The heat equation with singular potentials. II: Hypoelliptic case

This work is a continuation of the work arXiv:2004.11255v3. It is 21 pagesIn this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends a previous work , where the classical heat equation on $\mathbb R^n$ was considered

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.Comment: 26 pages, 7 figures. arXiv admin note: text overlap with arXiv:1812.0133

A parallel hybrid implementation of the 2D acoustic wave equation

In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the ADI method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on GPU, GPU+OpenMP, and Hybrid (GPU+OpenMP+MPI) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, CUDA, and CUDA+OpenMP implementations.Comment: 10 pages; 1 Chart; 1 Table; 1 Listing; 1 Algorith

Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for some fractional derivatives

In this paper we show different inequalities for fractional order differential operators. In particular, the Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for the Caputo, Riemann-Liouville and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities

Direct and Inverse problems for time-fractional pseudo-parabolic equations

18 pages18 pagesThe purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces

Bitsadze-Samarskii type problem for the integro-differential diffusion-wave equation on the Heisenberg group

This paper deals with the fractional generalization of the integro-differential diffusion-wave equation for the Heisenberg sub-Laplacian, with homogeneous Bitsadze-Samarskii type time-nonlocal conditions. For the considered problem, we show the existence, uniqueness and the explicit representation formulae for the solution

On nonlinear damped wave equations for positive operators. I. Discrete spectrum

In this paper we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on the interplay between the bottom of the operator’s spectrum and the mass term. Consequently, we prove global in time well-posedness results for semilinear and for more general nonlinear equations with small data. Examples are given for nonlinear damped wave equations for the harmonic oscillator, for the twisted Laplacian (Landau Hamiltonian), and for the Laplacians on compact manifolds

Nonharmonic analysis of boundary value problems without WZ condition

In this work we continue our research on nonharmonic analysis of boundary value problems as initiated in [RT16]. There, we assumed that the eigen- functions of the model operator on which the construction is based do not have zeros. In this paper we have weakened this condition extending the applicability of the developed pseudo-di erential analysis. Also, we do not assume that the underlying set is bounded