The Periodicity of the Accuracy of Numerical Integration Methods for the Solution of Different Engineering Problems

Newton-Cotes integration formulae have been researched for a long time, but the topic is still of interest since the correctness of the techniques has not yet been explicitly defined in a sequence for diverse engineering situations. The purpose of this paper is to give the readers an overview of the four numerical integration methods derived from Newton-Cotes formula, namely the Trapezoidal rule, Simpson's 1/3rd rule, Simpson's 3/8th rule, and Weddle's rule, as well as to demonstrate the periodicity of the most accurate methods for solving each engineering integral equation by varying the number of sub-divisions. The exact expressions by solving the numerical integral equations have been determined by Maple program and comparisons have been done using Python version 3.8.

A Non-Iterative Problem-Dependent Formula for Stiff Dynamic Problems

A novel one-step formula is proposed for solving initial value problems based on a concept of eigenmode. It is characterized by problem dependency since it has problem-dependent coefficients, which are functions of the product of the step size and the initial physical properties to define the problem under analysis. It can simultaneously combine A-stability, explicit formulation and second order accuracy. A-stability implies no limitation on step size based on stability consideration. An explicit formulation implies no nonlinear iterations for each step. The second order accuracy with an appropriate step size can have a good accuracy in numerical solutions. Thus, it seems promising for solving stiff dynamic problems. Numerical tests affirm that it can have the same performance as that of the trapezoidal rule for solving linear and nonlinear dynamic problems. It is evident that the most important advantage is of high computational efficiency in contrast to the trapezoidal rule due to no nonlinear iterations of each step.

The Taylor series method of order $p$ and Adams-Bashforth method on time scales

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order $p$ on time scales. The error and convergence analysis of the method is also obtained. The 2 step Adams-Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.

An improved Trapezoidal rule for numerical integration

Numerical Methods have attracted of research community for solving engineering problems. This interest is due to its practicality and the improvement of highspeed calculations done on current century processors. The increase in numerical method tools in engineering software, such as Matlab, is an example of the increased interest. In this paper, we are present a new improved numerical integration method, that is based on the well-known trapezoidal rule. The proposed method gives a great enhancement to the trapezoidal rule and overcomes the issue of the error value when dealing with some higher order functions even when solving for a single interval. After literature review, the proposed system is mathematically explained along with error analysis. Few examples are illustrated to prove improved accuracy of the proposed method over traditional trapezoidal method.

Numerical Integration Implementation Using Trapezoidal Rule Method To Calculate Aproximation Area Of West Java Province

An area can be shaped into a regular shape or an irregular shape. There is an area of irregular shape which is restricted by an unknown function, to determine that area must use a numerical integration. One of numerical integration methods is Trapezoidal Rule by replacing (????) with an integral approach function which can be evaluated, then let the (????) approximated by a linear polynomial in the certain interval, denoted as closed interval . This study is going to calculate the area of West Java Province by using this method with several different number of partitions in each quadrant such as, 9 partitions, 11 partitions, and 36 partitions in for different quadrants. This study provides the final result of the approximate area which will be compared with the actual area based in the error of result. The main finding is the approximate total area will be closer to the actual area followed by the increasing number of partitions.

New Kernel Function for the Approximation of Third Order Derivatives

A new kernel function is developed to approximate the third order derivative by mean of the Smoothed Particle Hydrodynamics (SPH) method. It has the advantage to be efficiently used with gridded data and random distributions. Due to the discrepancy of the particles in the former distribution, we extended the use of the CSPM method for the approximation of third order derivatives. Our new kernel function provides three accurate numerical schemes, in conjunction with the trapezoidal rule, Simpson’s rule and the CSPM method respectively.