Search

The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is very appropriate for images with flow-like structures. Two convolutions are used in the construction of diffusion tensor for such a model, see [Drblíková, O.—Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (2007), 37–60], [Weickert, J.: Coherence-enhancing diffusion filtering, Int. J. Comput. Vis. 31 (1999), 111–127]. In this paper we introduce the third supplemental convolution in order to enhance the diffusion strategy. First, we briefly present the classical coherence enhancing model and explain our modification. Then the discrete scheme is provided. The core of the paper consists in numerical experiments. Benefits of the additional convolution are discussed and illustrated in the figures.

In this paper, we present numerical solution for the fractional Bratu-type equation via fractional residual power series method (FRPSM). The fractional derivatives are described in Caputo sense. The main advantage of the FRPSM in comparison with the existing methods is that the method solves the nonlinear problems without using linearization, discretization, perturbation or any other restriction. Three numerical examples are given and the results are numerically and graphically compared with the exact solutions. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature. The results reveal that the FRPSM is a very effective, simple and efficient technique to handle a wide range of fractional differential equations.

The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-˜Q-representation. The infinite systems of functional equations f ( φ ˆ k ( x ) ) = β ˜ i k + 1 , k + 1 + p ˜ i k + 1 , k + 1 f ( φ ˆ k + 1 ( x ) ) , f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right), where x = Δ i 1 ( x ) i 2 ( x ) … i n ( x ) … - Q ˜ x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q} , and φ ̑ is the shift operator of the Q̃-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on [0, 1] functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable η = Δ ξ 1 ξ 2 … ξ n … Q ˜ \eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q} with independent Q̃-symbols are founded.

The aim of the paper is to study problem of image segmentation and missing boundaries completion introduced in [Mikula, K.—Sarti, A.––Sgallarri, A.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation, Comput. Vis. Sci. 9 (2006), 23–31], [Mikula, K.—Sarti, A.—Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation, in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York, 583–626, 2005], [Mikula, K.—Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. Math. 89 (2001), 561–590] and [Tibenský, M.: VyužitieMetód Založených na Level Set Rovnici v Spracovaní Obrazu, Faculty of mathematics, physics and informatics, Comenius University, Bratislava, 2016]. We generalize approach presented in [Eymard, R.—Handlovičová, A.—Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] and apply it in the field of image segmentation. The so called regularised Riemannian mean curvature flow equation is presented and the construction of the numerical scheme based on the finite volume method approach is explained. The principle of the level set, for the first time given in [Osher, S.—Sethian, J. A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), 12–49] is used. Based on the ideas from [Eymard, R.—Handlovičová, A.– –Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] we prove the stability estimates on the numerical solution and the uniqueness of the numerical solution. In the last section, there is a proof of the convergence of the numerical scheme to the weak solution of the regularised Riemannian mean curvature flow equation and the proof of the convergence of the approximation of the numerical gradient is mentioned as well.

The paper deal with a system of four nonlinear difference equations where the first equation is of a neutral type. We present sufficient conditions for the system to have property B. This means that any nonoscillatory solution is either strongly monotone or a Kneser solution and in addition, these solutions satisfy special asymptotic properties.

In this paper, we consider examples of the application of a certain higher order inversion theorem to demonstrate the existence of a solution of a certain class of integral equations and a class of boundary value problems equivalent to these equations

In this paper we construct a system of three stochastic differential equations, which has a solution composed of a generalized telegraph signal process and a basic process. This system enabled us to find the escape probability of the basic process from an interval through its endpoint.