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.

Oil price behaviour over the last 10 years has shown to be bimodal in character, displaying a strong tendency to congregate around one range of high oil prices and one range of low prices, indicating two distinct peaks in its frequency distribution. In this paper, we propose a new, single nonlinear stochastic process to model the bimodal behaviour, namely, dp=α(p1−p)(p2−p(p3−p)dt+σpγdZ, γ=0,0.5. Further, we find analytic approximations of oil price futures under this model in the cases where the stable fixed points of the corresponding deterministic model are (a) evenly spaced about the unstable fixed point and (b) are spaced in the ratio 1:2 about the unstable fixed point. The solutions are shown to produce accurate prices when compared to numerical solutions.

This paper looks at adapting a recent approach found in the literature for pricing short-term American options to price American straddle options with two free boundaries. We provide a series solution in which explicit formulas for the coefficients are given. Hence, no complicated, recursive systems or nonlinear integral equations need to be solved, and the method efficiently provides fast solutions. We also compare the method with a numerical method and find that it gives very accurate prices not only for the option value, but also for the critical stock prices.

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This work examines a new subclass of generalized bi-subordinate functions of complex order γ connected to the q-difference operator. We obtain the upper bounds ρm for generalized bi-subordinate functions of complex order γ using the Faber polynomial expansion technique. Additionally, we find coefficient bounds ρ2 and Feke–Sezgo problems ρ3−ρ22 for the functions in the newly defined class, subject to gap series conditions. Using the Faber polynomial expansion method, we show some results that illustrate diverse uses of the Ruschewey q differential operator. The findings in this paper generalize those from previous efforts by a number of prior researchers.