Approximate solution of inverse boundary problem for the heat exchange problem by A.N. Tikhonov regularization method
AAPPROXIMATE SOLUTION OF INVERSE BOUNDARY PROBLEM FOR THE HEAT EXCHANGE PROBLEM BY A.N. TIKHONOV’S REGULARI-ZATION METHOD
V.F. Mirasov , A.I. Sidikova Mirasov Vadim Faritovich - Post-graduate student, Calculating Mathematics Department, South Ural State University, Chelyabinsk, Russia Sidikova Anna Ivanovna – PhD. (Physics and Mathematics), associate professor, Calculating Mathematics Department, South Ural State University, Chelyabinsk, Russia Corresponding author: [email protected]
In this paper the approximate solution of the heat exchange problem by A.N. Tikhonov’s regularization method is presented. The errror estimation of this ap-proximate is obtained.
The direct problem definition
Let the thermal process is described by the equation , (1) , (2) , (3) , , (4) (5) – is a known number. , (6) Consider a classical solution of the problem (1-6), i.e. . The existence and uniqueness of the solution follows from the theorem formulated in [3], p. 190. The solution of the problem (1-6) has the form: (7) . (8)
The solution smoothness analysis:
It follows from (8) that (9) It follows from (7-9) that . (10) Proceed to the examination of the function continuity. For this purpose we differentiate the common term of the series (7) relative to t : . (11) By Abel sign and from (11) it follows that for any sufficiently small the series of the deriva-tives converges uniformly on the rectangle . Thereby . (12) Using (10) and (12) we get that the solution of the problem (1-6), which has the form (7) is classical. Using (4),(7), and(9) we get for any (13) The statement of the inverse boundary-value problem uppose, that in direct problem definition (1-6) the function , which defines boundary condition (3), isn’t known and it should be evaluated. Therefore the additional condition is entered: (14) Using (7) and (14) we get . (15) Suppose that if , then there is a decision , which satisfies conditions (4) and (5), but it is not known and instead of it we have and , for which (16) We need to find the approximate decision and get the estimation using and . Let's enter the linear operator , which transforms the space to and is given by following где (17) . (18) Note, that if and condition (4) is realized, the inverse boundary-value problem (1-2), (5), (6), (14), (16) is equivalent to the next integral equation: . (19) It is known that the task of theVolterr equation of the 1 st sort solution in space is incorrect, so for its decision we use the A.N.Tikhonov’s regularization method [2]. The 2 nd order of the Tikhonov’s regularization method This method consists in a reduction of the equation (19) to the variation problem, that depends on parameter . (20) The task (20) is equivalent to the integro-differential equation (21) - operator ,which is associated with , and . It is known (see [2]) that for any and there is an unambiguous solution of the equation (21).
The value of the regularization parameter can be defined from the residual principle [4], which satisfies the equation . (22) It is known that if , then the equation (22) has the unambiguous solution . Thus, the approximate solution of the equation (19) csn be determined by the formula: (23)
The estimation of the error
For the error estimation the continuity module will be entered: . (24) The estimation proof is presented in work [5] , (25) where is determined by (23). Consider the expansion of the inverse problem (1), (2), (5), (6), (14) on a half-line . For this pur-pose let’s enter the functions and for which и (26) и (27) The continuity of functions and follows from (10) and (13). Using (26) we get that the func-tion is the solution of the problem , (28) , (29) , (30) , (31) Function needs to be defined while (32) Let's designate as the linear variety such that if and only if , (33) where satisfies condition (4). Let's designate as linear operator, which acts from to and which is defined on the set by following: , (34) where , and is the solution of the problem (28), (29), (31) and (32). For the operator we enter the continuity module : (35) Using (24), (33-35), (13) we get . (36) For an upper estimation of the functions we solve the problem (28-31) uing Fourier trans-formation on t on a half-line .
Denote this transformation through . Thus the task (28-31) can be reduced to the following , (37) where [ ], (38) , (39) where [ ]. The solution of the problem (37) has the form , , (40) where . and need to be defined. It follows from (38) that . (41) It follows from (39) that . (42) Using (41) and (42) we get (43) It follows from (40-43) that (44) where , а
It follows from conditions (4) and (5) that (45) The operator can be dilatated on all space without designation changing: (46) where
It follows from (46) that is the injective linear bounded operator. The relation (45) defines the the addition operator , which transforms the space to , и (47) (48) Enter the correctness class , (49) where is the sphere in the space with the center in zero with the radius . Let’s designate as the subset of such that , (50) Using (47-49) and (50) we get (51) Enter the continuity module of the operator on the spase . . (52) Using (50-52), (35) and using the izometry of the transformation we can get . (53) The estimation, that follows from (25), (36) and (53) has the form . (54) Let’s proceed to the estimation of the functions . For this purpose we estimate the function . As this function is limited on any segment we get that there is the number such that (55) when (56) Let's define the number such a way that for any . (57) Using (57) we get for any (58) So, when we have (59) If , then from (58) and (59) it follows that when has the form , (60) using (60) and using the theorem, proved in [6] when , (61) Or . So, it follows from (60), (61) and (54) that for rather small values we get eference Alifanov O.M., Artyukhin E.A. The extremal methods of ill-defined problems solution // М.: Nauka, 1988, –287 p. 2.
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