AN ITERATIVE, BRACKETING & DERIVATIVE-FREE METHOD FOR NUMERICAL SOLUTION OF NON-LINEAR EQUATIONS USING STIRLING INTERPOLATION TECHNIQUE

In this article, an iterative, bracketing and derivative-free method have been proposed with the second-order of convergence for the solution of non-linear equations. The proposed method derives from the Stirling interpolation technique, Stirling interpolation technique is the process of using points with known values or sample points to estimate values at unknown points or polynomials. All types of problems (taken from literature) have been tested by the proposed method and compared with existing methods (regula falsi method, secant method and newton raphson method) and it’s noted that the proposed method is more rapidly converges as compared to all other existing methods. All problems were solved by using MATLAB Version: 8.3.0.532 (R2014a) on my personal computer with specification Intel(R) Core (TM) i3-4010U CPU @ 1.70GHz with RAM 4.00GB and Operating System: Microsoft Windows 10 Enterprise Version 10.0, 64-Bit Server, x64-based processor.

AN ASSESSMENT OF THE DISPLACEMENT OF A CAM FOLLOWER USING REGULA FALSI METHOD

Cam is a mechanical component that transforms circular motion to reciprocating motion by using mating component, called the follower. The principal aim of this work was to study and analyse the displacement of a cam-follower with Regula Falsi method and verify its input by using MATLAB and FORTRAN simulations. A study was conducted on angle of rotation and the displacement of the follower, which is equal to the radius of the cam given as transcendental equation to find the exact solution. The parameters such as initial guess, final guess, iteration counter and the desired displacement are involved in finding the angular displacement to the cam system in high speed rotation. The analysis was done using a computer programming that enables verification of the results obtained and ascertaining whether the inputs are correct or not for the displacement in cam follower system. The computer output showed results of the two data sets that yielded solutions and two that did not. The results revealed that the programme could be used to find the angular displacement corresponding to a given follower displacement for any cam; if the function CAMF is modified to include the appropriate radius function, r(x). The results further revealed that at a halve cycle of a rotating cam, which is equivalent to (x = 3.142 rad), is a solution that would provide the desired displacement of the follower (opening and closing of valves).

A NEW SECOND ORDER DERIVATIVE FREE METHOD FOR NUMERICAL SOLUTION OF NON-LINEAR ALGEBRAIC AND TRANSCENDENTAL EQUATIONS USING INTERPOLATION TECHNIQUE

Its most important task in numerical analysis to find roots of nonlinear equations, several methods already exist in literature to find roots but in this paper, we introduce a unique idea by using the interpolation technique. The proposed method derived from the newton backward interpolation technique and the convergence of the proposed method is quadratic, all types of problems (taken from literature) have been solved by this method and compared their results with another existing method (bisection method (BM), regula falsi method (RFM), secant method (SM) and newton raphson method (NRM)) it’s observed that the proposed method have fast convergence. MATLAB/C++ software is used to solve problems by different methods.

FREE VIBRATION OF TAPERED TIMOSHENKO BEAMS BY DEFORMATION DECOMPOSITION

This paper deals with the free vibration of tapered Timoshenko beams. The simultaneous differential equations governing the free vibration of tapered Timoshenko beams are derived by decomposing the deformations of the beam into components as transverse deflection, bending rotation and shear distortion. The governing differential equations are first integrated by the Runge–Kutta method and then solved by the determinant search method, combined with the Regula–Falsi method, to obtain the natural frequencies of the beam along with their corresponding mode shapes. In the numerical examples, the effects of various parameters on the frequencies and mode shapes of the beam are extensively discussed.

A Generalized Regula Falsi Method for Finding Zeros and Extrema of Real Functions

Many zero-finding numerical methods are based on the Intermediate Value Theorem, which states that a zero of a real function is bracketed in a given interval if and have opposite signs; that is, . But, some zeros cannot be bracketed this way because they do not satisfy the precondition . For example, local minima and maxima that annihilate may not be bracketed by the Intermediate Value Theorem. In this case, we can always use a numerical method for bracketing extrema, checking then whether it is a zero of or not. Instead, this paper introduces a single numerical method, calledgeneralized regula falsi(GRF) method to determine both zeros and extrema of a function. Consequently, it differs from the standardregula falsi methodin that it is capable of finding any function zero in a given interval even when the Intermediate Value Theorem is not satisfied.