Finite Difference Method with Dirichlet Problems of 2D Laplace’s Equation in Elliptic Domain

In this study finite difference method (FDM) is used with Dirichlet boundary conditions on rectangular domain to solve the 2D Laplace equation. The chosen body is elliptical, which is discretized into square grids. The finite difference method is applied for numerical differentiation of the observed example of rectangular domain with Dirichlet boundary conditions. The obtained numerical results arecompared with analytical solution. The obtained results show the efficiency of the FDM and settled with the obtained exact solution. The study objective is to check the accuracy of FDM for the numerical solutions of elliptical bodies of 2D Laplace equations. The study contributes to find the heat (temperature) distribution inside a regular rectangular elliptical discretized body

A Modified ODE Solver for Autonomous Initial Value Problems

In this work, modified version of a well-known variant of Euler method, known as the Improved Euler method,is proposed with a view to attain greater accuracy and efficiency. The attention is focused upon performance ofthe proposed method in autonomous initial value problems of ordinary differential equations. Order of accuracyof the proposed modified method is proved to be two using Taylor’s expansion. Numerical experiments areperformed using MS Excel 2010.Keywords: ODE solver, numerical solution, initial value problem

Analysis of Accuracy, Stability, Consistency and Convergence of an Explicit Iterative Algorithm

In this work, an analysis is carried out vis-à-vis an explicit iterative algorithm proposed by Qureshi et al (2013) for initial value problems in ordinary differential equations. The algorithm was constructed using the well – known Forward Euler’s method and its variants. Discussion carries with it an investigation for stability, consistency and convergence of the proposed algorithm-properties essential for an iterative algorithm to be of any use. The proposed algorithm is found to be second order accurate, consistent, stable and convergent. The regions and intervals of absolute stability for Forward Euler method and its variants have also been compared with that of the proposed algorithm. Numerical implementations have been carried out using MATLAB version 8.1 (R2013a) in double precision arithmetic. Further, the computation of approximate solutions, absolute and maximum global errors provided in accompanying figures and tables reveal equivalency of the algorithm to other second order algorithms taken from the literature. Keywords: Iterative Algorithm, Ordinary Differential Equations, Accuracy, Consistency, Convergence

On the Construction and Comparison of an Explicit Iterative Algorithm with Nonstandard Finite Difference Schemes

An explicit iterative algorithm to solve both linear and nonlinear problems of ordinary differential equations with initial conditions is formulated with main focus given on its comparison with some non-standard finite difference schemes. Two first order linear initial value problems (IVPs) with periodic behavior are used to analyze the performance of the proposed algorithm with respect to maximum absolute error and computational effort where proposed algorithm performs better in both cases. The proposed algorithm efficiently follows the oscillatory behavior of models like Lotka-Volterra predator-prey and mass-spring system (damped case) in comparison to the nonstandard schemes. All necessary computations have been carried out through MATLAB version 8.1 (R2013a) in double precision arithmetic. Numerical results obtained by the proposed algorithm are found to be computationally reliable and practical in comparison with two nonstandard finite difference schemes discussed in literature. Keywords: Iterative algorithm, nonstandard finite difference scheme, Initial conditions, Maximum absolute error

Some New Time and Cost Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases

Some New Time and Cost Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases

A new three step derivative free method using weight function for numerical solution of non-linear equations arises in application problems

Abstract In this paper a three-step numerical method, using weight function, has been derived for finding the root of non-linear equations. The proposed method possesses the accuracy of order eight with four functional evaluations.The efficiency index of the derived scheme is 1.682. Numerical examples, application problems are used to demonstrate the performance of the presented schemes and compare them to other available methods in the literature of the same order. Matlab, Mathematica 2021 & Maple 2021 software were used for numerical results

A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function

It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in  literature to find root but their cost are very high In this regard we introduce a two-step  fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods