K-Step Block Predictor-Corrector Methods for Solving First Order Ordinary Differential Equations

A K-step block Predictor-Corrector Methods for solving first order ordinary differential equations are formulated and applied on non-stiff and mildly stiff problems using variable step size technique. In this method, collocation and interpolation of the power series as the approximate solution is carried out with aim of generating the continuous scheme. The investigation of some selected theoretical properties of the method is analysed as well as determination of the region of absolute stability of the method. In addition, the implementation of the proposed method is done by applying variable step size techniqu

Solving Continuous and Weakly Singular Linear Volterra Integral Equations of the second kind by Laplace Transform Method

This work provides solutions to some continuous and weakly singular linear Volterra integral equations of the second kind by Laplace transform method. With the basic definition of convolution integral of two functions and Volterra fundamental theorems, the Laplace transform method gives an efficient and remarkable performance. Test problems are presented to show the efficiency and reliability of the method. Keywords: Volterra Integral equations, continuous and weakly singular Kernels , Laplace metho

Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations

Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictor corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the chief local truncation error] (CLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels

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

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

Formulating Mathematica pseudocodes of block-Milne's device for accomplishing third-order ODEs

Formulating Mathematica pseudocodes for carrying out third-order ordinary differential equations (ODEs) is of essence necessary for proficient computation. This research paper is prepared to formulate Mathematica Pseudocodes block Milne’s device (FMPBMD) for accomplishing third-order ODEs. The coming together of Mathematica pseudocodes and proficient computing using block Milne’s device will bring about ease in ciphering, proficiency, acceleration and better accuracy. Side by side estimation and extrapolation is considered with successive function approximation gives rise to FMPBMD. This FMPBMD turns out to bring about the star local truncation error thereby finding the degree of the scheme. FMPBMD will be implemented on some numerical examples to corroborate the superiority over other block methods established by employing fixed step size and handled computation

Multiprocessing Suited Pace Size Proficiency for Ciphering First Order ODEs

Abstract-Appraising computed error in forecasting-adjusting system is all essential for purposeful acquiring suited pace size. Diverse schemata for controlling/estimating error bank on forecasting-adjusting system. This study examines multiprocessing suited pace size (adaptive) proficiency for ciphering first order ordinary differential equations (ODEs). This involves compounding Newton’s back difference interpolating multinomial with numeral consolidation method. This is valuated at more or less preferred grid points to invent multiprocessing forecasting-adjusting system. Moreover, process progresses to produce main local truncation error (MLTE) of multiprocessing forecasting-adjusting system after showing degree of the system. Numeral resolutions manifest effectiveness of varying pace size in working out first order ODEs. Accomplished resolution rendered is aided using mathematical software program. Mathematical resolutions show-case adaptive proficiency is effectual and function better than subsisting systems with respect to the maximal computed errors in the least time-tested tolerance bounds

A Variable-stpe-size Block Predictor-corrector Method for Ordinary Differential Equations

A Variable-stpe-size Block Predictor-corrector Method for Ordinary Differential Equations Background and Objective: Over the years, block predictor-corrector method has been limited to predicting and correcting methods without further use. Predictor-corrector method possesses other attributes that utilize the Principal Local Truncation Error (PLTE) to design a suitable step size, tolerance level and control error. This study examined a variable step-size block predictor-corrector method for solving first-order Ordinary Differential Equations (ODEs). Materials and Methods: The combination of Newton’s backward difference interpolation polynomial and numerical integration methods were applied and evaluated at some selected grid points to formulate the block predictor-corrector method. Nevertheless, this process advances to generate the PLTE of the block predictor-corrector method after establishing the order of the method. Results: The numerical results were shown to demonstrate the performance of the variable step-size block predictor-corrector method in solving first-order ODEs. The complete results were incurred with the aid of Mathematica 9 kernel for Microsoft windows (64bit). Conclusion: Numerical results showed that the variable-step-size block predictor-corrector method is more effective and perform better than existing methods in terms of the maximum errors at all tested tolerance levels as well as designing a suitable step size to control error

3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form =(,,),(0)=,(0)=. The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy

Block Solver for Multidimensional Systems of Ordinary Differential Equations

This research study aimed at developing block solver for multidimensional systems (BSMS) of ordinary differential equations. This method will be formulated via interpolation and collocation techniques with multinomial as the basis function approximate. The block solver has the capacity to utilize each principal local truncation errors to generate the convergence criteria that will ensure convergence. Some theoretical properties will be stated. The process for executing the block solver will be done via the idea of the convergence criteria introduced. Step by step implementation algorithm will be specified. Some selected model applications will be worked out and a suitable step size will be determined to satisfy the convergence criteria in order to enhance the accuracy and efficiency of the method. The implementation of BSMS is coded in Mathematica and executed under the platform of Mathematica Kernel 9