scholarly journals BBDF-Alpha for solving stiff ordinary differential equations with oscillating solutions

2020 ◽  
Vol 51 (2) ◽  
pp. 123-136
Author(s):  
Iskandar Shah Mohd Zawawi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Experiments ◽  
Stability Region ◽  
Newton Iteration ◽  
Computational Time ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Oscillating Solutions ◽  
The Stability

In this paper, the block backward differentiation α formulas (BBDF-α) is derived for solving first order stiff ordinary differential equations with oscillating solutions. The consistency and zero stability conditions are investigated to prove the convergence of the method. The stability region in the entire negative half plane shows that the derived method is A-stable for certain values of α. The implementation of the method using Newton iteration is also discussed. Several numerical experiments are conducted to demonstrate the performance of the method in terms of accuracy and computational time.

scholarly journals An Accurate Implicit Quarter Step First Derivative Block Hybrid Method (AIQSFDBHM) for Solving Ordinary Differential Equations

2019 ◽  
pp. 1-13
Author(s):  
P. Tumba ◽  
J. Sabo ◽  
A. A. Okeke ◽  
D. I. Yakoko
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Numerical Experiments ◽  
Stability Region ◽  
Numerical Solutions ◽  
Initial Value ◽  
First Order ◽  
First Derivative ◽  
Stiff Odes

The new accurate implicit quarter step first derivative blocks hybrid method for solving ordinary differential equations have been proposed in this paper via interpolation and collocation method for the solution of stiff ODEs. The analysis of the method was study and it was found to be consistent, convergent, zero-stability, We further compute the region of absolute stability region and it was found to be Aα − stable . It is obvious that, the numerical experiments considered showed that the methods compete favorably with existing ones. Thus, the pair of numerical methods developed in this research is computationally reliable in solving first order initial value problems, as the results from numerical solutions of stiff ODEs shows that this method is superior and best to solve such problems as in tables and figures.

scholarly journals Fixed Coefficient A ( α ) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 846 ◽  
Author(s):  
Zarina Ibrahim ◽  
Nursyazwani Mohd Noor ◽  
Khairil Othman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Execution Time ◽  
Stability Region ◽  
Scientific Literature ◽  
Necessary Conditions ◽  
Computational Time ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

The main contribution in this paper is to construct an implicit fixed coefficient Block Backward Differentiation Formulas denoted as A ( α ) -BBDF with equal intervals for solving stiff ordinary differential equations (ODEs). To avoid calculating the differentiation coefficients at each step of the integration, the coefficients of the formulas will be stored, with the intention of optimizing the performance in terms of precision and computational time. The plots of their A ( α ) stability region are provided, and the order of the method is also verified. The necessary conditions for convergence, such as the consistency and zero stability of the method, are also discussed. The numerical results clearly showed the efficiency of the method in terms of accuracy and execution time as compared to other existing methods in the scientific literature.

MODELLING TUMOR GROWTH WITH IMMUNE RESPONSE AND DRUG USING ORDINARY DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 79 (5) ◽  
Author(s):  
Mohd Rashid Admon ◽  
Normah Maan
Keyword(s):  
Immune Response ◽  
Differential Equations ◽  
Tumor Growth ◽  
Ordinary Differential Equations ◽  
Stability Region ◽  
Cycle Phase ◽  
Specific Drug ◽  
Mathematical Study ◽  
Growth Region ◽  
The Stability

This is a mathematical study about tumor growth from a different perspective, with the aim of predicting and/or controlling the disease. The focus is on the effect and interaction of tumor cell with immune and drug. This paper presents a mathematical model of immune response and a cycle phase specific drug using a system of ordinary differential equations.  Stability analysis is used to produce stability regions for various values of certain parameters during mitosis. The stability region of the graph shows that the curve splits the tumor decay and growth regions in the absence of immune response. However, when immune response is present, the tumor growth region is decreased. When drugs are considered in the system, the stability region remains unchanged as the system with the presence of immune response but the population of tumor cells at interphase and metaphase is reduced with percentage differences of 1.27 and 1.53 respectively. The combination of immunity and drug to fight cancer provides a better method to reduce tumor population compared to immunity alone.

scholarly journals Qualitative behavior of stiff ODEs through a stochastic approach

2020 ◽  
Vol 10 (2) ◽  
pp. 181-187
Author(s):  
Hande Uslu ◽  
Murat Sari ◽  
Tahir Cosgun
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Life ◽  
Stochastic Approach ◽  
Computational Time ◽  
Qualitative Behavior ◽  
Computational Error ◽  
Linear Ordinary Differential Equations ◽  
Stiff Ordinary Differential Equations ◽  
Simulation Techniques

In the last few decades, stiff differential equations have attracted a great deal of interest from academic society, because much of the real life is covered by stiff behavior. In addition to importance of producing model equations, capturing an exact behavior of the problem by dealing with a solution method is also handling issue. Although there are many explicit and implicit numerical methods for solving them, those methods cannot be properly applied due to their computational time, computational error or effort spent for construction of a structure. Therefore, simulation techniques can be taken into account in capturing the stiff behavior. In this respect, this study aims at analyzing stiff processes through stochastic approaches. Thus, a Monte Carlo based algorithm has been presented for solving some stiff ordinary differential equations and system of stiff linear ordinary differential equations. The produced results have been qualitatively and quantitatively discussed.

DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 11 (09) ◽  
pp. 2339-2357 ◽  
Author(s):  
V. N. GOVORUKHIN ◽  
V. G. TSYBULIN ◽  
B. KARASÖZEN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Equilibrium Points ◽  
Two Dimensions ◽  
Model Systems ◽  
Continuous Family ◽  
First Order ◽  
Spurious Solutions ◽  
Overall Performance ◽  
The Stability

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

scholarly journals A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 806
Author(s):  
Ali Shokri ◽  
Beny Neta ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Mohammad Mehdi Rashidi ◽  
Hamid Mohammad-Sedighi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Stability Property ◽  
Stability Region ◽  
Mathieu Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Implicit Schemes ◽  
The Stability

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

STABILITY ANALYSIS OF A PROPOSED SCHEME OF ORDER FIVE FOR FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 898
Author(s):  
Sunday Emmanuel Fadugba ◽  
Roseline Bosede Ogunrinde ◽  
Rowland Rotimi Ogunrinde
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Step Length ◽  
Step Method ◽  
First Order ◽  
Absolute Relative Error ◽  
One Step ◽  
Test Equation ◽  
The Stability

This paper presents the stability analysis of a proposed scheme of order five (FCM) for first order Ordinary Differential Equations (ODEs). The proposed FCM is derived by means of an interpolating function of polynomial and exponential forms. The properties of FCM were discussed extensively. The linear stability of FCM in the context of the Third Order One-Step Method (TCM) and Second Order One-Step Method (SCM) for the solution of initial value problems of first order differential equations is presented. The stability region of FCM, TCM and SCM is investigated using the Dahlquist’s test equation. The numerical results obtained via FCM are compared with TCM and SCM. Moreover, by varying the step length, the accuracy and convergence of the methods in terms of the final absolute relative error are measured. The results show that FCM converges faster and more stable than its counterparts.

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