sobolevskiĭ, “The linear operator interpolation theory and the stability of difference schemes,” Doklady Akademii Nauk SSSR, vol. 275, no. 6, pp. 1289–1291, 1984 (Russian).

A. Ashyralyev, I. Karatay, and P. E. Sobolevskii, “Well-posedness of the nonlocal boundary value problem for parabolic difference equations,” Discrete Dynamics in Nature and Society, no. 2, pp. 273–286, 2004.

A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, “Coercive solvability of the nonlocal boundary value problem for parabolic differential equations,” Abstract and Applied Analysis, vol. 6, no. 1, pp. 53–61, 2001.

A. Ashyralyev, “High-accuracy stable difference schemes for well-posed NBVP,” in Modern Analysis and Applications, vol. 191 of Operator Theory: Advances and Applications, pp. 229–252, 2009.

A. Ashyralyev, “A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 557–573, 2008.

Y. S. Eidelman, “Two-point boundary value problem for a differential equation with a parameter,” Dopovidi Akademii Nauk Ukrainskoi RSR Seriya A-Fiziko-Matematichni ta Technichni Nauki, no. 4, pp. 15–18, 1983 (Russian).

A. I. Prilepko, “Inverse problems of potential theory,” Matematicheskie Zametki, vol. 14, pp. 755–767, 1973, English translation Mathematical Notes, vol. 14, 1973

A. D. Iskenderov and R. G. Tagiev, “ The inverse problem of determining the right-hand sides of evolution equations in Banach space,” Nauchnyye Trudy Azerbaidzhanskogo Gosudarstvennogo Universiteta, no. 1, pp. 51–56, 1979 (Russian).

W. Rundell, “Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,” Applicable Analysis, vol. 10, no. 3, pp. 231–242, 1980.

A. I. Prilepko and I. A. Vasin, “Some time-dependent inverse problems of hydrodynamics with final observation,” Doklady Akademii Nauk SSSR, vol. 314, no. 5, pp. 1075–1078, 1990, English translation Soviet Mathematics. Doklady, vol. 42, 1991

A. I. Prilepko and A. B. Kostin, “On certain inverse problems for parabolic equations with final and integral observation,” Matematicheskii Sbornik, vol. 183, no. 4, pp. 49–68, 1992, English translation Russian Academy of Sciences. Sbornik Mathematics, vol. 75, 1993

A. I. Prilepko and I. V. Tikhonov, “Uniqueness of the solution of an inverse problem for an evolution equation and applications to the transfer equation,” Matematicheskie Zametki, vol. 51, no. 2, pp. 77–87, 1992, English translation Mathematical Notes, vol. 2, pp. 77–87, 1992

D. G. Orlovskii, “On a problem of determining the parameter of an evolution equation,” Differentsial'nye Uravneniya, vol. 26, no. 9, pp. 1614–1621, 1990, English translation Differential Equations, vol. 26, 1990

J. R. Cannon, Y. L. Lin, and S. Xu, “Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations,” Inverse Problems, vol. 10, no. 2, pp. 227–243, 1994.

M. Dehghan, “Finding a control parameter in one-dimensional parabolic equations,” Applied Mathematics and Computation, vol. 135, no. 1-2, pp. 491–503, 2003.

A. Ashyralyev, “On the problem of determining the parameter of a parabolic equation,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397–1408, 2011.

A. Ashyralyev and O. Demirdag, “A note on boundary value parabolic problems,” Vestnik of Odessa National University. Mathematics and Mechanics, vol. 16, no. 16, pp. 131–143, 2010.

A. Ashyralyev and O. Demirdağ, “A note on a problem of obtaining the parameter of a parabolic equation,” International Journal of Mathematics and Computation, vol. 11, no. 11, pp. 10–20, 2011.

C.-R. Ye and Z.-Z. Sun, “On the stability and convergence of a difference scheme for an one-dimensional parabolic inverse problem,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 214–225, 2007.

A. Ashyralyev and O. Demirdağ, “On the numerical solution of parabolic equation with the Neumann condition arising in determination of a control parameter,” AIP Conference Proceedings, vol. 1389, pp. 613–616, 2011.

A. Ashyralyev and A. S. Erdoğan, “On the numerical solution of a parabolic inverse problem with the Dirichlet condition,” International Journal of Mathematics and Computation, vol. 11, no. 11, pp. 73–81, 2011.

A. Ashyralyev and A. S. Erdogan, “Well-posedness of the inverse problem of a multidimensional parabolic equation,” Vestnik of Odessa National University. Mathematicsand Mechanics, vol. 15, no. 18, pp. 129–135, 2010.

A. Ashyralyev, “Fractional spaces generated by the positivite di¤erential and difference operator in a Banach space,” in Proceedings of the Mathematical Methods and Engineering, K. Taş, Ed., pp. 13–22, Springer, The Netherlands, 2007.

A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, Operator Theory Advances and Applications, Birkhäuser, Boston, Berlin, 1994.

Y. A. Smirnitskii and P. E. Sobolevskii, “Positivity of multidimensional di¤erence operators in the C-norm,” Uspekhi Matematicheskikh Nauk, vol. 36, no. 4, pp. 202–203, 1981 (Russian).

Y. A. Smirnitskii, Fractional powers of elliptic di¤erence operators [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1983.

P. E. Sobolevskiĭ, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971 (Russian).