A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation
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The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.
Keywordsfourth-order pseudoparabolic equation Cauchy problem equations with discontinuous coefficients fundamental solution
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- 2.V. D. Gilev and G. A. Shadrin, “Construction of a Fundamental Solution to the Equation Governing the Flow of Fluid in Fissured Media,” in Computational Mathematics and Programming (Izd. Lenin Mosk. Gos. Ped. Inst.I, Moscow, 1976), No. 4, pp. 102–111 [in Russian].Google Scholar
- 3.N. I. Yurchuk, S. N. Baranovskaya, and V. I. Yashkin, “On Classical and Weak Classical Solutions to Hyperbolic Equations,” in Abstracts of the Int. Conf. Devoted to the 75th Anniversary of L. D. Kudryavtsev, Moscow, 1998, p. 71 [in Russian].Google Scholar
- 6.D. Mangeron, “New Methods for Determining Solutions of Mathematical Models Governing Polyvibrating Phenomena,” Bul. Inst. Polit. Din. Zasi 14(18), 433–436 (1968).Google Scholar
- 8.M. Kh. Shkhanukov and A. P. Soldatov, “Boundary Value Problems subject to the General Nonlocal Samarskii,s Condition for Pseudoparabolic Equations of a High Order,” Dokl. Akad. Nauk SSSR 297, 547–552 (1987).Google Scholar
- 11.V. I. Zhegalov and E. A. Utkina, “On a Pseudoparabolic Third-Order Equation,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 10, pp. 73–76 (1999).Google Scholar
- 13.I. G. Mamedov, “A New Type Cauchy Problem for a Pseudoparabolic Equation with a Fourth-Order Dominating Derivative with Nonsmooth Coefficients,” in Abstracts of the XII Int. Conf. on Mathematics and Mechanics Devoted to the 70th Anniversary of the Birth of B. A. Iskenderov, Baku, 2006, p. 108 [in Russian].Google Scholar
- 14.I. G. Mamedov, “A New Type Cauchy Problem for a Pseudoparabolic Equation with a Fourth Order Dominating Derivative with Non-Smooth Coefficients,” Mathematics, No. 5, pp. 34–40 (2007).Google Scholar
- 15.I. G. Mamedov, “A New Type Goursat Problem for Loaded Volterra Hyperbolic Integro-Differential Vector Equations of the Fourth Order with Nonsmooth Matrix Coefficients,” Izv. Nat. Acad. Nauk Azerbaijana, Ser. FTMN. 26(2), 74–79 (2006).Google Scholar