Abstract
In this paper, we study an inverse source problem for the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. The problem is severely ill-posed in the sense of Hadamard. To regularize the unstable solution, we apply the Tikhonov method regularization solution and obtain an a priori error estimate between the exact solution and regularized solutions. We also propose methods for both a priori and a posteriori parameter choice rules. In addition, we verify the proposed regularized methods by numerical experiments to estimate the errors between the regularized and exact solutions.
Highlights
Numerical solutions of Rayleigh–Stokes problem for a heated generalized second-grade fluid with fractional derivatives have been considered and developed in some previous papers by Dehghan et al [4,5,6,7]
In this paper, we consider the Rayleigh–Stokes problem for a generalized second-grade fluid model with fractional derivative ⎧⎪⎪⎪⎪⎪⎨∂ut(ux,t)(=1 +γ 0, ∂tα ) u = f (x)χ(t),(x, t) ∈ Ω × (0, T), x ∈ ∂Ω,⎪⎪⎪⎪⎪⎩uu((xx, 0) = u0(x), T) = g(x), x ∈ Ω, x ∈ Ω, (1.1)where Ω ⊂ Rd (d = 1, 2, 3) is a smooth domain with boundary ∂Ω, and T > 0 is a given time
Γ > 0 is a constant, u0 is the initial data in L2(Ω), ∂t = ∂/∂t, and ∂tα is the Riemann–Liouville fractional derivative of order α ∈ (0, 1) defined by [1, 2]
Summary
Numerical solutions of Rayleigh–Stokes problem for a heated generalized second-grade fluid with fractional derivatives have been considered and developed in some previous papers by Dehghan et al [4,5,6,7]. The main results are given, including the Tikhonov regularization method and its stability estimates under a priori and a posteriori parameters. 2 Regularization of the inverse source problem by the Tikhonov method 2.1 Preliminaries we introduce some useful definitions and preliminary results.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
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 call
