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Finite Difference Methods
Operator Theory
Proceedings of the International Conference held at the University of Oulu, July 3-11, 2017
Proceedings of the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, May 29–June 1, 2018

Abstract

In this paper, we give some approximation properties by Stancu–Chlodowsky type λ-Bernstein operators in the polynomial weighted space and obtain the convergence properties of these operators by using Korovkin’s theorem. We also establish the direct result and the Voronovskaja type asymptotic formula.

Abstract

In this paper, we introduce the class of asymptotically demicontractive multivalued mappings and establish a strong convergence theorem of the modified Mann iteration to a common fixed point of a finite family of asymptotically demicontractive multivalued mappings in a complete CAT(0) space. We also give a numerical example of our iterative method to show its applicability.

Abstract

Delaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

Abstract

In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on (0,π) with the Robin boundary conditions and the jump conditions at the point π2. We prove that the potential M(x) on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential M(x) is given on (0,(1+α)π4); (ii) the potential M(x) is given on ((1+α)π4,π), where 0<α<1, respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.

Abstract

Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.