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This study focused mainly on the derivation of the 2 and 3-point block methods with constant coefficients for solving special second order ordinary differential equations directly based on Newton-Gregory backward interpolation formula. The performance of the new methods was compared with the conventional 1-point method using a standard set of test problems. Numerical results were presented to illustrate the effectiveness of the methods in terms of total number of steps taken, maximum error and execution time. The results suggested a significant improvement in efficiency of the r-point block method. AMS Subject Classification: 65L05. Key word: Initial value problems, special second order ordinary differential equations, block method, INTRODUCTION Special second order Ordinary Differential Equations (ODEs) arises naturally in describing mechanics and electrical systems, wave oscillations and a variety of other physical problems. Such equations can be written [...]

In this paper, we proposed the numerical solution of Volterra integro-differential equations of the second kind using Improved Runge-Kutta method of order three and four with 2 stages and 4 stages, respectively. The improved Runge-kutta method is considered as two-step numerical method for solving the ordinary differential equation part and the integral operator in Volterra integro-differential equation is approximated using quadrature rule and Lagrange interpolation polynomials. To illustrate the efficiency of proposed methods, the test problems are carried out and the numerical results are compared with existing third and fourth order classical Runge-Kutta method with 3 and 4 stages, respectively. The numerical results showed that the Improved Runge-Kutta method by achieving the higher accuracy performed better results than existing methods.

In this paper, fourth-order improved Runge–Kutta method (IRKD) for directly solving a special third-order ordinary differential equation is constructed. The fourth-order IRKD method has a lower number of function evaluations compared with the fourth-order Runge–Kutta method. The stability polynomial of the method is given. Numerical comparisons are also performed using the existing Runge–Kutta method after reducing the problems into a system of first-order equations and solving them, and direct RKD method for solving special third-order ordinary differential equations. Numerical examples are presented to illustrate the efficiency and the accuracy of the new method in terms of number of function evaluations as well as max absolute error.

In the current work, the synthesis of carbon nanotubes (CNTs) and carbon nanospheres (CNS’s) has been investigated by applying the chemical vapor deposition method in a one-step sample preparation. In this method, iron nitrate non-hydrate (Fe(NO3)3.9H2O) and acetylene (C2H2) have been used as the catalyst source and carbon source, respectively, to grow CNT directly on the CF surface at 700°C and then CNS’s were synthesized on the CNT layers at 900°C under a 250sccm gas flow rate (40%N2, 40%H2, 20% C2H2). According to the SEM and TEM micrographs from the resultant carbon nanoparticles, the diameters of the CNTs and CNS’s have been estimated about 30-50nm and 300-400nm, respectively.

Although the wide applications of the fractional order of dierential equations hasbeen reported in the physics, astronomy, and bioengineering elds, little attention has beenpaid to the fractional kinetic models in the literature. In this research, we are conned with theapplication of Jacobi polynomials for solving fractional order of l dierential equation arising inchemical reaction model into the Palm oil in the sense of fuzzy context. The results demonstratethe capability of the method which can be a reliable numerical technique to be utilized in thesimilar reaction models.

In this paper, we present a mathematical model for Marangoni convection boundary layer flow with radiation and different types of nanoparticles, namely, Cu, Al2O3, and TiO2in a water-based fluid. The governing equations in the form of partial differential equations have been reduced to a set of ordinary differential equations by applying suitable similarity transformations, which is then solved numerically using the shooting method. Numerical results are obtained for the surface-temperature gradient or the heat transfer rate as well as the temperature profiles for some values of the governing parameters, namely, the nanoparticle volume fraction φ, the constant exponent β, and thermal radiation parameter Nr. The results indicate that the heat transfer rate at the surface decreases as the thermal radiation parameter Nrincreases. MSC:76N20, fluid mechanics.

The stability properties for singly diagonally implicit runge-kutta method fourth order five stages embedded in fifth six stages combined with lagrange interpolsation when applied to the linear delay differential equation. [image omitted]  [image omitted]  are investigated. Their regions of stability for [image omitted]  are determined and numerical results when some problems are solved using the method are presented.

In this paper, fourth order, five-stage embedded in fifth order six-stage Singly Diagonally Implicit Runge-Kutta (SDIRK) methods are derived, and their regions of stability are determined. Numerical results of these methods are presented and compared with the results obtained when the same systems are solved using Rabeh's third order three-stage embedded in fourth order four-stage SDIRK method. Also presented are the stability regions of the methods.

This numerical study exclusively focused on the direct two point diagonally multistep block method of order four (2DDM4) in the form of Adams-type formulas. The proposed predictor–corrector scheme was applied in this study to compute two equally spaced numerical solutions for the third-order two point and multipoint boundary value problems (BVPs) subject to Robin boundary conditions concurrently at each step. The optimization of the computational costs was taken into consideration by not resolving the equation into a set of first-order differential equations. Instead, its implementation involved the use of shooting technique, which included the Newton divided difference formula employed for the iterative part, for the estimation of the initial guess. Apart from studying the local truncation error, the study also included the method analysis, including the order, stability, and convergence. The results of eight numerical problems demonstrated and highlighted competitive computational cost attained by the scheme, as compared to the existing method. [ABSTRACT FROM AUTHOR]

A set of order condition for block explicit hybrid method up to order five is presented and, based on the order conditions, two-point block explicit hybrid method of order five for the approximation of special second order delay differential equations is derived. The method is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutions. The efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted, which clearly demonstrated the superiority of the trigonometrically fitted block hybrid method.