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• "integral observation"
Clear All  1 Introduction The purpose of our study will be the questions of existence and uniqueness of solution { u ⁢ ( t , x ) ; p ⁢ ( x ) } {\{u(t,x);p(x)\}} of inverse problems for nonuniformly parabolic equations with additional condition of integral observation (1.1) ∫ 0 T u ⁢ ( t , x ) ⁢ χ ⁢ ( t ) ⁢ 𝑑 t = φ ⁢ ( x ) , x ∈ Ω . \int_{0}^{T}u(t,x)\chi(t)\,dt=\varphi(x),\quad x\in\Omega. For the first time, the inverse problems for nonstationary differential equations with condition ( 1.1 ) were considered in the paper [ 25 ] for uniformly parabolic equations u t - ∑ i

_{t}-a(x,t)u_{xx}-b(x,t)u_{x}-\mathrm{d}(x,t)u=p(x)g(x,t)+r(x,t),\quad(x,t)% \in Q\mathrel{\mathop{:}}=[0,l]\times[0,T], with the initial and boundary conditions (1.2) u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) , x ∈ [ 0 , l ] ; u ⁢ ( 0 , t ) = u ⁢ ( l , t ) = 0 , t ∈ [ 0 , T ] . u(x,0)=u_{0}(x),\quad x\in[0,l];\qquad u(0,t)=u(l,t)=0,\quad t\in[0,T]. In the inverse problem, the additional information is given by the integral observation (1.3) ∫ 0 T u ⁢ ( x , t ) ⁢ χ ⁢ ( t ) ⁢ d ⁢ t = φ ⁢ ( x ) , x ∈ [ 0 , l ] , \int_{0}^{T}u(x,t)\chi(t)\mathop{}\!dt=\varphi(x),\quad x\in[0,l], which is more practically realistic/feasible and general than the specification

derivatives Comput. Math. Appl. 51 2006 9–10 1539 1550  A. I. Prilepko and A. B. Kostin, On certain inverse problems for parabolic equations with final and integral observation, Russ. Acad. Sci. Siberian Math. 75 (1993), 473–490. Prilepko A. I. Kostin A. B. On certain inverse problems for parabolic equations with final and integral observation Russ. Acad. Sci. Siberian Math. 75 1993 473 490  A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textb. Pure Appl. Math. 231, Marcel Dekker, New

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method 501 Aleksey I. Prilepko, Vitaly L. Kamynin, Andrew B. Kostin Inverse source problem for parabolic equation with the condition of integral observation in time 523 V. P. Tanana A comparison of error estimates at a point and on a set when solving ill-posed problems 541 Alexandra Smirnova On TSVD regularization for a Broyden-type algorithm 551

hence we have used Tikhonov’s regularizations. The principle of solution was based on the dual representation of the square residual functional and this representation was used for the study and construction of solution algorithms. 1. Formulation of the problem of minimization of pollution concentration in Moscow region In this section we present the statement of the problem of minimization of pollu- tion concentration in the Moscow region based on the turbulent diffusion equa- tion with ‘local’ sources and integral observation, and the question of its unique

inverse problems for parabolic equations with final and integral observation. Mat. Sbornik (1992) 183, No. 4, 49–68 (in Russian). 5. A. I. Prilepko and A. B. Kostin, On inverse problems of finding coefficients in a parabolic equation. II. Sib. Mat. Zh. (1993) 34, No. 1, 147–162 (in Russian). 6. W. Rundell, Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data. Appl. Anal. (1980) 10, 231–242. 7. W. Greenberg, C. V. M. van der Mee, and P.F. Zweifel, Generalized kinetic equations. Integral Equations Oper

-Posed Problems 11, 5 (2003), 439–473. 4. A. Hurwitz and R. Courant, Theory of Functions. Nauka, Moscow, 1968 (in Russian). 5. A. I. Kozhanov, Composite Type Equations and Inverse Problems. VSP, Utrecht, 1999. 6. A. I. Prilepko and A. B. Kostin, On certain inverse problems for parabolic equations with final and integral observation. Russ. Acad. Sci., Sb., Math. 75, 2 (1993), 473–490. 7. A. I. Prilepko and A. B. Kostin, Inverse problems of the determination of the coefficient in parabolic equations. II. Sib. Math. J. 34, 5 (1993), 923–937. 8. S. G. Pyatkov, Solvability of some

term for differential equations. Izv. Acad. Nauk Az. SSR (1976) 2, 35–44 (in Russian). 7. D.G. Orlovskii, Determination of a parameter for a parabolic equation in a Hilbert structure. Mat. Zametki (1994) 55, No. 3, 109–117 (in Russian). 8. A. I. Prilepko, I. V. Tikhonov, Uniqueness of a solution to an inverse prob- lem for the evolution equation and its application to the transport equa- tion. Mat. Zametki (1992) 51, No. 2, 77–87 (in Russian). 9. A. I. Prilepko, A.B. Kostin, On some inverse problems for parabolic equa- tions with final and integral observation. Mat

integral observation. Mat. Sb. (1992) 183, No. 4, 49–68 (in Russian). 9. A. I. Prilepko and A.B. Kostin, On inverse problems of determining a co- efficient in a parabolic equation. I. Siberian Math. J. (1992) 33, No. 3, 146–155. 10. A. I. Prilepko and A.B. Kostin, On inverse problems of determining a co- efficient in a parabolic equation. II. Siberian Math. J. (1993) 34, No. 5, 923–937. 11. M.P. Vishnevskii, Solvability of an inverse problem for the parabolic equa- tion with convergence. Siberian Math. J. (1992) 33, No. 3, 402–408. 12. V.V. Solov′ev, Existence of a

. Prilepko, Inverse problems in the potential theory (elliptic, parabolic, and hyperbolic equations and transfer equation). Mat. Zametki (1973) 14, No. 6, 755{767 (in Russian). 35. A. I. Prilepko, Prediction-control; inverse and nonlocal problems for non- stationary equations. In: Distributed Systems: Optimization and Appli- cations in Economics and Environmental Sciences. Ural Branch of the Russian Acad. Sci., Ekaterinburg, 2000, 22{23 (in Russian). 36. A. I. Prilepko and A. B. Kostin, On some inverse problems for parabolic equations with ­ nal and integral observation  