scholarly journals Solution for nonlinear Duffing oscillator using variable order variable stepsize block method

MATEMATIKA ◽  
2017 ◽  
Vol 33 (2) ◽  
pp. 165 ◽  
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohamad Hasan Abdul Sathar ◽  
Norizarina Ishak ◽  
Nur Shuhada Kamarudin ◽  
Muhamad Azrin Nazri ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Duffing Oscillator ◽  
Computational Cost ◽  
Real Life ◽  
Multistep Method ◽  
Variable Order ◽  
Block Method ◽  
Step Size ◽  
Variable Stepsize

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

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.

scholarly journals Two-Point Block variable order step size Multistep Method for Solving Higher Order Ordinary Differential Equations Directly

2021 ◽  
pp. 101376
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak ◽  
Wong Tze Jin ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Multistep Method ◽  
Variable Order ◽  
Step Size

scholarly journals Numerical Approach for Solving Delay Differential Equations with Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1073
Author(s):  
Nur Tasnem Jaaffar ◽  
Zanariah Abdul Majid ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Numerical Approach ◽  
Block Method ◽  
Data Simulation ◽  
Step Size ◽  
Function Calls ◽  
Delay Differential

In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena with data simulation. Thus, an efficient numerical method is needed for the numerical treatment of time delay in the applications. The proposed direct block method computes the numerical solutions at two points concurrently at each computed step along the interval. The types of delays involved in this research are constant delay, pantograph delay, and time-dependent delay. The shooting technique is utilized to deal with the boundary conditions by applying a Newton-like method to guess the next initial values. The analysis of the proposed method based on the order, consistency, convergence, and stability of the method are discussed in detail. Four tested problems are presented to measure the efficiency of the developed direct multistep block method. The numerical simulation indicates that the proposed direct multistep block method performs better than existing methods in terms of accuracy, total function calls, and execution times.

Revisiting the numerical solution of stochastic differential equations

2019 ◽  
Vol 9 (3) ◽  
pp. 312-323
Author(s):  
Stan Hurn ◽  
Kenneth A. Lindsay ◽  
Lina Xu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Bias Reduction ◽  
Step Size ◽  
True Solution ◽  
Content Type ◽  
Reduction Methods ◽  
Improved Accuracy

Purpose The purpose of this paper is to revisit the numerical solutions of stochastic differential equations (SDEs). An important drawback when integrating SDEs numerically is the number of steps required to attain acceptable accuracy of convergence to the true solution. Design/methodology/approach This paper develops a bias reducing method based loosely on extrapolation. Findings The method is seen to perform acceptably well and for realistic steps sizes provides improved accuracy at no significant additional computational cost. In addition, the optimal step size of the bias reduction methods is shown to be consistent with theoretical analysis. Originality/value Overall, there is evidence to suggest that the proposed method is a viable, easy to implement competitor for other commonly used numerical schemes.

scholarly journals Approximating non linear higher order ODEs by a three point block algorithm

PLoS ONE ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. e0246904
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Khairil Iskandar Othman ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Algorithm ◽  
Computational Cost ◽  
Real Life ◽  
Higher Order ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Block Algorithm ◽  
Life Problems

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

scholarly journals Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunye Gong ◽  
Weimin Bao ◽  
Guojian Tang ◽  
Yuewen Jiang ◽  
Jie Liu
Keyword(s):  
Parallel Computing ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Computer Science ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Variable Order ◽  
Fractional Equations ◽  
Fractional Differential

We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

Highly accurate technique for studying some chaotic models described by ABC-fractional differential equations of variable-order

2020 ◽  
pp. 2150018
Author(s):  
M. M. Khader ◽  
Ibrahim Al-Dayel
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Polynomial Interpolation ◽  
Numerical Technique ◽  
Variable Order ◽  
Lagrange Polynomial ◽  
Efficient Tool ◽  
Fractional Differential ◽  
The Given

The propose of this paper is to introduce and investigate a highly accurate technique for solving the fractional Logistic and Ricatti differential equations of variable-order. We consider these models with the most common nonsingular Atangana–Baleanu–Caputo (ABC) fractional derivative which depends on the Mittag–Leffler kernel. The proposed numerical technique is based upon the fundamental theorem of the fractional calculus as well as the Lagrange polynomial interpolation. We satisfy the efficiency and the accuracy of the given procedure; and study the effect of the variation of the fractional-order [Formula: see text] on the behavior of the solutions due to the presence of ABC-operator by evaluating the solution with different values of [Formula: see text]. The results show that the given procedure is an easy and efficient tool to investigate the solution for such models. We compare the numerical solutions with the exact solution, thereby showing excellent agreement which we have found by applying the ABC-derivatives. We observe the chaotic solutions with some fractional-variable-order functions.

scholarly journals A Spectral Deferred Correction Method for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jia Xin ◽  
Jianfei Huang ◽  
Weijia Zhao ◽  
Jiang Zhu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Correction Method ◽  
Initial Value ◽  
Deferred Correction ◽  
Fractional Differential ◽  
Spectral Deferred Correction

A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Finally, preliminary numerical experiments are given to verify the efficiency and accuracy of this method.

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