In this paper, we analyze the three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions. The differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second order accurate in time and fourth convergent in space, while it is second order convergent in both time and space for high Reynolds numbers problems. This shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature for solving the two-dimensional time dependent nonlinear coupled Burgers' equations. Numerical examples which confirm the theoretical results are presented and discussed.Keywords: two-dimensional unsteady nonlinear coupled Burgers' equations, one-dimensional difference operators, two-step MacCormack scheme, three-level explicit time-split MacCormack method, stability and convergence rate.

In this paper, we show that given a nontrivial concircular vector field u on a Riemannian manifold (M, g) with potential function f , there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇f , which we call the connecting function of the concircular vector field u. Then this connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere S n (c) and the Euclidean space E n . We also show that the connecting function influences topology of the Riemannian manifold.2000 Mathematics Subject Classification. Primary 53C20, 53C21.

The main target of this work is presenting two efficient accurate algorithms for solving numerically one of the most important models in physics and engineering mathematics, Fisher–Kolmogorov–Petrovsky–Piskunov’s equation (Fisher-KPP) with fractional order, where the derivative operator is defined and studied by the fractional derivative in the sense of Liouville–Caputo (LC). There are two main processes; in the first one, we use the compact finite difference technique (CFDT) to discretize the derivative operator and generate a semidiscrete time derivative and then implement the Vieta–Lucas spectral collocation method (VLSCM) to discretize the spatial fractional derivative. The presented approach helps us to transform the studied problem into a simple system of algebraic equations that can be easily resolved. Some theoretical studies are provided with their evidence to analyze the convergence and stability analysis of the presented algorithm. To test the accuracy and applicability of our presented algorithm a numerical simulation is given.

In this paper, we study rectifying curves arising through the dilation of unit speed curves on the unit sphere S 3 and conclude that arcs of great circles on S 3 do not dilate to rectifying curves, which develope previously obtained results for rectifying curves in Eucidean spaces. This fact allows us to prove that there exists an associated rectifying curve for each Frenet curve in the Euclidean space E 4 and a result of the fact rectifying curves in the Euclidean space E 4 are ample , indeed as an appication, we present an ordinary differential equation satisfied by the distance function of a Frenet curve in E 4 which alows us to characterize the spherical curves and rectifying curves in E 4. Furthermore, we study ccr-curves in the Euclidean space E 4 which are generalizations of helices in E 3 and show that the property of a helix that its tangent vector field makes a constant angel with a fixed vector (axis of helix) does not go through for a ccr-curve.

<abstract><p>In Euclidean 3-space $ {\mathrm{E}}^3 $, a canonical subject is the focal surface of such a cliched space curve, which would be a two-dimensional corrosive with Lagrangian discontinuities. The tubular surfaces with respect to the B-Darboux frame and type-2 Bishop frame in $ {\mathrm{E}}^3 $ are given. These tubular surfaces' focal surfaces are then defined. For such types of surfaces, we acquire some results becoming Weingarten, flat, linear Weingarten conditions and we demonstrate that in $ {\mathrm{E}}^3 $, a tubular surface has no minimal focal surface. We also provide some examples of these types of surfaces.</p></abstract>