Hybrid Block Method Research Articles

Block Hybrid Method for the Numerical solution of Fourth order Boundary Value Problems

A linear Multistep Block Hybrid Method with four off-grid points is presented for the direct approximation of the solution of fourth order Boundary Value Problems. Multiple Finite Difference formulas are derived and grouped into a unique block to form a numerical integrator to solve directly the fourth order problem, without the need to reduce it previously to a first-order system. The convergence of the proposed method is discussed. The superiority of this method over existing methods is established numerically considering different problems that have appeared in the literature.

Hybrid-Block Numerical Method for Solving Second Order Ordinary Differential Equations

In this work, a new hybrid-block method for the solution of second order ordinary differential equation is developed using power series as the basis function. The developed scheme was used to solve some problems and the result compare with existing results to ascertain the superiority of the new method

Order six generalised hybrid block method for the solution of third order initial value problems

Order six generalised hybrid block method for the solution of third order initial value problems

Implementation of the One-Step One-Hybrid Block Method on the Nonlinear Equation of a Circular Sector Oscillator

Solving nonlinear differential equation of a circular sector oscillator is of a scientific importance. Thus, to solve such equations, a single- step implicit block method involving one hybrid point with the introduction of a third derivative is proposed. To derive this method, the approximate basis solution is interpolated at {xn, xn + 3/5} while its second and third derivatives are collocated at all points {xn, xn + 3/5, xn + 1}on the integrated interval of approximation. Numerical results are presented in the form of table and graphs for the variation of different physical parameters. The study reveals that the proposed hybrid block method is zero stable, which proves that it is convergent beside a significant interval of absolute stability, thus making it suitable for solving stiff ODEs.

Development and Implementation of Block Unification Multi-step Methods for the Solution of Second Order Ordinary Differential Equations

In this paper, linear multi-step hybrid block methods with three-, four- and five-step numbers are developed for approximating directly the solution of second order Initial and Boundary Value Problems (IBVPs). Multiple finite difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to second order IBVPs over sub-intervals. A new class of orthogonal polynomials constructed as basis function to develop the hybrid block methods adopting collocation technique with a non-negative weight function. The scheme is applied as simultaneous integrator to second order initial value and boundary value problems of ODEs. The properties and convergence of the proposed method are discussed. The derived schemes were used to solve some problems and the numerical result shows the effectiveness, accuracy and superiority of the method over the existing methods found 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.

An Accurate Continuous Hybrid Block Method for the Direct Solution of General Second Initial Value Problem

In this research, a new hybrid block method of order 5 for solving second order Initial Value Problems (IVPs) is determined. One step with 3 off step points is used to generate the coefficients of the new method. This method approximates the solutions of the second order IVPS at xn+1/6, xn+2/6, xn+3/6, xn+1, concurrently by using interpolation and collocation approach with fixed step size. The developed method don’t need to reduce the IVPs to its equivalent system of first order IVPs. The numerical results obtained show that the new method outperforms the existing methods in terms of error.

The Double Step Hybrid Linear Multistep Method for Solving Second Order Initial Value Problems

In this paper, we develop the double step hybrid linear multistep method for solving second order initial value problems via interpolation and collocation method of power series approximate solution to give a system of nonlinear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method.
 The basic numerical properties of the hybrid block method was established and found to be zero-stable, consistent and convergent. The efficiency of the new method was conformed on some initial value problems and found to give better approximation than the existing methods.

The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

Order Six Generalized Hybrid Block Method for the Solution of Third Order Initial Value Problems

In this article, a generalise two step method with two off step points that gives better approximate solution for third order IVP of ODEs is examined. Method of collocation and interpolation is used where approximated power series is considered as basis function. The stability properties of the method is established and the method is applied to variety of test problems. The results generated proved the superiority of the new method over existing methods in terms of error.

Hybrid Third Derivative Block Method for the Solution of General Second Order Initial Value Problems with Generalized One Step Point

This paper deals with two-step hybrid block method with one generalized off-step points for solving second order initial value problem. In derivation of this method, power series of order nine are interpolated at the first two step points while its second and third derivatives are collocated at all point in the selected interval. The new developed method is employed to solve several problems of second order initial value problems. Convergence analysis of the new method alongside numerical procedure is established. The performance of the proposed method is found to be more accurate than existing method available in the literature when solving the same problems.

An efficient optimized adaptive step-size hybrid block method for integrating differential systems

This paper deals with the development, analysis and implementation of an optimized hybrid block method having different features, for integrating numerically initial value ordinary differential systems. The hybrid nature of the proposed one-step scheme allows us to bypass the first Dahlquist’s barrier on linear multi-step methods. The theory of interpolation and collocation has been used in the development of the method. We assume an appropriate polynomial representation of the theoretical solution of the problem and consider three off-step points in a one-step block. One of these three off-step points is fixed and the other two off-step points are optimized in order to minimize the local truncation errors of the main method and other additional formula. The resulting scheme is of order five having the property of A-stability. An embedded-type approach is used in order to formulate the proposed method in adaptive form, showing a high efficiency. The adaptive method is tested on well-known differential systems viz. the Robertson’s system, a Gear’s system, a system related with Jacobi elliptic functions, the Brusselator system, and the Van der Pol system, and compared with some well-known numerical codes in the scientific literature.

Numerical solution of boundary value problems by using an optimized two-step block method

This paper aims at the application of an optimized two-step hybrid block method for solving boundary value problems with different types of boundary conditions. The proposed approach produces simultaneously approximations at all the grid points after solving an algebraic system of equations. The final approximate solution is obtained through a homotopy-type strategy which is used in order to get starting values for Newton’s method. The convergence analysis shows that the proposed method has at least fifth order of convergence. Some numerical experiments such as Bratu’s problem, singularly perturbed, and nonlinear system of BVPs are presented to illustrate the better performance of the proposed approach in comparison with other methods available in the recent literature.

An Accurate Implicit Quarter Step First Derivative Block Hybrid Method (AIQSFDBHM) for Solving Ordinary Differential Equations

The new accurate implicit quarter step first derivative blocks hybrid method for solving ordinary differential equations have been proposed in this paper via interpolation and collocation method for the solution of stiff ODEs. The analysis of the method was study and it was found to be consistent, convergent, zero-stability, We further compute the region of absolute stability region and it was found to be Aα − stable . It is obvious that, the numerical experiments considered showed that the methods compete favorably with existing ones. Thus, the pair of numerical methods developed in this research is computationally reliable in solving first order initial value problems, as the results from numerical solutions of stiff ODEs shows that this method is superior and best to solve such problems as in tables and figures.

Heat transfer study of convective fin with temperature‐dependent internal heat generation by hybrid block method

AbstractPurposeIn this study, an implicit one‐step hybrid block method using two off‐step points involving the presence of a third derivative for solving second‐order boundary value problems are subjected to Dirichlet‐mixed conditions.MethodologyTo derive this method, the approximate power series solution is interpolated at while its second and third derivatives are collocated at all points on the integrated interval of approximation.FindingsThe new derived method not only performs better compared with the existing methods when solving the same problems but also obtains better properties of the numerical method. Afterward, the proposed method is applied to solve the problem of a convective fin with temperature‐dependent internal heat generation. The effects of various physical parameters on temperature distribution are also examined.

Numerical solution of third order boundary value problems using one-step hybrid block method

Abstract Numerical hybrid block methods have been thought to be an appropriate method for solving ordinary differential equation. Therefore, this paper introduces a one-step hybrid block method of order five for directly solving third order boundary value problems. Power series of order seven is used for the development of the new method by using collocation and interpolation approach. The proposed method together with shooting technique will solve third order BVPs. Some numerical examples are involved to show the efficiency of the method. The numerical results clearly show that the developed method is able to generate good results when it is compared with existing method in terms of error. It has been noted, the new proposed method outperforms the existing methods in terms of accuracy.

A Block Hybrid Method for Non-linear Second Order Boundary Value Problems

In this paper, we propose a new method of hybrid type, for non-linear second order BVPs, to overcome some drawbacks of classic difference finite methods. An estimation of the local and global error is also given. Numerical tests on BVPs which is well known in the literature confirm the accuracy of the theoretical results.

An Orthogonal-based Self-starting Numerical Integrator for Second Order Initial and Boundary Value Problems ODEs

The direct integration of second order initial and boundary value problems is considered in this paper. We employ a new class of orthogonal polynomials constructed as basis function to develop a two-step hybrid block method (2SHBM) adopting collocation technique. The recursive formula of the class of polynomials have been constructed, and then we give analysis of the basic properties of 2SHBM as findings show that the method is accurate and convergent. The boundary locus of the proposed 2SHBM shows that the new scheme is A-stable.

A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were investigated and found to be zero stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing method.

Two-Step Hybrid Block Method for Solving Nonlinear Jerk Equations

In this paper, a block method with one hybrid point for solving Jerk equations is presented. The hybrid point is chosen to optimize the local truncation errors of the main formulas for the solution and the derivative at the end of the block. Analysis of the method is discussed, and some numerical examples show that the proposed method is efficient and accurate.