Parallel Block Methods for Solving Higher Order Ordinary Differential Equations Directly

Numerous problems that are encountered in various branches of science and engineering involve ordinary differential equations (ODEs). Some of these problems require lengthy computation and immediate solutions. With the availability of parallel computers nowadays, the demands can be achieved. However, most of the existing methods for solving ODEs directly, particularly of higher order, are sequential in nature. These methods approximate numerical solution at one point at a time and therefore do not fully exploit the capability of parallel computers. Hence, the development of parallel algorithms to suit these machines becomes essential. In this thesis, new explicit and implicit parallel block methods for solving a single equation of ODE directly using constant step size and back values are developed. These methods, which calculate the numerical solution at more than one point simultaneously, are parallel in nature. The programs of the methods employed are run on a shared memory Sequent Symmetry S27 parallel computer. The numerical results show that the new methods reduce the total number of steps and execution time. The accuracy of the parallel block and 1-point methods is comparable particularly when finer step sizes are used. A new parallel algorithm for solving systems of ODEs using variable step size and order is also developed. The strategies used to design this method are based on both the Direct Integration (DI) and parallel block methods. The results demonstrate the superiority of the new method in terms of the total number of steps and execution times especially with finer tolerances. In conclusion, the new methods developed can be used as viable alternatives for solving higher order ODEs directly

Numerical solution of first order initial value problems using a self-starting implicit two-step Obrechkoff-type block method

The conventional two-step implicit Obrechkoff method is a discrete scheme that requires additional starting values when implemented for the numerical solution of first order initial value problems. This paper therefore presents a two-step implicit Obrechkoff-type block method which is self-starting for solving first order initial value problems, hence bypassing the rigour of developing and implementing new starting values for the method. Numerical examples are considered to show the new method performing better when com-pared with previously existing methods in literature

Implicit one-step block hybrid third-derivative method for the direct solution of initial value problems of second-order ordinary differential equations

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {xn,xn+r} while its second and third derivatives are collocated at all points {xn,xn+r,xn+s,xn+t,xn+1} in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy

Parallel R-point implicit block method for solving higher order ordinary differential equations directly

Most of the existing methods for solving ordinary differential equations (ODEs) of higher order are sequential in nature. These methods approximate numerical solution at one point at a time and therefore do not fully exploit the capability of parallel computers. Hence, the development of parallel algorithms to suit these machines becomes essential. In this paper, a new method called parallel R-point implicit block method for solving higher order ODEs directly using constant step size is developed. This method calculates the numerical solution at more than one point simultaneously and is parallel in nature, thus suitable for parallel computation. Computational advantages are presented comparing the results obtained by the new method with that of conventional 1-point method. The numerical results show that the new method reduces the total number of steps and execution time. The accuracy of the parallel block and the conventional 1-point methods are comparable particularly when finer step sizes are used

Numerical solution of third order ordinary differential equations using a seven-step block method

This paper aims to provide a direct solution to third order initial value problems of ordinary differential equations.Multistep collocation approach is adopted in the derivation of the method.The new block method is zero-stable, consistent and convergent.The application of the new method to solving differential equations gives better results when compared with the existing methods

Developing parallel 3-point implicit block method for solving second order ordinary differential equations directly

Ordinary differential equations are commonly used for mathematical modeling in many diverse fields such as engineering, industrial mathematics, operation research, artificial intelligence, management, sociology and behavioural sciences. Numerous problems encountered in these fields require lengthy computation and immediate solution. In this paper, a new method called parallel 3-point implicit block method for solving second order ODES is developed. This method takes full advantage of parallel computers because the numerical solution can be computed at three points simultaneously. As a result, the solution can be obtained faster if compared to the conventional methods where the numerical solution is computed at one point at a time. Computational advantages are presented comparing the results obtained by the new method with that of 1-point and 2-point implicit block methods. The numerical results show that parallel 3-point implicit block method reduces the total number of steps and execution time without sacrificing the accuracy

Generalized one-step third derivative implicit hybrid block method for the direct solution of second order ordinary differential equation

In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary differential equations using collocation and interpolation approach.To derive this method, the approximate solution power series is interpolated at the first and off-step points and its second and third derivatives are collocated at all points in the given interval.Besides having good numerical method properties, the new developed method is also superior to the existing methods in terms of accuracy when solving the same problems

Surveying the best volatility measurements to forecast stock market

This paper investigates methods to forecast future adjusted price of S&P 500 by using geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) for better investment decision. Four types of formulas are used to find the appropriate volatility measurement that may provide forecast value which closely resembling to actual movement of stock price. The evaluation of forecasting methods is computed by the mean absolute percentage error (MAPE).The findings showed high accuracy in all forecasting methods, with all MAPE are less than 10%, with the best forecasting method is GFBM with stochastic volatility which follow fractional Ornstein – Uhlenbeck (FOU) process

The division free parallel algorithm for finding determinant

A cross multiplication method for determinant was generalized for any size of square matrices using a new permutation strategy.The permutation is generated based on starter sets.However, via permutation, the time execution of sequential algorithm became longer.Thus, in order to reduce the computation time, a parallel strategy was developed which is suited for master and slave paradigm of the high performance computer.A parallel algorithm is integrated with message passing interface.The numerical results showed that the parallel methods computed the determinants faster than the sequential counterparts particularly when the tasks were equally allocated

(γ, β)-PS-irresolute and (γ, β)-PS-continuous functions

This paper introduces some new types of functions called (γ, β)-PS-irresolute and (γ, β)-PS-continuous by using γ-PS-open sets in topological spaces (X, τ).From γ-PS-open and γ-PS-closed sets, some other types of γ-PS functions can also be defined. Moreover, some basic properties and preservation theorems of these functions are obtained. In addition, we investigate basic characterizations and properties of these γ-PS- functions. Finally, some compositions of these γ-PS- functions are given