Developing Clean Technology through Approximate Solutions of Mathematical Models

In this paper, the role of mathematical modeling in the development of clean technology has been considered. One method each for obtaining approximate solutions of mathematical models by ordinary differential equations and partial differential equations respectively arising from the modeling of systems and physical phenomena has been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate analytical method, for the solution of nonlinear partial differential equations are discussed

A One Step Method for the Solution of General Second Order Ordinary Differential Equations

In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better

Two Steps Block Method for the Solution of General Second Order Initial Value Problems of Ordinary Differential Equation

In this paper, an implicit block linear multistep method for the solution of ordinary differential equation was extended to the general form of differential equation. This method is self starting and does not need a predictor to solve for the unknown in the corrector. The method can also be extended to boundary value problems without additional cost. The method was found to be efficient after being tested with numerical problems of second order

Solving General Second Order Ordinary Differential Equations by a One-Step Hybrid Collocation Method

A one-step hybrid method is developed for the numerical approximation of second order initial value problems of ordinary differential equations by interpolation and collocation at nonstop and step points respectively. The method is zero stable and consistent with very small error term. Numerical experiment of the method on sample problem shows that the method is more efficient and accurate than the results obtained from our earlier methods

Modified Block Method for the Direct Solution of Second Order Ordinary Differential Equations

The direct solution of general second order ordinary differential equations is considered in this paper. The method is based on the collocation and interpolation of the power series approximate solution to generate a continuous linear multistep method. We modified the existing block method in order to accommodate the general nth order ordinary differential equation. The method was found to be efficient when tested on second order ordinary differential equation

Four Steps Implicit Method for the Solution of General Second Order Ordinary Differential Equations

Four steps implicit scheme for the solution of second order ordinary differential equation was derived through interpolation and collocation method. Newton polynomial approximation method was used to generate the unknown parameters in the corrector. The method was tested with numerical examples and it was found to be efficient in solving second order ordinary differential equations

A Four Point Block Integration Method for the Solutions of IVP in ODE

A one step block integration method for initial value problems of general second order ordinary differential equations which combine the Runge-Kutta type one step procedure and the Adam’s type multistep procedure is proposed in this paper. Convergence of this sixth order method is established by the consistency and zero stability properties. The method is also characterized by the region of absolute stability. Comparison with existing methods obtained with step number k>1 shows that the new method is comparatively accurate

On a Fractional Beta-Exponential Distribution

A four parameter distribution representing the ratio of two independent Beta-Exponential variates is defined. An expression for the probability density function and the cumulative density function is given. The resulting distribution has the Quotient of Beta-Weibull distribution and the Pareto distribution as special cases. Its statistical properties were investigated and the method of Maximum Likelihood Estimation (MLE) has been proposed for estimating the parameters of the model. Based on the behavior of its hazard function, the model is appropriate in modeling the occurrence of infant mortality failures

Coefficient Bounds for Certain Classes of Analytic and Univalent functions as Related to Sigmoid Function

In this work, the authors investigated certain classes of analytic and univalent functions in terms of their coefficient bounds as related to activation sig- moid function with respect to symmetric and conjugate points

CONTINUOUS IMPLICIT HYBRID ONE-STEP METHODS FOR THE SOLUTION OF INITIAL VALUE PROBLEMS OF GENERAL SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

The numerical solutions of initial value problems of general second order ordinary differential equations have been studied in this work. A new class of continuous implicit hybrid one step methods capable of solving initial value problems of general second order ordinary differential equations has been developed using the collocation and interpolation technique on the power series approximate solution. The one step method was augmented by the introduction of offstep points in order to circumvent Dahlquist zero stability barrier and upgrade the order of consistency of the methods. The new class of continuous implicit hybrid one step methods has the advantage of easy change of step length and evaluation of functions at offstep points. The Block method used to implement the main method guarantees that each discrete method obtained from the simultaneous solution of the block has the same order of accuracy as the main method. Hence, the new class of one step methods gives high order of accuracy with very low error constants, gives large intervals of absolute stability, are zero stable and converge. Sample examples of linear, nonlinear and stiff problems have been used to test the performance of the methods as well as to compare computed results and the associated errors with the exact solutions and errors of results obtained from existing methods, respectively, in terms of step number and order of accuracy, using written effcient computer codes