A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages

A Functionally-Fitted Block Numerov Method for Solving Second-Order Initial-Value Problems with Oscillatory Solutions

[EN] A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, y′′=f(x,y,y′), it is a fourth order convergent method for the special second-order ordinary differential equation, y′′=f(x,y). Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

COMPUTING OSCILLATING VIBRATIONS EMPLOYING EXPONENTIALLY FITTED BLOCK MILNE’S DEVICE

Background and Objectives: The idea of estimating oscillating vibration problems via multinomial basis function haven been seen by some authors as a convenient approach but not appropriate. This is as result of the behavior of the problem and as such depends largely on the step size and frequency. This research article is geared towards computing oscillating vibrations employing exponentially fitted block Milne’s device (COVEFBMD). Materials and Methods: This is specifically designed using interpolation and collocation via exponentially fitted method as the approximate solution to generate COVEFBMD, thereby finding the tolerance level of the method. Results: Some numerical examples were selected and implemented on Mathematica kernel 9 to show speed, technicality and accuracy. Conclusion: The completed solutions show that COVEFBMD performs better than the existing methods because of its ability to design a worthy step size; decide the tolerance level resulting to maximized errors

Expanded Trigonometrically Matched Block Variable-Step-Size Technics for Computing Oscillating Vibrations

The expanded trigonometrically matched block variable-step-size technics for computing oscillatory vibrations are considered. The combination of both is of import for determining a suited step-size and yielding better error estimates. Versatile schemes to approximate the error procedure bank on the choice of block variable-step-size technics. This field of study employs an expanded trigonometrically matched block variablestep- size-technics for computing oscillating vibrations. This expanded trigonometrically matched is interpolated and collocated at some selected grid points to form the system of equations and simplifying as well as subbing the unknowns values into the expanded trigonometrically matched will produce continuous block variable-step-size technic. Valuating the continuous block variable-step size technics at solution points of will lead to the block variable-step-size technics. Moreover, this operation will give rise to the principal local truncation error (PLTE) of the block variable-step-size technics after showing the order of the method. Numeral final results demonstrate that the expanded trigonometrically matched block variable-step-size technics are more efficient and execute better than existent methods in terms of the maximum errors at all examined convergence criteria. In addition, this is the direct consequence of designing a suited step-size to fit the acknowledged frequence thereby bettering the block variable-step-size with controlled errors

A Dilated Trigonometrically Equipped Algorithm to Compute Periodic Vibrations through Block Milne’s Implementation

This paper intends to investigate the use of a dilated trigonometrically equipped algorithm to compute periodic vibrations in block Milne's implementation. The block-Milne implementation is established by developing a block variable-step-size predictor-corrector method of Adam’s family using a dilated trigonometrically equipped algorithm. The execution is carried out using a block variable-step-size predictor-corrector method. This system has significant advantages that include the varying step-size and finding out the convergence-criteria and error control. Convergence-criteria and operational mode are discussed to showcase the accuracy and effectuality of the proposed approach

A Dilated Trigonometrically Equipped Algorithm to Compute Periodic Vibrations through Block Milne’s Implementation

This paper intends to investigate the use of a dilated trigonometrically equipped algorithm to compute periodic vibrations in block Milne’s implementation. The block-Milne implementation is established by developing a block variablestep-size predictor-corrector method of Adam’s family using a dilated trigonometrically equipped algorithm. The execution is carried out using a block variable-step-size predictor-corrector method. This system has significant advantages that include the varying step-size and finding out the convergence-criteria and error control. Convergence-criteria and operational mode are discussed to showcase the accuracy and effectuality of the proposed approach

Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages

The Mathematica Kernel Programming Codes Designed for Implementing Block Milne’s Device

In this article, we propose the Mathematica kernel programming codes designed for implementing block Milne’s device employing an expounded trigonometrically fitted method. Block Milne’s device is an extraction of Adam’s family constructed via expounded trigonometrically fitted method. We execute the Mathematica kernel programming codes of block Milne’s device in a block by block mode. This proficiencies of scientific computing have great advantages of easy computation, speed, faster convergence and accuracy. Other numerical gains of block Milne’s device includes; changing the step-size, deciding the convergence criteria and control errors. Additionally, the Mathematica kernel programming codes for implementing block Milne’s device is performed on some special problems to demonstrate the accuracy and efficiency

Adapted block hybrid method for the numerical solution of duffing equations and related problems

Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established