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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 xxt t uu bxtuxtsxtRs u (1) where ()δ x ,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 [1] (Chapter 2) and [2] (Chapter 1). The solvability problems for multidimen- sional inverse problems were considered in [2] (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 [5], 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 }, T QtsstT Ω:{(,)|0|| ||}, 0 Txtxt TxT , 1() tT CQ is the class of function continuous in s , con- tinuously differentiable in t , and defined on T Q. We let B denote the set of function )( tx,b such that (,) (Ω) T bxt C , (,) (,)bxt bxt . Theorem 1. If at a 0T 1 (,) () T f tsCQ and the condition 1 (0,) 2 fs s (3) is met, then for all 00 (0,_0), (1/40),TT Tα 0 α () 4(,) T ' tCQ fts 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, k fts k and let (,), 1,2, k bxt k be the solutions to the inverse problems with the data (,), 1,2, k fts 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) 1 12 12 (Ω)() 4 (,)(,)(,)(,) 1- TtT CCQ bxtb xtftsf ts ρ , (4) D. K. DURDIEV ET AL. Copyright © 2010 SciRes. AM 125 where . T T 0 2. Construction of a System Integral Equations for Equivalent Inverse Problems We represent the solution of problem (1) as 1 (,,)(| |)(,,). 2 uxts θts xvxts (5) where 1)( t for ,0t ,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 xx uu δxδts , and obtain the problem for the function v : 2 ,0 1 (,)(||)(,,), 2 (,) , 0, 0. tt xxt t vv bxtδts xvxts xt Rs v (6) It follows from the d’Alembert formula that the solu- tion of problem (6) satisfies the integral equation Δ(,) 2 11 (,,)(,) (||)(,,) 22 , (,) , 0, t xt vxts bξτδτ sξvξτ s dξdτxt 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 (,,)(, ) 4 1 (, )(,,), 2 , xts xts t Υxts vxts bξsξdξ bξτvξτsdτdξ ts x (8) where the domain (,,) x ts is defined by () (,,) (,),2 ,0,. 2 x ts xtsst x xts s ts const By differentiating the equality (8), we obtain () 2 () 2 1 (,,) , 82 2 , 22 1(, )(, , ), . 2 t xts t xts xtsxts vxts b xts xts b btxv txsdtsx (9) It is obvious that 1 (,)(0,,)(0,,) 2 f tsu tsv ts for 0t. Moreover, the function (, ) f ts 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 2 2 1 (,), 422 ,, , , (,) . t ts t ts T tsts ftsb bξtξvξtξsdξ ts Q We rewrite this equality, replacing ()/2ts with || x and ()/2ts with t, and solve it for (,).bxt We obtain - (,)4(||, ||)4, (, ||, ||), ||. x ' t x t bxtftx txbξtxξ vξtxξtxdξtx (10) Let (,,) ,0 T x tsxs t Txs tT The domain T in the space of the variables ,, x t and s is a pyramid with the base t and vertex (0,,/2)TT . To find the value of the function b at (,) x t, it is hence necessary to integrate (,)bxt over the inter- val with the endpoints (| |,) x t and (||, ) x t and to integrate the function (,,) t vxtsover the interval with the endpoints (||, ,||) x tt x and (||,,||), x tt 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 , 2 2 , 28 1 ),,( ),( ),,( 2 1 txb stxstx b stxstx bstxv tx stx t The operator A is defined on the set of functions T C and, according to (9), (10), has the form 12 (, ), A AA where () 2 12 1 () 2 2 2 22 12 1,{(,) 2 1, 82 2 ,}, 22 4(||, ||)4(,) 1 (,,), 8 xts xts x ' t x Atxtx txstxs txstxs d Aftxtx tx tx 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 12 max ,. TT TCC Let s be the set of TT T C that satisfy the following conditions: TT 00 , where 0 01 02 , (0, 4(||, ||). ' t ψψψ f txtx It is obvi- ously, that 0 0 4(,) T ' tTT TCQ fts 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 .20 00 TTT Furthermore, one has 2 1012 2 2 2 2 0 0 202 2 12 2 1, 2 1 ,, , 82 , 22 5, 28 4, 1 ,,, 8 ,10 xts xts T x x Atx txs tx s tx stx s tx sT d Aбtx tx txxt xt d 2 0 0. T T Therefore, if * 0 1/10T , 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 , 2 8 1 ,, ,, 2 , 22 , 28 1 ,, , 2 1 2 2 1 2 2 1 1 1 2 2 2 )1( 2 1 1 2 2 2 1 2 2 1 1 1 2 2 sxtsxt sxt sxt sxt sxtsxt sxt sxt xt AA stx stx D. K. DURDIEV ET AL. Copyright © 2010 SciRes. AM 127 12 22 12 0 , 22 5, 2T tx stx s T d 12 12 22 22 1 1 1(1) 22 212 211 11 22 11 22 12 0 4 , ,, 1 , , 8 , 1 ,, 8 , , 40 . x x T AA txtxtx xt xt tx tx tx xt xt dT It follows from the preceding estimates that if 0 0401 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 0 10 n nn converges not slower than the series T n n T 0 010 We now prove Theorem 2. Since the conditions Theorem 1 hold, the solution belong to the set S and .2,1,2 0 i T i Let 2,1, k k be vector functions which are the solution of the Equation (11) with the data ,, 1,2, k fts k respectively, i. e., kk A From the previous results in the proof of Theorem 1, it follows that 12 12 1 012 ,,4 ,, 40,1, 2. kk Q CtT T xtsf tsfts Tk Therefore, one has 1 12 12 12 4, , TT t fts ftsQ CT The last inequality gives 1 12 12 4,, 1 Tt fts ftsQ CT (12) The stability estimate (4) follows from the inequality (12). 4. References [1] V. G. Romanov, “Inverse Problems of Mathematical Physics,” in Russian, Publishing House “Nauka”, Mos- cow, 1984. [2] V. G. Romanov, “Stability in Inverse Problems,” in Rus- sian, Nauchnyi Mir, Moscow, 2005. [3] D. K. Durdiev, “A Multidimensional Inverse Problem for an Equation with Memory,” Siberian Mathematical Jour- nal, Vol. 35, No. 3, 1994, pp. 514-521. [4] 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. [5] 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. |