In this paper, we focus on designing a well-conditioned Glarkin spectral methods for solving a two-sided fractional diffusion equations with drift, in which the fractional operators are defined neither in Riemann-Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann-Liouville fractional integral operators on Lp([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well-conditioned—the condition number of the stiff matrix can be reduced from O(N2α) to O(Nα), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion.

In this paper, we deal with the wave equation with acoustic boundary conditions. The exponential stabilization is obtained by Lyapunov approach and Riemannian geometry method. We then apply our main theorem to the wave equations with memory type acoustic boundary conditions, which is not available in the literature and give an example in the end.

The Dirichlet, Neumann and mixed boundary value problems for the linear second-order scalar elliptic differential equation with variable coefficient in a bounded three-dimensional domain are considered. The PDE right-hand side belongs to $H^{-1}(\Omega)$ or $\widetilde{H}^{-1}(\Omega),$ when neither classical nor canonical co-normal derivatives are well defined. Using the two-operator approach and appropriate parametrix (Levi function) each problem is reduced to different systems of boundary domain integral equations (BDIEs). Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and as well as invertibility of the BDIE operators are analysed in appropriate Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.

We consider a linear RLC network, modeled by differential-algebraic equations, containing distributed semiconductor devices, modeled by multi-dimensional parabolic-elliptic equations. With appropriate coupling conditions, we obtain a system of multidimensional partial differential-algebraic equations. For the resulting system, we prove an existence and uniqueness result, and study the asymptotic behavior of the solutions.

A linear manifold ${\mathcal K}_2$ of evolutionary equations for a pseudovector field on ${\Bbb R}^3$ is described. An infinitisimal shift of each equation is determined by a second-order differential operator of divergent type. All operators are invariant with respect to space translations in ${\Bbb R}^3$, relative to time translations, and they are transformed by covariant way relative to rotations of ${\Bbb R}^3$. It is proved that the linear space ${\mathcal M}_2 \subset {\mathcal K}_2$ of differential operators preserving solenoidal property and unimodularity of the field is one-dimensional and an explicit form of such operators is found.