İntegral Geometri Problemleri ve Transport Denklemleri için Ters Problemler

Bu çalışmada, son zamanlarda özellikle X-ışını, MRI, ultrason, sismik tomografi ve elektron mikroskobu gibi görüntüleme tekniklerinde ortaya çıkan gelişmelere paralel olarak büyük ilgi gören integral geometri problemleri IGP ve bu problemlerle ilgili transport denklemler için bazı ters problemler ele alınmış, bu alanda yapılan başlıca çalışmalar kısaca sunulmuştur

Integral Geometry Problems and Inverse Problems for Transport Equations

Bu çalışmada, son zamanlarda özellikle X-ışını, MRI, ultrason, sismik tomografi ve elektron mikroskobu gibi görüntüleme tekniklerinde ortaya çıkan gelişmelere bağlı olarak büyük ilgi gören integral geometri problemleri IGP ve bu problemlerle ilgili transport denklemler için bazı ters problemler ele alınmış, bu alanda yapılan başlıca çalışmalar kısaca sunulmuştur.

___

  • Agoshkov, VI. 1998. Boundary Value Problems for Transport Equations, Birkhauser, Boston.
  • Alekseev, AS. 1967. Inverse dynamical problems of seismology. In Some Methods and Algorithms for Interpretation of Geophysical Data. Nauka, Moscow, pp. 9-84.
  • Alekseev, AS., Dobrinsky, VI. 1975. Questions of practical application of dynamical inverse problems of seismology. Mathematical Problems of Geophysics, 6 (2):7-53.
  • Ambarzumian, VA. 1929. Über eine Frage der Eigenwerttheorie. Zeitschrift für Physik, 53: 690-695.
  • Amirov, A. 1986. Existence and uniqueness theorems for the solution of an inverse problem for the transport equation. Sib. Math. J., 27: 785-800.
  • Amirov, A. 2001. Integral Geometry and Inverse Problems for Kinetic Equations, VSP, Utrecht The Netherlands.
  • Amirov, A., Yıldız, M., Ustaoğlu, Z. 2009. Solvability of a problem of integral geometry via an inverse problem for a transport-like equation and a numerical method. Inverse Problems, 25: 095002.
  • Anikonov, YE. 1976. The solvability of a certain problem of integral geometry. Mat. Sb., 101(143): No: 2, 271–279.
  • Anikonov, YE. 2001. Inverse Problems for Kinetic and other Evolution Equations, VSP, Utrecht The Netherlands.
  • Anikonov, YE., Amirov, A. 1983. A uniqueness theorem for the solution of an inverse problem for the kinetic equation. Dokl. Akad. Nauk SSSR., 272 (6): 1292-1293.
  • Anlı, F., Yaşa, F. 2003. Nötron Transportu için Küresel Geometride Özdeğer Hesaplaması. KSÜ Fen ve Mühendislik Dergisi 6(2):28-33.
  • Antonenko, OF. 1967. Finite-difference scheme inversion for solving a one-dimensional dynamical inverse problem of seismology. In: Some Methods and Algorithms for Interpretation of Geophysical Data. Nauka, Moscow, 92-98.
  • Bal, G. 2009. Inverse Transport Theory and Applications. Inverse Problems, 25: 055006.
  • Bal, G., Tamasan, A. 2007. Inverse source problems in transport equations. SIAM J. Math. Anal., 39(1): 57-76.
  • iωU(x) - ∇. D∇U + μ(x)U(x) = 0, x∈Ω, α (6) v
  • U+3εLv(x) . D∇U = Λ(f)(x), x∈дΩ. з
  • Kireitov, VR. 1975. The problem of determining an optical surface from its representations. Funktsional. Anal. i Prilozhen. 10 (3): 45–54.
  • Klibanov, MV., Timonov, A. 2004. Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht The Netherlands.
  • Kostelyanets, PO., Reshetnyak, YG. 1954. Defining an absolutely additive function by specifying its values on half-spaces. Uspekhi Mat. Nauk, 9(3): 135–141.
  • Krein, MG. 1951. Solution of the inverse Sturm-Liouville problem. Dokl. Akad. Nauk SSSR, 76: 21–24.
  • Krein, MG. 1954. On a method of the effective solution of an inverse boundary value problem. Dokl. Akad. Nauk SSSR, 95: 767-770.
  • Kunetz, G. 1964. Generalization of the antiresonance operators to a certain number of reflections. Geophys. Prospect., 12: 283- 289.
  • Marchenko, VA. 1950. Some problems in the theory of second- order differential operators. Dokl. Akad. Nauk SSSR, 72: 457–560.
  • Marchenko, VA. 1955. Reconstruction of the potential energy from the phase of scattered waves. Dokl. Akad. Nauk SSR, 104: 635-698.
  • Mukhometov, RG. 1977. The problem of the recovery of a two- dimensional Riemannmetric and integral geometry. Dokl. Akad. Nauk SSSR 232 (1): 32-35.
  • Novikov, P. 1938. Sur le probleme inverse du potentiel. Dokl. Akad. Nauk SSSR, 18: 165-168.
  • Pestov, LN., Sharafutdinov, VA. 1988. Integral geometry of tensor fields on a manifold of negative curvature. Soviet Math. Dokl. 36(1): 203-204.
  • Plaksin, GI. 1966. On a problem of Gelfland. Dokl. Akad. Nauk SSSR, 170 (4) 783-785.
  • Romanov, VG. 1972. Some Inverse Problems for of Hyperbolic Type Equation,. Nauka, Novosibirsk.
  • Romanov, VG. 1978. Integral geometry on geodesics of an isotropic Riemannian metric, Dokl. Akad. Nauk SSSR, 241(2): 298-293.
  • Romanov, VG. 1987. Inverse Problems of Mathematical Physics, VSP, Utrecht The Netherlands.
  • Santalo, LA. 1976. Integral geometry and geometric probability, Addison-Wesley Publishing Company, Reading, Mass.
  • Santosa, F. 1982. Numerical scheme for the inversion of acoustical impedance profile based on the Gelfand-Levitan method. Geophys. J. Roy. Astr. Soc. 70:229-244.
  • Sharafutdinov, VA. 1981. Determining the characteristics of an optical body in a homogeneous medium from its images. In: Mathematical methods of Solving Direct and Inverse Problems of Geophysics, 123-148, Novosibirsk.
  • Sharafutdinov, VA. 1994. Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands.
  • Problem and Properties of the Misfit Functional. J. Inverse
  • Ill-Posed Problems, 6 (5): 431-452.
  • Uspenskii, SV. 1972. Reconstruction of a function specified by integrals over a family of ellipsoids. Sibirsk. Mat. Zh., 13 (6): 1374-1382.
  • Uspenskii, SV. 1977. The reconstruction of a function given by integrals over a family of conical surfaces. Sibirsk. Mat. Zh., 18 (3): 675-684.
  • Wiechert, E., Zoeppritz, K. 1907. Uber Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss, Goettingen, 4: 415-549.