Continuous Linear Multistep Method Research Articles

A New Hybrid Block Method via Combined Hermite Polynomials and Exponential Functions as Basis Function

This paper presents the derivation and implementation of a hybrid block method for solving stiff and oscillatory first-order initial value problems of ordinary differential equations (ODEs). The hybrid block method was derived by continuous collocation and interpolation using combined Hermite polynomials and exponential functions as the basis function to produce a continuous implicit Linear Multistep Method (LMM) of order nine and implement in block form. The basic properties of the derived method were studied, and the hybrid block integrator was demonstrated to be zero-stable, convergent, consistent, and to have an A-stable region of absolute stability, which made it suitable for stiff and oscillatory ordinary differential equations. The use of a combined basis in the generation of LMMs is worthy of universal acceptance. The technique indicates that, utilizing an interpolation and collocation approach, continuous LMMs can be derived from combinations of any polynomials and exponential functions. On two sampled stiff and oscillatory problems, the new integrator was tested. The numerical results indicate that our new hybrid block integrator is computationally efficient and outperforms existing methods in terms of accuracy

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

This research work considers derivation of three step four point optimized hybrid block method for solving general third order differential equations (odes) without reduction to systems of lower order odes. A combination of power series and exponential function is used as an approximate solution to the general third order ode problems. Continuous linear multistep method is developed by interpolating the basis function at both grid and off-grid points and collocating the differential function at only grid points. The unknown parameters in the system of linear equations arising from the collocation and interpolation functions were determined and the values substituted in the approximate solution to the problem. The required continuous method is obtained after necessary simplification. The derived method is tested and found to be consistent, symmetric and of low error constant. The results obtained showed a better performance than the existing methods in literature under review.

Direct Solution of Second Order Ordinary Differential Equations with a OneStep Hybrid Numerical Model

This study shows how to solve general second order ordinary differential equations directly using a one-step hybrid block technique. By means of collocation and interpolation of power series approximation, a continuous linear multistep method was developed. The developed method was evaluated at several grid and off-grid points to generate a discrete block approach. Order, error-constant, zero stability, consistency, and convergence were all investigated as basic features of the approach. The method was found to be more efficient and produce better approximations than previous approaches for solving some second order ordinary differential equations with initial value problems.

Numerical Solution of Stiff and Oscillatory Problems using Third Derivative Trigonometrically Fitted Block Method

This paper considered the formulation of continuous third derivative trigonometrically fitted method for the solution of oscillatory first order initial value problems using the technique of interpolation and collocation of the approximate solution by combining polynomial and trigonometric functions. Solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which was evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. The blocks are convergent and stable. Numerical experiments show that the methods compete favorably with existing method.This paper considered the formulation of continuous third derivative trigonometrically fitted block method for the solution of stiff and oscillatory problems. The development of the technique involved the interpolation and collocation of the approximate solution which is the combination of polynomial and trigonometric functions. Solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which is evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. The blocks are convergent and stable. Numerical experiments show that the methods compete favorably with existing method and efficient for the solution of stiff and oscillatory problems.

DERIVATION AND IMPLEMENTATION OF A THREE-STEP HYBRID BLOCK ALGORITHM (TSHBA) FOR DIRECT SOLUTION OF LINEAR AND NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations

Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order

An Efficient Seven-Step Block Method for Numerical Solution of SIR and Growth Model

In this article, a new implicit continuous block method is developed using the interpolation and collocation techniques via Power series as the basis function. A constant step length within a seven-step interval of integration was adopted. The selected grid points were evaluated to get a continuous linear multistep method. The evaluation of the continuous method at the non-interpolation points produces the discrete schemes which form the block. The basic properties of the block method were investigated and found to be consistent, zero stable and hence convergent. The new method was tested on real life problems namely: SIR and Growth model. The results were found to compare favourably with the existing methods in terms of accuracy and efficiency. Keywords: Block method, Growth Model, implicit, power series and SIR model.

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.

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.

Continuous linear multistep method for the general solution of first order initial value problems for Volterra integro-differential equations

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

Symmetric 2-Step 4-Point Hybrid Method for the Solution of General Third Order Differential Equations

This research considers a symmetric hybrid continuous linear multistep method for the solution of general third order ordinary differential equations. The method is generated by interpolation and collocation approach using a combination of power series and exponential function as basis function. The approximate basis function is interpolated at both grid and off-grid points but the collocation of the differential function is only at the grid points. The derived method was found to be symmetric, consistent, zero stable and of order six with low error constant. Accuracy of the method was confirmed by implementing the method on linear and non-linear test problems. The results show better performance over known existing methods solved with the same third order problems. AMS 2010 Subject Classification: 65D05; 65L05; 65L06

Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations

We construct a continuous linear multistep method with trigonometric coefficients from which a symmetric main method as well as additional methods are reproduced. The main and additional methods whose coefficients depend on the frequency and step length are then applied as a trigonometric symmetric boundary value method (SBVM) to solve systems of second-order initial and boundary value problems of the form y″=f(x,y) without first reducing the ordinary differential equation into an equivalent first-order system. Moreover, the method is successfully applied to solve hyperbolic and elliptic partial differential equations, such as the sine-Gordon and the Poisson equations. The stability property of the SBVM is discussed and numerical experiments are performed to show the accuracy of the method.

Derivation of One-Sixth Hybrid Block Method for Solving General First Order Ordinary Differential Equations

This paper focuses on the derivation of one-sixth hybrid block method for the general solution of first order initial values problems of ordinary differential equations. The new proposed method was derived by using the approach of collocation and interpolation of Chebyshev polynomials, approximate solution at some selected points to get a continuous linear multistep method, which was evaluated at some off-grid points to generate hybrid linear multistep methods. Basic properties of the proposed method wasexamined and the method found to be zero-stable, consistent and convergent. The efficiency of the method was tested on some numerical examples and in particular, on well-known SIR Model, Prothero-Robinson oscillatory problem and highly stiff oscillatory problem. On comparison, the new proposed method performed favourably when compare with the existing method proposed by other researchers in the area of Numerical Analysis.

A NEW NUMERICAL INTEGRATOR FOR THE SOLUTION OF GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper considered the development of numerical integrator for the solution of second order initial value problems. The method was derived through the interpolation and collocation of the basis polynomial which is com- bination of power series and exponential function to derive a continuous linear multistep method. The method was implemented in block method which gave solution at a non overlapping interval. The method was found to be conver- gence and A stable. The efficiency of the method was tested on some numerical examples.

English

In this paper, we derived a continuous linear multistep method (LMM) with step number k = 4 through collocation and interpolation techniques using power series as basis function for approximate solution. An order-seven scheme is developed which was used to solve the third-order initial value problems (IVPS) in ordinary differential equation without first reducing to a system first-order. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained was compared favourably with existing methods. Key words: Continuous collocation, multistep methods, interpolation, third-order, power series.

A Note on the Construction of Constant Order Predictor Corrector Algorithm for the Solution of y′ = f(x,y)

This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compare favorably with the existing methods.

Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator

This paper presents the derivation and implementation of a block integrator for the solution of stiff and oscillatory first-order initial value problems of Ordinary Differential Equations (ODEs). The integrator was derived by collocation and interpolation of the combination of power series and exponential function to generate a continuous implicit Linear Multistep Method (LMM). The basic properties of the derived integrator were investigated and the integrator was implemented on some sampled stiff and oscillatory problems. From the results obtained, it is obvious that the block integrator gives better approximation than some existing ones.

A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations

In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.

Starting Order Seven Method Accurately for the Solution of First Initial Value Problems of First Order Ordinary Differential Equations

In this paper, we developed an order seven linear multistep method, which is implemented in predictor corrector-method. The corrector is developed by method of collocation and interpolation of power series, approximate solutions at some selected grid points, to give a continuous linear multistep method, which is evaluated at some selected grid points to give a discrete linear multistep method of order seven. The predictors were also developed by method of collocation and interpolation of power series approximate solution, to give a continuous linear multistep method. The continuous linear multistep method is then solved for the independent solution to give a continuous block formula, which is evaluated at some selected grid points to give discrete block method. Basic properties of the corrector was investigated and found to be zero stable, consistent and convergent. The efficiency of the method was tested on some numerical experiments and found to compare favorably with the existing methods.

A NEW HYBRID BLOCK METHOD FOR THE SOLUTION OF GENERAL THIRD ORDER INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we develop an order six block method using method of collocation and interpolation of power series approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method . The continous hybrid linear multistep method is solved for the independent solutions to give a continous 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 was 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.