Jimevwo G. Oghonyon

Gumbel Distribution: Ordinary Differential Equations

In this chapter, homogenous ordinary differential equations (ODES) of different orders were obtained for the probability density function, quantile function, survival function inverse survival function, hazard function and reversed hazard functions of Gumbel distribution. This is possible since the aforementioned probability functions are differentiable. Differentiation and modified product rule were used to obtain the required ordinary differential equations, whose solutions are the respective probability functions. The different conditions necessary for the existence of the ODEs were obtained and it is almost in consistent with the support that defined the various probability functions considered. The parameters that defined each distribution greatly affect the nature of the ODEs obtained. This method provides new ways of classifying and approximating other probability distributions apart from Gumbel distribution considered in this chapter. In addition, the result of the quantile function can be compared with quantile approximation using the quantile mechanics.

3-Parameter Weibull Distribution: Ordinary Differential Equations

In this chapter, homogenous ordinary differential equations (ODE) of different orders were obtained for the probability density function, quantile function, survival function inverse survival function, hazard function and reversed hazard functions of 3-parameter Weibull distribution. This is possible since the aforementioned probability functions are differentiable. Differentiation and modified product rule were used to obtain the required ordinary differential equations, whose solutions are the respective probability functions. 3-parameter Weibull distribution is an extension of the Weibull distribution with an extra parameter. The different conditions necessary for the existence of the ODEs were obtained and it is almost in consistent with the support that defined the various probability functions considered. The parameters that defined each distribution greatly affect the nature of the ODE obtained. This method provides new ways of classifying and approximating other probability distributions apart from 3-parameter Weibull distribution considered in this chapter.

Natural Frequencies of an Euler-Bernoulli Beam with Special Attention to the Higher Modes via Variational Iteration Method

In this chapter, natural frequencies of an Euler-Bernoulli prismatic beam on different supports are analyzed. The variational iteration method (VIM) is employed to compute the said natural frequencies especially for the higher modes of vibration. Some numerical examples are presented with the view to demonstrating excellent agreement between the results obtained using VIM and other methods.

Implementing a Type of Block Predictor-corrector Mode for Solving General Second Order Ordinary Differential Equations

The paper is geared towards implementing a type of block predictor-corrector mode capable of integratinggeneral second order ordinary differential equations using variable step size. This technique will be carried out on nonstiff problems. The mode which emanated from Milne’s estimate has many computation advantages such as changing and designing a suitable step size, correcting to convergence, error control/minimization with better accuracy compare to other methods with fixed step size. Moreover, the approach will adopt the estimates of the principal local truncation error on a pair of explicit (predictor) and implicit (corrector) Adams family which are implemented in P(CE) m mode. Numerical examples are given to examine the efficiency of the method and compared with subsisting methods.