Volume 80 (2021): Issue 3 (December 2021)

Some new oscillation criteria are obtained for a class of thirdorder quasi-linear Emden-Fowler differential equations with unbounded neutral coefficients of the form (a(t)(z″(t))α)′+f(t)yλ(g(t))=0,\[(a(t){(z(t))^\alpha })' + f(t){y^\lambda }(g(t)) = 0,\] where z(t) = y(t) + p(t)y(σ(t)) and α, λ are ratios of odd positive integers. The established results generalize, improve and complement to known results.

In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the backward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders O(Δt + h³) for the backward Euler method and O(Δt² + h³) for the Crank-Nicolson method, where Δt and h are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.

In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L^∞ and weighted L² norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.

Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter θ, 0 ≤ θ ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (θ = 0), semi-implicit, fully-implicit (θ = 1) and Crank-Nicolson (θ = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.

We introduce a novel method for map registration and apply it to transformation of the river Ister from Strabo’s map of the World to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that Strabo’s Suevi in the Hercynian Forest corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.

The use of balloon models to address the problems of “snakes” based models was introduced by Laurent D. Cohen. This paper presents a geodesic active contours model with a modified external force term that includes a balloon model. This balloon model makes the segmentation surface to behave like a balloon inflated by the external forces. In this paper, we show an automatic way to control the behaviour of the external force with respect to the segmentation evolution. The external forces, comprised of edge and inflation terms, push the segmentation surface to edges, while curvature regularizes the evolution. As segmentation evolves, the influence of the applied inflation force is determined by how close we are to the edges. With this setup, the initial segmentation does not need to be close to the object’s edges, instead it is inflated by the balloon model towards the edges. Closer to the edges, the influence of the inflation force is adjusted accordingly. The force’s influence is completely turned off when the evolution is stable (reached the edges), then only the curvature and edge information is used to evolve the segmentation.

This approach solves the issues associated with inclusion of balloon model. These issues are that the inflation force can overpower forces from weak edges, or they can cause the contour to be slightly larger than the actual minima. We present examples of the improved model for segmentation of human bladder images. Weak edges are more prevalent in medical images, and the automated handling of the inflation forces gives promising results for this kind of images.

In this paper, we analyse stability of survival of red blood cells in animal fractional order model with time delay. Results have been illustrated by numerical simulations.