Hybrid Block Method Research Articles

A ninth-order first derivative method for numerical integration

In this paper, we present a ninth–order block hybrid method for the numerical solution of stiff and non–stiff systems of first–order differential equations. The method is based on an interpolation and collocation approach which results in a single continuous formulation; from which eight discrete schemes that make the block method were obtained. A convergence analysis of our method illustrated that it is A–stable, consistent, and convergent. We applied our method to some numerical examples which showed that the new method not only outperformed a second derivative method of order fourteen in the literature but also compared well with the exact solution.

A Six-Point y-Function Hybrid Block Method for Direct Solution of Third Order Ordinary Differential Equations

In this article, a new continuous numerical method was developed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method was derived by adopting interpolation and collocation approach using a combination of power series and exponential functions as the basis function for the approximate solution to the ODEs. The approximate solution was interpolated at both nodal and off-nodal points while the differential systems from the approximate solution was collocated at all nodal points to obtain the set of methods with given step-numbers. This derived approach ensured that the hybrid points are obtained at the y-values of the methods. This class of hybrid linear multi-step method satisfied the conditions for usability and accuracy of any given method. These conditions were confirmed by testing the consistency, stability and convergence of the developed method. The method was applied to solve directly linear and non-linear third order ODEs without reduction to system of first-order ordinary differential equations. Results from the computation compared favorably with some existing numerical methods in literature with comparable properties to confirm the superiority of the newly developed method.

The Efficiency of Block Hybrid Method for Solving Malthusian Growth Model and Prothero-Robinson Oscillatory Differential Equations

The efficiency of block hybrid method for solving Malthusian Growth Model, Prothero-Robinson equation and highly stiff oscillatory differential equations was proposed using a power series polynomial through interpolation and collocation. The new method's basic properties, including order, error constant, consistency, zero-stability, and stability regions, were comprehensively analyzed and satisfied all necessary conditions for analysis. Tested on various real-life problems, the new method demonstrated superior performance compared to existing techniques. The study highlights the innovative approach's enhanced convergence and stability properties, providing a more reliable numerical analysis tool for researchers and practitioners. Practical applications validate the method's effectiveness, showcasing its superior performance across different examples and establishing it as a highly effective solution for Malthusian growth model and oscillatory differential equations.

Hybrid block formulae whose eigenvalues of Jacobian matrices are close to the imaginary axis of the complex plane

Methods for integrating stiff initial value problems are required to be A-stable. Of great interest are A-stable methods whose Jacobian matrices have their eigenvalues close to the imaginary axis of the complex plane. This class of A-stable methods are very rare. This paper is on the development of a new family of A-stable hybrid block method whose Jacobian matrices possess eigenvalues on the imaginary of the complex plane via interpolation and collocation techniques. The family of methods developed herein are A-stable for order p ≤ 18. Numerical solutions generated by the new method are compared with existing methods in the literature. The numerical results show that the new class of methods are more efficient and accurate.

Two-Step Hybrid Block Method for Solving Second Order Initial Value Problem of Ordinary Differential Equations

Two-Step Hybrid Block Method for Solving Second Order Initial Value Problem of Ordinary Differential Equations

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.

Numerical integration of third-order BVPs using a fourth-order hybrid block method

This research paper introduces a new hybrid block method to solve the third-order boundary value problems (BVPs). The method combines interpolation and collocation and uses a power series polynomial to find an approximate solution to the considered third-order BVPs. Some third-order BVP models are numerically solved to verify the performance and efficiency of the proposed method, and the approximate solution from the proposed method is more efficient when compared to some existing numerical methods. In summary, the proposed method provides reliable and efficient accuracy for solving third-order BVPs, making it a valuable contribution to the fields of numerical analysis and computational mathematics. The advantages of the proposed method include improved computational time efficiency and accuracy in terms of maximum absolute errors for solving third-order BVPs.

A new block method with variable stepsize implementation for solving third‐order differential systems

This article presents an efficient one‐step hybrid block method (OHBM) to solve third‐order initial value problems (IVPs). The proposed method is derived using interpolation and collocation techniques of the assumed exact solution and its derivatives. The theoretical properties of the OHBM are analyzed. An embedding‐like technique is utilized and executed in an adaptive mode to improve the performance of the proposed method. The OHBM's efficiency and accuracy are tested by solving practical application problems in applied sciences and engineering. The numerical solutions demonstrate that the proposed OHBM is highly efficient and provides very accurate approximations.

An Algorithm for Approximating Third-Order Ordinary Differential Equations with Applications to Thin Film Flow and Nonlinear Genesio Problems

Third-order problems have been found to model real-life phenomena such as thick film fluid flow, boundary layer problems, and nonlinear Genesio problems, to name a few. This research focuses on the development of an algorithm using a basis function that combines an exponential function with a power series. This algorithm, called the Exponential Function Induced Algorithm (EFIA), was formulated using the collocation and interpolation techniques. The theoretical analysis of the algorithm was also carried out in this research. The outcome of the analysis showed that the algorithm is consistent, convergent, and zero-stable. The EFIA was applied to solving some real-life third-order problems like thin film flow and nonlinear Genesio problems. The numerical and graphical results obtained showed that the EFIA is accurate, has high precision, and is easy to implement. Algorithm, Exponentially-fitted function, Hybrid block method, Nonlinear Genesio equations, Thin film flow problem, Third-order

A conservative algorithm based on a hybrid block method and tension B‐spline differential quadrature method for Rosenau–KdV–RLW equation

In this study, the authors present a uniform algebraic trigonometric tension B‐spline‐based differential quadrature method combined with an optimized hybrid block method to numerically solve the Rosenau–KdV–RLW equation. The discrete mass and energy have been calculated, showing that they are conserved, thus indicating the efficiency and accuracy of the present approach. The method outperforms other methods and does not require linearization, allowing it to be directly implemented for solving nonlinear partial differential equations.

A One-Step Hybrid Obrechkoff-Type Block Method for First-Order Initial Value Problems in Ordinary Differential Equations

In this paper, we propose a one-step hybrid block method of the Obrechkoff-type block method for solving initial value problems in ordinary differential equations using Taylor series expansion. The new method was implemented with some selected initial value problems to determine the efficiency and accuracy of the method. The convergence and stability properties were also examined. It was discovered that the new method compared favorably with other existing methods in the selected literature.

A Block Hybrid Method with Equally Spaced Grid Points for Third-Order Initial Value Problems

In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

Multi-derivative hybrid block methods for singular initial value problems with application

This study presents a novel class of multi-derivative integrating techniques designed to effectively address singular initial value problems. This study not only delves into the application of these methods but also explores their potential in solving parabolic partial differential equations. We introduce the underlying mathematical framework of the multi-derivative method, derived from power series polynomials. This approach encompasses the strategic incorporation of collocation and interpolation techniques, utilizing a combination of hybrid and non-hybrid points within the problem domain. This multi-derivatives method combines the advantages of derivative-based methods, which offer high accuracy, and finite difference methods, which provide simplicity and stability. By utilizing multiple derivative approximations at various points within the problem domain, the multi-derivative method effectively handles the singularities and captures the discountinuity of the solutions. Furthermore, our investigation includes comprehensive qualitative analysis, demonstrating the method's inherent stability, consistency, and convergence when applied to practical problems. The numerical results obtained through the application of the multi-derivative method attest to its reliability and efficiency, particularly in tackling singular and stiff problems.

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

A one-step three-parameter optimized hybrid block method and second derivative hybrid block method with optimized points were proposed to solve first-order ordinary differential equations. The techniques of interpolation and collocation were adopted for the derivation of the methods using a three-parameter approximation. The hybrid points were obtained by optimizing the local truncation error of the method. The schemes obtained were reformulated to reduce the number of occurrences of the source term. The hybrid points were used in the derivation of the second derivative hybrid block method. The discrete schemeswere produced as a by-product of the continuous scheme and used to simultaneously solve initial value problems (IVPs) in block mode. The resulting schemes are self-starting, do not require the creation of individual predictors, and are consistent, zero-stable, and convergent. The accuracy and efficiency of the methods were ascertained using several numerical test problems. The numerical results were favourably compared to some techniques from the cited literature.

Numerical integration of stiff problems using a new time-efficient hybrid block solver based on collocation and interpolation techniques

In this study, an optimal L-stable time-efficient hybrid block method with a relative measure of stability is developed for solving stiff systems in ordinary differential equations. The derivation resorts to interpolation and collocation techniques over a single step with two intermediate points, resulting in an efficient one-step method. The optimization of the two off-grid points is achieved by means of the local truncation error (LTE) of the main formula. The theoretical analysis shows that the method is consistent, zero-stable, seventh-order convergent for the main formula, and L-stable. The highly stiff systems solved with the proposed and other algorithms (even of higher-order than the proposed one) proved the efficiency of the former in the context of several types of errors, precision factors, and computational time.

One‐Step Family of Three Optimized Second‐Derivative Hybrid Block Methods for Solving First‐Order Stiff Problems

This paper introduces a novel approach for solving first‐order stiff initial value problems through the development of a one‐step family of three optimized second‐derivative hybrid block methods. The optimization process was integrated into the derivation of the methods to achieve maximal accuracy. Through a rigorous analysis, it was determined that the methods exhibit properties of consistency, zero‐stability, convergence, and A‐stability. The proposed methods were implemented using the waveform relaxation technique, and the computed results demonstrated the superiority of these schemes over certain existing methods investigated in the study.

Family of One-Step A-Stable Optimized Third Derivative Hybrid Block Methods for Solving General Second-Order IVPs

Family of One-Step A-Stable Optimized Third Derivative Hybrid Block Methods for Solving General Second-Order IVPs

A New Two-Step Hybrid Block Method for the FitzHugh–Nagumo Model Equation

This paper presents an efficient two-step hybrid block method (ETHBM) to obtain an approximate solution to the FitzHugh–Nagumo problem. The considered partial differential equation model problems are semi-discretized, reducing them to equivalent ordinary differential equations using the method of lines. In order to evaluate the effectiveness of the proposed ETHBM, three numerical examples are presented and compared with the results obtained through existing methods. The results demonstrate that the proposed ETHBM produces more efficient results than some other numerical approaches in the literature.

A coupled scheme based on uniform algebraic trigonometric tension B-spline and a hybrid block method for Camassa-Holm and Degasperis-Procesi equations

In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points.

A class of single-step hybrid block methods with equally spaced points for general third-order ordinary differential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary differential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the effectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.