Volume 10, Issue 4

We have developed an approximation to the solution of the Schrödinger equation in abstract setting. The accuracy of our approximation depends on the smoothness of this solution. We show that for the analytical initial vectors our approximation possesses a super exponential convergence rate.

In this paper we present a method for the numerical solution of Cauchy type singular integral equations of the first kind on a finite segment which is unbounded at the end points of the segment. Chebyshev polynomials of the first and second kinds are used to derive an approximate solution. Moreover, an estimation error is computed for the approximate solution.

In this study, we develop an optimal family of derivative-free iterative methods. Convergence analysis shows that the methods are fourth order convergent, which is also verified numerically. The methods require three functional evaluations during each iteration. Though the methods are independent of derivatives, computa- tional results demonstrate that the family of methods are efficient and demonstrate equal or better performance as compared with many well-known methods and the clas- sical Newton method. Through optimization we derive an optimal value for the free parameter and implement it adaptively, which enhances the convergence order without increasing functional evaluations.

We investigate the convergence rate of the quantics-TT (QTT) stochas- tic collocation tensor approximations to solutions of multiparametric elliptic PDEs and construct efficient iterative methods for solving arising high-dimensional parameter- dependent algebraic systems of equations. Such PDEs arise, for example, in the para- metric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the M-term truncated Karhunen-Loève expansion. We consider both the case of additive and log-additive dependence on the multivariate parameter. The local-global versions of the QTT-rank estimates for the system matrix in terms of the parameter space dimension is proven. Similar rank bounds are observed in numerics for the solutions of the discrete linear system. We propose QTT-truncated iteration based on the construction of solution-adaptive preconditioner that provides robust conver- gence in both additive and log-additive cases. Various numerical tests indicate that the numerical complexity scales almost linearly in the dimension of parametric space M.

We have studied the stability of finite-difference schemes approximating boundary value problems for parabolic equations with a nonlinear and nonmonotonic source of the power type. We have obtained simple sufficient input data conditions, in which the solution of the differential problem is globally stable for all 0 ≤ t ≤ +∞. It is shown that if these conditions fail, then the solution can blow up (go to infinity) in finite time. The lower bound of the blow up time has been determined. The stability of the solution of BVP for the nonlinear convection-diffusion equation has been investigated. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear differential equations, Bihari-type inequalities and their discrete analogs.

In an earlier paper the last two authors studied spatially semidiscrete piecewise linear finite element approximations of the heat equation and showed that, in the case of the standard Galerkin method, the solution operator of the initial-value problem is neither positive nor contractive in the maximum-norm for small time, but that for the lumped mass method these properties hold, if the triangulations are essentially of Delaunay type. In this paper we continue the study by considering fully discrete analogues obtained by discretization also in time. The above properties then carry over to the backward Euler time stepping method, but for other methods the results are more restrictive. We discuss in particular the θ-method and the (0; 2) Padé approximation in one space dimension.

The paper considers a method for solving nonlinear ill-posed problems with monotone operators. The approach combines the Lavrentiev method, the fixedpoint method, and the balancing principle for selection of the regularization parameter. The method’s optimality has been proved for some set of smooth solutions. A test example proves the efficiency of the proposed method.