One-step Block Method Research Articles

An optimized hybrid block method for solving nonlinear singularly perturbed problems using non-uniform Jain-type meshes

This work aims to implement a one-step block method with three optimized hybrid points for solving singularly perturbed nonlinear initial value problems on non-uniform meshes. The basic properties of the method are investigated and it is shown that the proposed method is consistent, zero-stable, convergent and has a large region of absolute stability. The presented numerical results confirm the efficiency and accuracy of the proposed method.

A variable step-size implementation of a new one-step block method for integrating Burgers' model equation

This article proposes a variable stepsize implementation of an improved one-step block method (OBM) that uses two intermediate points to solve Burgers' equation. The model's problems under consideration are semi-discretized by reducing them into an equivalent differential system using the method of lines. The OBM is improved with an embedding-like method executed in variable stepsize mode. A few of Burgers' real-life model problems are numerically solved to test the accuracy of our error estimation and mesh selection strategy. The numerical results confirmed that the variable stepsize implementation used in this manuscript performed better than the fixed stepsize implementation.

The Direct Two-Point Block One-Step Method Which is efficient and Suitable for Solving Second-Order Differential Equation

Abstract: A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size.Two point four step direct implicit block method is developed for solving directly the second order system of ordinary differential equations (ODEs) using variable step size. The method will estimate the solutions of Initial Value Problems (IVPs) at two points simultaneously by using four backward steps. The method developed is suitable for the numerical integration of non stiff and mildly stiff differential systems. Numerical results are given to compare the efficiency of the developed method to the existence block method.

Optimization of One-Step Block Method for Solving Second-Order Fuzzy Initial Value Problems

In this article, we present a one-step hybrid block method for approximating the solutions of second-order fuzzy initial value problems. We prove the stability and convergence results of the method and present several examples to illustrate the efficiency and accuracy of the proposed method. The numerical results are compared with the existing ones in the literature.

Numerical solution to Volterra Integro-Differential Equations of the second kind by hybrid one-step block method

The hybrid block one-step method of order four is presented and implemented to solve first order Volterra Integro-Differential Equations (VIDEs). The technique is developed using Lagrange interpolation method. The numerical solutions of VIDEs will be solved at two-point concurrently using the proposed numerical method. Properties of the method such as consistency, order and region of absolute stability are explored. After that, the linear and nonlinear of VIDEs has been solved and the numerical results show that the accuracy and efficiency of the hybrid block method is better than the other methods which is RK4 and VIM.

Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {xn,xn+r} while its second and third derivatives are collocated at all points {xn,xn+r,xn+s,xn+t,xn+1} in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy.

Generalized Hybrid One-Step Block Method Involving Fifth Derivative for Solving Fourth-Order Ordinary Differential Equation Directly

A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated atxn,xn+r,xn+s,xn+twhile its fourth and fifth derivatives are collocated at all pointsxi,i=0,r,s,t,1, in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error.

DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In this article, boundary value problems of second order linear ordinary differen- tial equations are solved directly using hybrid block one-step method. Power series of order six is adopted as basis function to derive this method through collocation and interpolation approach. Shooting method is employed to transform boundary value problem into initial value problem. Then hybrid block one step is used to approximate the solution. The perfor- mance of the new method is shown by solving some boundary value problems examined by previous methods.

SOLVING STIFF DIFFERENTIAL EQUATIONS USING A-STABLE BLOCK METHOD

This paper will present the two-point block one-step method for solving stiff ordinary differential equations (ODE s). The propose block method is A-stable and the order is three. The solutions will be obtain simultaneously in block and produces two approximate solutions using constant step size. The method is similar as the one-step method and it is self-starting but the im- plementation is based on the predictor and corrector formulae. The order of the method will be discussed. The numerical results is presented to illustrate the applicability of the propose method. The results clearly shown that the propose method is able to produce comparable and better results compared to the existing method when solving stiff differential equations.

Solving delay differential equations by the five-point one-step block method using Neville's interpolation

A five-point one-step block method based on the Newton backward divided difference formulae for the solution of first-order delay differential equations is derived. The proposed block method will approximate the solutions of initial value problems at five points simultaneously using variable step size. The approximation of the delay term is calculated using Neville's interpolation. The block method will be formulated in terms of linear multistep method, but the method is equivalent to one-step method. The order of the block method will be discussed. The P-stability and Q-stability regions of the block method using Neville's interpolation for the delay term are presented for a fixed step size. Numerical results are given to show the efficiency of the proposed method and compared with the existing method.

NUMERICAL SOLUTION OF LINEAR DIRICHLET TWO-POINT BOUNDARY VALUE PROBLEMS USING BLOCK METHOD

This paper presents a direct two-point block one-step method for solving linear Dirichlet boundary value problems (BVPs) directly. The block method is formulated using Lagrange interpolating polynomial. Mathematical problems which involve higher order ordinary differential equations (ODEs) were likely to be reduced into the system of first order equations in order to solve it. However, this method will solve the second order linear Dirichlet BVPs directly without reducing it to the system of first order equations. The direct solution of the linear Dirichlet BVPs will be calculated at the two-points simultaneously using constant step size. This method will be used together with the linear shooting technique to construct the numerical solution. The implementation is based on the predictor and corrector formulas in the PE(CE)r mode. Numerical results are given to show the efficiency and performance of this method compared to the existing methods.

Solving second order delay differential equations by direct two and three point one-step block method

In this paper we present a two point and three point one-step block method for solving second order delay differential equations (DDEs). The one-step block method will solve directly the second order DDEs without reducing to first order equations. The two point and three point one-step block method will compute the solutions for the DDEs at two and three points simultaneously along the interval. These methods will solve the retarded type of DDE of single delay using constant step size. The P- stability and Q-stability are also discussed. The numerical results are presented to illustrate the performance of those block method for solving delay differential equations.

NUMERICAL SOLUTION FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS USING BLOCK METHOD

The purpose of this paper is to present a four point direct block one-step method for solving directly the general second order nonstiff initial value problems (IVPs) of ordinary differential equations (ODEs). The mathematical problems in real world can be written in the form of differential equations and arise in the fields of science and engineering such as fluid dynamic, electric circuit, motion of rocket or satellite and other area of application. The proposed method will estimate the approximation solutions at four points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed method.

2-Point Implicit Block One-Step Method Half Gauss-Seidel For Solving First Order Ordinary Differential Equations

In this paper, a 2 point implicit block one-step method for solving a system of ordinary differential equations (ODEs) using variable step size is developed. This method will estimate the solutions of initial value problems at 2 point simultaneously based on equidistant block method. Numerical results are given to compare the efficiency of the new method and that of the 2 point implicit block one-step method by Rosser (1967). Keywords: Block Method; One-Step Method; Ordinary Differential Equations