Speciality:
10.01.03 (Literature of peoples of foreign countries (with indication of a literature))
Birth date:
12.09.1972
E-mail: Keywords: PDE of composite type, initial-boundary value problems.
Subject:
Ph.D. thesis is devoted to investigation of non classical PDE of a composite type. Such equations appear in mathematical models of waves processes in mediums with a non-isotropic dispersion such as ionic-acoustical waves in plasma, spin waves in ferromagnetics and antiferromagnetics etc. Symbolical methods was applied to solving equations of this type and corresponding initial-boundary value problems. Classical and generalised resolvability of initial-boundary value problems was investigated for some cases. The existence and uniqueness of solution of such initial-boundary value problems is proved by methods of the theory of dynamic potentials. Was shown, that the side conditions are necessary for uniqueness of a solution of tasks with moving boundaries. Such conditions were formulated, and with the help of method of parametric prolongation the solution of such problem was constructed. These researches give theoretical justification of series of nontrivial physical phenomena and also allow to find out the boundaries of applicability of considered models. Recent investigations are devoted to construction numerical methods for initial-boundary value problems for PDE of composite type describing waves processes in mediums with non-isotropic dispersion. Numerical method for calculation such problems in unlimited area was constructed in 2001. The new method was successfully tested on some initial-boundary value problems for PDE of composite type including non-linear PDE.
Main publications:
Alshin A. B., Pletner Yu. D., Sveshnikov A. G. Odnoznachnaya razreshimost zadachi Dirikhle dlya uravneniya ionno-zvukovykh voln v plazme // Doklady Akademii nauk, 1998 g., t. 361, # 6, s. 749–751.
Alshin A. B., Pletner Yu. D., Sveshnikov A. G. Razreshimost odnoi nachalno-kraevoi zadachi dlya uravneniya ionno-zvukovykh voln. // Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 1996 g., t. 36, # 10, s. 180–189.
Alshin A. B., Pletner Yu. D., Sveshnikov A. G. O razreshimosti nestatsionarnogo analoga zadachi Lavrenteva–Sedogo–Keldysha dlya uravneniya ionno-zvukovykh voln // Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 1997 g., t. 37, # 12, s. 1520–1525.
Alshin A. B., Pletner Yu. D., Sveshnikov A. G. K zadache Dirikhle dlya dvumernogo uravneniya ionno-zvukovykh voln // Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 1998 g., t. 38, # 10, s. 1672–1679.
Alshin A. B. Nachalno-kraevye zadachi dlya uravneniya sostavnogo tipa s dvizhuschimisya granitsami // Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 2002, t. 42, # 2, s. 171-184.
