Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems

[EN]This paper is devoted to the development and analysis of a modified family of Falkner- type methods for solving differential systems of second-order initial-value problems. The approaches of collocation and interpolation are adopted to derive the new methods. These modified methods are implemented in block form to obtain the numerical solutions to the considered problems. The study of the properties of the proposed block Falkner-type methods reveals that they are consistent and zero-stable, and thus, convergent. From the stability analysis, it could be seen that the proposed Falkner methods have non-empty sta- bility regions for k = 2 , 3 , 4 . Some numerical test are presented to illustrate the efficiency of the proposed family

An Improved Oscillation Result for a Class of Higher Order Non-canonical Delay Differential Equations.

[EN]In this work, by obtaining a new condition that excludes a class of positive solutions of a type of higher order delay differential equations, we were able to construct an oscillation criterion that simplifies, improves and complements the previous results in the literature. The adopted approach extends those commonly used in the study of second-order equations. The simplification lies in obtaining an oscillation criterion with two conditions, unlike the previous results, which required at least three conditions. In addition, we illustrate the improvement with the new criterion, applying it to some examples and comparing the results obtained with previous results in the literature

On the asymptotic and oscillatory behavior of the solutions of a class of higher-order differential equations with middle term

[EN]In this paper, we deal with the asymptotic and oscillatory behavior of the solutions of higher-order differential equations with middle term of a particular form. By using generalized Riccati transformations we study the asymptotic behavior and derive a new oscillation criterion. The results obtained here extend and improve some well-known results which have been published recently in the literature. An example is given to illustrate the applicability of the obtained results

An Optimized Two-Step Hybrid Block Method Formulated in Variable Step-Size Mode for Integrating y′′=f(x,y,y′)y''=f(x,y,y') Numerically.

[EN]An optimized two-step hybrid block method is presented for integrating general second order initial value problems numerically. The method considers two intra-step points which are selected adequately in order to optimize the local truncation errors of the main formulas for the solution and the first derivative at the final point of the block. The new proposed method is consistent, zero-stable and has seventh algebraic order of convergence. To illustrate the performance of the method, some numerical experiments are presented for solving this kind of problems, in comparison with methods of similar characteristics in the literature

A two-step hybrid block method with fourth derivatives for solving third-order boundary value problems

Abstract This manuscript proposes an implicit two-step hybrid block method which incorporates fourth derivatives, for solving linear and non-linear third-order boundary value problems in ODEs. The derivation of the present method is based on collocation and interpolation techniques, and the convergence analysis of the new strategy is proved to be seventh-order convergent. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. Numerical experiments are studied to show the performance and viability of the proposed approach. The numerical results demonstrated that the new technique gives accurate approximations, which are better than some existing strategies in the available literature and also found to be in good agreement with known analytical solutions

Numerical solution of third‐order boundary value problems by using a two‐step hybrid block method with a fourth derivative

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Solving SIVPs of Lane–Emden–Fowler Type Using a Pair of Optimized Nyström Methods with a Variable Step Size.

[EN]This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation

A new one-step method with three intermediate points in a variable step-size mode for stiff differential systems.

[EN]This work introduces a new one-step method with three intermediate points for solving stiff differential systems. These types of problems appear in different disciplines and, in particular, in problems derived from chemical reactions. In fact, the term “stiff”’ was coined by Curtiss and Hirschfelder in an article on problems of chemical kinetics (Hirschfelder, Proc Natl Acad Sci USA 38:235–243, 1952). The techniques of interpolation and collocation are used in the construction of the scheme. We consider a suitable polynomial to approximate the theoretical solution of the problem under consideration. The basic properties of the new scheme are analyzed. An embedded strategy is adopted to formulate the proposed scheme in a variable stepsize mode to get better performance. Finally, some models of initial-value problems, including ordinary and time-dependent partial differential equations, are solved numerically to assess the performance and efficiency of the proposed technique, with applications to real-world problems

An adaptive pair of one-step hybrid block Nyström methods for singular initial-value problems of Lane–Emden–Fowler type.

[EN]In this paper, an optimized pair of hybrid block techniques is presented and successfully applied to integrate second-order singular initial value problems of Lane–Emden–Fowler type emanating from applied sciences and engineering. An adaptive technique implementation is considered. One of the proposed one-step hybrid block techniques is obtained by using three intermediate points. The obtained block formulas are then paired with a suitable set of formulas applied at the first step to avoid the singularity issue at the left end of the integration interval. Some real-world application problems, including the well-known isothermal gas sphere’s equations, are integrated numerically to ascertain our developed error estimation and control strategy impact. The presented numerical simulations confirm the superiority and robust performance of the proposed scheme

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

[EN]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