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

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Differential Equations ◽  
Stability Property ◽  
Telegraph Equation ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Value ◽  
Step Size ◽  
Grid Points ◽  
The Stability ◽  
Predictor Corrector ◽  
Trigonometrically Fitted Method

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.

scholarly journals Functionally Fitted Block Method for Solving the General Oscillatory Second-Order Initial Value Problems and Hyperbolic Partial Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
S. N. Jator ◽  
F. F. Ngwane ◽  
N. O. Kirby
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Hyperbolic Partial Differential Equations ◽  
Fixed Frequency ◽  
Block Method ◽  
Initial Value ◽  
Partial Differential ◽  
Predictor Corrector

We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.

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

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Hamiltonian Systems ◽  
Initial Value Problems ◽  
Grid Point ◽  
Second Order ◽  
Initial Value ◽  
Step Size ◽  
Energy Conserving ◽  
One Step ◽  
The Stability ◽  
Predictor Corrector

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.

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

2018 ◽  
Vol 8 (3) ◽  
pp. 2943-2948
Author(s):  
J. G. Oghonyon ◽  
S. A. Okunuga ◽  
K. S. Eke ◽  
O. A. Odetunmibi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Interpolation Polynomial ◽  
Numerical Results ◽  
Convergence Criteria ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Grid Points ◽  
Predictor Corrector

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 principal local truncation error (PLTE) 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.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

A Parallel Block Predictor-Corrector Method by Python-Based Distributed Computing

2012 ◽  
Vol 263-266 ◽  
pp. 1315-1318
Author(s):  
Kun Ming Yu ◽  
Ming Gong Lee
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Initial Value Problem ◽  
Message Passing ◽  
Message Passing Interface ◽  
Parallel Structure ◽  
Initial Value ◽  
Speed Up ◽  
Predictor Corrector

This paper is to discuss how Python can be used in designing a cluster parallel computation environment in numerical solution of some block predictor-corrector method for ordinary differential equations. In the parallel process, MPI-2(message passing interface) is used as a standard of MPICH2 to communicate between CPUs. The operation of data receiving and sending are operated and controlled by mpi4py which is based on Python. Implementation of a block predictor-corrector numerical method with one and two CPUs respectively is used to test the performance of some initial value problem. Minor speed up is obtained due to small size problems and few CPUs used in the scheme, though the establishment of this scheme by Python is valuable due to very few research has been carried in this kind of parallel structure under Python.

scholarly journals Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

2019 ◽  
Vol 52 (1) ◽  
pp. 437-450 ◽  
Author(s):  
Mouffak Benchohra ◽  
Soufyane Bouriah ◽  
Juan J. Nieto
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
The Stability ◽  
The Banach Contraction Principle ◽  
Implicit Fractional Differential Equations

AbstractIn this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Solving third order ordinary differential equations directly using hybrid numerical models

2020 ◽  
pp. 69-76
Author(s):  
J. O. Kuboye ◽  
O. F. Quadri ◽  
O. R. Elusakin
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Numerical Models ◽  
Gaussian Elimination ◽  
Initial Value ◽  
Third Order ◽  
Block Methods ◽  
Grid Points

In this work, numerical methods for solving third order initial value problems of ordinary differential equations are developed. Multi-step collocation is used in deriving the methods, where power series approximate solution is employed as a basis function. Gaussian elimination approach is considered in finding the unknown variables $a_j, j=0(1)8$ in interpolation and collocation equations, which are substituted into the power series to give the continuous implicit schemes. The discrete schemes and its derivatives are derived by evaluating the grid and non-grid points. These schemes are arranged in a matrix form to produce block methods. The order of the developed methods are found to be six. The numerical results proved the efficiency of the methods over the existing methods.

scholarly journals Numerical behavior of a fractional order dynamical model of RNA silencing

2016 ◽  
Vol 4 (2) ◽  
pp. 52
Author(s):  
A.M.A. El-Sayed ◽  
M. Khalil ◽  
A.A.M. Arafa ◽  
Amaal Sayed
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Detailed Analysis ◽  
Fractional Order ◽  
Rna Silencing ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
The Stability ◽  
Predictor Corrector ◽  
With Memory

A class of fractional-order differential models of RNA silencing with memory is presented in this paper. We also carry out a detailed analysis on the stability of equilibrium and we show that the model established in this paper possesses non-negative solutions. Numerical solutions are obtained using a predictor-corrector method to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. Numerical simulations are presented to illustrate the results. Also, the numerical simulations show that, modeling the phenomena of RNA silencing by fractional ordinary differential equations (FODE) has more advantages than classical integer-order modeling.

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