 Applied Mathematics, 2010, 1, 124-127 doi:10.4236/am.2010.12016 Published Online July 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Problem of Determining the Two-Dimensional Absorption Coefficient in a Hyperbolic-Type Equation Durdimurat K. Durdiev Bukhara State University, Bukhara, Uzbekistan E-mail: durdiev65@mail.ru Received March 25, 2010; revised May 16, 2010; accepted May 29, 2010 Abstract The problem of determining the hyperbolic equation coefficient on two variables is considered. Some addi-tional information is given by the trace of the direct problem solution on the hyperplane x = 0. The theorems of local solvability and stability of the solution of the inverse problem are proved. Keywords: Inverse Problem, Hyperbolic Equation, Delta Function, Local Solvability 1. Statement of the Problem and the Main Results We consider the generalized Cauchy problem 2<0(, )=(, ), (, ), >0, 0,tt xxttuu bxtuxtsxtRsu  (1) where ()δx,t is the two-dimensional Dirac delta func- tion, ()bx,t is a continuous function, s is a problem parameter, and ()ux,t,s . We pose the inverse problem as follows: it is required to find absorption coefficient ()bx,t if the values of the solution for are known, i.e., if the function (0)( )00u,t,sft,s, t, s. (2) Definition. A function ()bx,t such that the solution of problem (1) corresponding to this function satisfies rela-tion (2) is called a solution of inverse problem (1), (2). The inverse problem posed in this paper is two-dim- ensional. For the case where (,) ()bxt bx the solv-ability problems for different statements of problems close to (1), (2) were studied in  (Chapter 2) and  (Chapter 1). The solvability problems for multidimen-sional inverse problems were considered in  (Chapter 3), [3,4], where the local existence theorems were proved in the class of functions smooth one of the variables and analytic in the other variables. In , the problems of stability and global uniqueness were investigated for inverse problem of determining the nonstationary poten-tial in hyperbolic-type equation. In this paper, we prove the local solvability theorem and stability of the solution of the inverse problem (1), (2). Let :{(,)|0 }, TQtsstT Ω:{(,)|0|| ||}, 0Txtxt TxT , 1()tTCQ is the class of function continuous in s, con-tinuously differentiable in t, and defined on TQ. We let B denote the set of function )( tx,b such that (,) (Ω)Tbxt C, (,) (,)bxt bxt . Theorem 1. If at a 0T 1(,) ()TftsCQ and the condition 1(0,) 2fs s (3) is met, then for all 00(0,_0), (1/40),TT Tα 0α ()4(,)T'tCQfts the solution to the inverse problems (1), (2) in the class of function (,)bxt B exists and is unique. Theorem 2. Let the conditions in Theorem 1 hold for the functions (,), 1,2,kfts k and let (,), 1,2,kbxt k be the solutions to the inverse problems with the data (,), 1,2,kfts k respectively. Then the following esti-mate is valid for 00(0, ), ( () TTT is defined in the same way as in proof of the Theorem 1) 112 12(Ω)()4 (,)(,)(,)(,)1-TtTCCQbxtb xtftsf tsρ, (4) D. K. DURDIEV ET AL. Copyright © 2010 SciRes. AM 125where .TT0 2. Construction of a System Integral Equations for Equivalent Inverse Problems We represent the solution of problem (1) as 1(,,)(| |)(,,).2uxts θts xvxts (5) where 1)( t for ,0t ,0)( t for 0t, (,vx ,)ts is a some regular function. We substitute the Expression (5) in (1), take into ac-count that (||)/2ts x satisfies (in the generalized sense) the equation () ()tt xxuu δxδts , and obtain the problem for the function v: 2,01(,)(||)(,,), 2 (,) , 0, 0. tt xxttvv bxtδts xvxtsxt Rsv  (6) It follows from the d’Alembert formula that the solu-tion of problem (6) satisfies the integral equation Δ(,)211(,,)(,) (||)(,,)22 , (,) , 0,txtvxts bξτδτ sξvξτsdξdτxt R s (7) where .,0),(),( txtxxttx  We use the properties of the - function and easily ob-tain the relation in a different form: ()2()2(,,)1(,,)(, )41 (, )(,,),2 ,xtsxtstΥxtsvxts bξsξdξbξτvξτsdτdξts x (8) where the domain (,,)xts is defined by ()(,,) (,),2 ,0,.2xtsxtsst xxtssts const    By differentiating the equality (8), we obtain ()2()21(,,) ,82 2 ,221(, )(, , ), .2txtstxtsxtsxtsvxts bxts xtsbbtxv txsdtsx   (9) It is obvious that 1(,)(0,,)(0,,) 2ftsu tsv ts for 0t. Moreover, the function (, )fts be must sat-isfy the condition (9). We set 0x in the equality (9), use the fact that the function (,)bxt is even in x, and obtain the relation 221 (,), 422 ,, , , (,) .ttsttsTtstsftsbbξtξvξtξsdξts Q We rewrite this equality, replacing ()/2ts with ||x and ()/2ts with t, and solve it for (,).bxt We obtain -(,)4(||, ||)4, (, ||, ||), ||. x'txtbxtftx txbξtxξvξtxξtxdξtx  (10) Let (,,) ,0Txtsxs t Txs tT  The domain T in the space of the variables ,,xt and s is a pyramid with the base t and vertex (0,,/2)TT . To find the value of the function b at (,)xt, it is hence necessary to integrate (,)bxt over the inter-val with the endpoints (| |,)xt and (||, )xt and to integrate the function (,,) tvxtsover the interval with the endpoints (||, ,||)xtt x and (||,,||), xtt x which belong to the domain T. One can rewrite the system of Equations (9) and (10) in the nonlinear operator form, ,A (11) where D. K. DURDIEV ET AL. Copyright © 2010 SciRes. AM 126 ),(2,22,281),,(),(),,(21txb stxstxb stxstxbstxvtxstxt The operator A is defined on the set of functions TC and, according to (9), (10), has the form 12(, ),AAA where ()212 1()22222121,{(,)21,82 2,},224(||, ||)4(,)1 (,,),8 xtsxtsx'txAtxtxtxstxstxstxs dAftxtx txtx txxt     2 ,d.xt  At fulfillment of the condition (3) the inverse problem (1), (2) is equivalent to the operator Equation (11). 3. Proofs of the Theorems Define  12max ,.TTTCC Let s be the set of  TT TC that satisfy the following conditions: TT00, where 001 02, (0, 4(||, ||).'tψψψftxtx It is obvi-ously, that 004(,) T'tTTTCQfts Q. Now we can show that if T is small enough, A is a contraction mapping operator in S. The local theorem of existence and uniqueness then follows immediately from the con-traction mapping principle. First let us prove that A has the first property of a contraction mapping operator, i.e., if ,S then SA  when T is small enough. Let S. It is then easy to see that .2000 TTT Furthermore, one has 21012222200202 21221,21,, ,82,225,284,1,,,8,10xtsxtsTxxAtxtxstx stx stx stx sTdAбtxtx txxtxt d          200.TT Therefore, if *01/10T, then for 0,0 TT  the operator A satisfies the condition SA . Consider next the second property of contraction mapping operator for A i.e., if SS21 ,, then 11AA  11 with 1, when T is small enough. Let 12,.SS Then one has      2,281,,,,2,22,281,,,2122122111222)1(21122212211122sxtsxt sxt sxtsxtsxtsxtsxtsxt xt AAstxstx D. K. DURDIEV ET AL. Copyright © 2010 SciRes. AM 127  1222120,225,2Ttx stx sTd        12 1222 22111(1)22212211112211221204 , ,, 1 , , 8 , 1,, 8 , , 40 . xxTAAtxtxtxxt xttxtx txxt xtdT         It follows from the preceding estimates that if 00401T, then for 0,0 TT  the operator A is a contraction operator with 0/TT on the set S. Therefore, the Equation (11) has a unique solution which belongs to S according to the contraction mapping prin-ciple. The solution is the limit of the sequence n, 0,1,2,...,n where nn A 10 ,0, and the series  010nnn converges not slower than the series   TnnT 0010  We now prove Theorem 2. Since the conditions Theorem 1 hold, the solution belong to the set S and .2,1,2 0 iTi Let 2,1, kk be vector functions which are the solution of the Equation (11) with the data ,, 1,2,kfts k respectively, i. e., kk A From the previous results in the proof of Theorem 1, it follows that   1212 1012,,4 ,,40,1, 2.kkQCtTTxtsf tsftsTk Therefore, one has 11212 124, ,TTtfts ftsQCT   The last inequality gives  112124,,1Ttfts ftsQCT   (12) The stability estimate (4) follows from the inequality (12). 4. References  V. G. Romanov, “Inverse Problems of Mathematical Physics,” in Russian, Publishing House “Nauka”, Mos-cow, 1984.  V. G. Romanov, “Stability in Inverse Problems,” in Rus-sian, Nauchnyi Mir, Moscow, 2005.  D. K. Durdiev, “A Multidimensional Inverse Problem for an Equation with Memory,” Siberian Mathematical Jour-nal, Vol. 35, No. 3, 1994, pp. 514-521.  D. K. Durdiev, “Some Multidimensional Inverse Prob-lems of Memory Determination in Hyperbolic Equa-tions,” Journal of Mathematical Physics, Analysis, Ge-ometry, Vol. 3, No. 4, 2007, pp. 411-423.  D. K. Durdiev, “Problem of Determining the Nonstation-ary Potential in a Hyperbolic-Type Equation,” Journal of Theoretical and Mathematical Physics, Vol. 2, No. 156, 2008, pp. 1154-1158.