Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions

On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors

“…In order to show that the constructed three-level discrete scheme on this special nonuniform time grid ω t is stable and additional grid points really reduce the global error of the discrete solution, we give also the errors Z 3 of the classical three-level discrete scheme (14) when the time grid is uniform and it has N discrete points. The results of the computational experiments are presented in Table 1, where Z 1 is the error for the discrete solution of the scheme ( 12) and ( 13), Z 2 is the error for the discrete solution of the classical finite-difference scheme (14) on non-uniform time grids, and Z 3 is the error of the solution of the symmetrical scheme (7) when the time grid is uniform in [0, 1] and it has N points.…”

“…Errors ∥Z 1 ∥ ∞ and experimental convergence rates ρ(τ) at T = 1 for the discrete solution of the scheme ( 12) and ( 13) and errors ∥Z 2 ∥ ∞ and experimental convergence rates ρ(τ) for the discrete solution of the finite-difference scheme (14) for a sequence of time steps τ. ∥Z 3 ∥ ∞ is the error of the discrete solution of the the symmetrical scheme (7) when the time grid is uniform and it has N points. As expected, the new three-level discrete scheme is stable and preserves the second order of convergence.…”

“…The general convergence framework states that the stability and approximation properties guarantee the convergence of discrete solutions [1,2]. The deep, broad, and constructive theories of approximation and stability are developed, and they cover various important topics dealing with non-smooth data [3][4][5], weak solutions [6][7][8], energy and maximum principle stability estimates [1,2,9,10], ill-posed and inverse problems [11,12], and nonlocal mathematical models including fractional derivatives [13][14][15][16].…”

Identification of a Time-Dependent Source Term in a Nonlocal Problem for Time Fractional Diffusion Equation

“…Inverse source problems for TFDe have been intensively investigated by many researchers under various initial, boundary and overdetermination conditions. The inverse problem of finding of a space-dependent source term from final temperature distribution was considered in [4,6,14,15,21,22,30,37] and recovering a space-dependent source term from total energy measurement has been discussed in [8,24,25]. For a TFDe inverse problem of identification a time-dependent source term from temperature measurement at the selected point in the spatial domain was considered in [5,19] and determining a time-dependent source term from integral type overdetermination condition was studied in [2,3,16,18].…”

Time-periodic Poiseuille-type solution with minimally regular flow rate

“…However, in real applications, one usually does not have data defined by smooth functions, and it is important to study the case of minimal regularity of data. The nonstationary Poiseuille-type solution with a prescribed initial condition and given flow rate F (t) belonging to L 2 (0, T ) was studied in [22], where a new class of weak solutions was introduced, and the unique existence of the solution in such class was proved. The goal of the present paper is to extend the result obtained in [22] to the case of time-periodic Poiseuille-type solutions.…”