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In this paper, we consider the backward problem for fractional in time evolution equations $ \partial _t^\alpha u(t)= A u(t) $ ∂ t α u (t) = Au (t) with the Caputo derivative of order $ 0 0 < α ≤ 1 , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results. [ABSTRACT FROM AUTHOR]

We investigate the inverse problem of numerically identifying unknown initial temperatures in a heat equation with dynamic boundary conditions whenever some overdetermination data are provided after a final time. This is a backward parabolic problem which is severely ill-posed. As a first step, the inverse problem is reformulated as a minimization problem for an associated Tikhonov functional. Using the weak solution approach, an explicit formula for the Fréchet gradient of the cost functional is derived from the corresponding sensitivity and adjoint problems. Then, the Lipschitz continuity of the gradient is proved. Next, further spectral properties of the input–output operator are established. Finally, the numerical results for noisy measured data are performed using the regularization framework and the conjugate gradient method. We consider both one- and two-dimensional numerical experiments using finite difference discretization to illustrate the efficiency of the designed algorithm. Aside from dealing with a time derivative on the boundary, the presence of a boundary diffusion makes the analysis more complicated. This issue is handled in the 2-D case by considering the polar coordinate system. The presented method implies fast numerical results. [ABSTRACT FROM AUTHOR]

The p-version of the finite element method is used in conjunction with the blending function method to investigate the nonlinear free vibration of a doubly curved shallow shell of elliptical plan-form. The effects of transverse shear deformations, rotary inertia, and geometrical nonlinearity are taken into account. The harmonic balance method is used to derive the equations of free motion. The resultant nonlinear equations are solved iteratively using the linearized updated mode method. The efficiency of the method is demonstrated through convergence study and comparison with published results. The effects of geometric parameters such as thickness, ellipse aspects, and curvatures on the backbone curves of a clamped doubly curved shallow shell of elliptical plan-form are studied. It is shown that the increase or decrease of the hardening behavior is dependent upon these parameters. [ABSTRACT FROM AUTHOR]