Abstract
For parabolic Shilov equations with continuous coefficients, the problem of finding classical solutions that satisfy a modified initial condition with generalized data such as the Gelfand and Shilov distributions is considered. This condition arises in the approximate solution of parabolic problems inverse in time. It linearly combines the meaning of the solution at the initial and some intermediate points in time. The conditions for the correct solvability of this problem are clarified and the formula for its solution is found. Using the results obtained, the corresponding problems with impulse action were solved.
Highlights
In the medium Rn, we consider a certain process whose evolution u(t; x) during the time (0; T] is described by the partial differential equation: ztu(t; x) P t; izxu(t; x), (t; x) ∈ Π(0;T] ≔ (0; T] × Rn, (1) in whichP t; izx ak(t)i|k|zkx, (2)|k|≤p differential expression of order p > 1 with continuous coefficients ak(·) on the set [0; T]
I is an imaginary unit, zkx is a partial derivative of a variable x of order k, and Rn is a real Euclidean space of dimension n with a scalar product (·, ·) and norm ‖x‖ ≔ (x, x)1/2
Similar problems for parabolic Shilov equations with p ≠ h still remain in the state of expectation
Summary
In the medium Rn, we consider a certain process whose evolution u(t; x) during the time (0; T] is described by the partial differential equation: ztu(t; x) P t; izxu(t; x), (t; x) ∈ Π(0;T] ≔ E results of these studies naturally complement and generalize the classical theory of the Cauchy problem for parabolic Petrovsky equations [17,18,19]. In [26], developing Tikhonov’s idea, a method for finding an approximate solution of the inverse thermal conductivity problem is proposed It is based on the replacement of this problem by a corresponding problem with a nonlocal by time condition, in which a partial shift of the condition is carried out at t T at the initial moment t 0: u(t; ·)|t 0 + ]u(t; ·)|t T f,. Various questions concerning the correctness of such problems and methods of solving them under certain conditions on the input data are considered In this case, similar problems for parabolic Shilov equations with p ≠ h still remain in the state of expectation.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Mathematics and Mathematical Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
