This article investigates a function f (x), constructed from the Nikol'skii class in S-2. The estimation obtained will show that the Riesz mean of the spectral expansions is unable to be strengthened due to absence of localization caused by a singualrity at a definite point f (x), on the sphere.

Wilkie investment model is a stochastic investment model that was built by A. D Wilkie in 1984 and was updated in 1995. The model building objective is forecasting. Box-Jenkins method was the basic structure of Wilkie model. It involves various type of forecasting model. Some model handle stationary time series such as autoregressive moving average (ARMA) model while some of them handle non-stationary time series such as autoregressive integrated moving average (ARIMA) model. There are four sub-models in the Wilkie model which is retail price index model, share dividend yield model, share dividend index model and Consols yield model.

Previous numerical methods for solving systems of higher order ordinary differential equations (ODEs) directly require calculating the integration coefficients at every step. This research provides a block multi step method for solving orbital problems with periodic solutions in the form of higher order ODEs directly. The advantage of the proposed method is, it requires calculating the integration coefficients only once at the beginning of the integration is presented. The derived formulae is then validated by running simulations with known higher order orbital equations. To provide further efficiency, a relationship between integration coefficients of various order is obtained.

The study of chaotic motion in periodic self-excited oscillators are an area of interest in science and engineering. In the current research, a numerical solution hi backward difference form is proposed for solving these chaotic motions in periodic-self excited oscillators. Study conducted in this article focuses on chaotic motions in the form of Duffing-Van Der Pol Oscillators. A backward difference formulation in predictor-corrector (PeCe) mode is introduced for solving these Duffing-Van Der Pol directly. Numerical simulations provided will show the accuracy of the PeCe backward difference formulation.

Purpose The purpose of this paper is to examine incentives for risk shifting in debt- and equity-based contracts based on the critiques of the similarities between sukuk and bonds. Design/methodology/approach This paper uses a theoretical and mathematical model to investigate whether incentives for risk taking exist in: debt contracts; and equity contracts. Findings Based on this theoretical model, it argues that risk shifting behaviour exists in debt contracts only because debt naturally gives rise to risk shifting behaviour when the transaction takes place. In contrast, equity contracts, by their very nature, involve sharing transactional risk and returns and are thus thought to make risk shifting behaviour undesirable. Nonetheless, previous researchers have found that equity-based financing also might carry risk shifting incentives. Even so, this paper argues that the amount of capital provided and the underlying assets must be considered, especially in the event of default. Through mathematical modelling, this element of equity financing can make risk shifting unattractive, thus making equity financing more distinct than debt financing. Research limitations/implications Global awareness of the dangers of debt should be increased as a means of reducing the amount of debt outstanding globally. Although some regulators suggest that sukuk replaces debt, they must also be aware that imitative sukuk poses the same threat to efforts to avoid debt. In short, efforts to ensure future financial stability cannot address only debts or bonds but must also address those types of sukuk that mirrors bonds in their operation. In the wake of the global financial crisis, amid the frantic search for ways of protecting against future financial shocks, this analysis aims to help create future stability by encouraging market players to avoid debt-based activities and promoting equity-based instruments. Practical implications This paper's findings are relevant for countries that feature more than one type of financial market (e.g. Islamic and conventional) because risk shifting behaviour can degrade economic and financial stability. Originality/value This paper differs from the previous literature in two important ways, viewing risk shifting behaviour not only in relation to debt or bonds but also when set against debt-based sukuk, which has been subjected to similar criticism. Indeed, to the extent that debts and bonds encourage risk shifting behaviour and threaten the entire financial system, so, too, can imitation sukuk or debt-based sukuk. Second, this paper is unique in exploring the ability of equity features to curb equityholders' incentive to engage in risk shifting behaviour. Such an examination is necessary for the wake of the global financial crisis, for researchers and economists now agree that risk shifting must be controlled.

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

A two-point block backward difference technique is established for solving nonlinear Riccati differential equations directly. The proposed method is coded using a variable order step size (VOS) algorithm. The advantage of the two-point block method is its programmability to implement parallel programming techniques. Combination of the block method and VOS algorithm allows for a significant reduction of computation cost in comparison to conventional methods. With an added advantage of the recursive relationship between integration coefficients of different orders, the proposed two-point block method provides efficient computation without loss of accuracy.

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy.