scholarly journals A New Eighth Order Runge-Kutta Family Method

2019 ◽  
Vol 11 (2) ◽  
pp. 190
Author(s):  
Séka Hippolyte ◽  
Assui Kouassi Richard
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stability Region ◽  
Eighth Order ◽  
Stability Regions ◽  
New Family ◽  
Runge Kutta ◽  
The Stability

In this article, a new family of Runge-Kutta methods of 8^th order for solving ordinary differential equations is discovered and depends on the parameters b_8 and a_10;5. For b8 = 49/180 and a10;5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a_10;5. We compare the stability regions according to the values of a_10;5 with respect to the stability region of Cooper-Verner.

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 High order explicit Runge-Kutta pairs for ephemerides of the Solar System and the Moon

2000 ◽  
Vol 4 (2) ◽  
pp. 183-192 ◽  
Author(s):  
Philip W. Sharp
Keyword(s):  
Solar System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Test Problem ◽  
Second Order ◽  
High Order ◽  
The Moon ◽  
Initial Value ◽  
New Family ◽  
Runge Kutta

Numerically integrated ephemerides of the Solar System and the Moon require very accurate integrations of systems of second order ordinary differential equations. We present a new family of 8-9 explicit Runge-Kutta pairs and assess the performance of two new 8-9 pairs on the equations used to create the ephemeris DE102. Part of this work is the introduction of these equations as a test problem for integrators of initial value ordinary differential equations.

Stability regions for multi-parameter systems of periodic second order ordinary differential equations

1979 ◽  
Vol 84 (3-4) ◽  
pp. 249-257 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Parameter System ◽  
Stability Regions ◽  
The Stability ◽  
Image Position

SynopsisThis paper studies the stability regions associated with the multi-parameter systemwhere the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

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 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 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.

scholarly journals ORDER OF THE RUNGE-KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION

2019 ◽  
Vol 5 (2) ◽  
pp. 64
Author(s):  
Hippolyte Séka ◽  
Kouassi Richard Assui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Absolute Stability ◽  
Stability Region ◽  
Kutta Method ◽  
Stability Regions ◽  
Runge Kutta Method ◽  
The Absolute ◽  
Runge Kutta ◽  
The Stability

In this article, we demonstrate through specific examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary differential equation does not depend on the order of methods.

scholarly journals Regions of Absolute Stability

1979 ◽  
Vol 8 (102) ◽  
Author(s):  
Jørgen Sand ◽  
Ole Østerby
Keyword(s):  
Differential Equations ◽  
Computer Science ◽  
Ordinary Differential Equations ◽  
Large Class ◽  
Absolute Stability ◽  
Stability Regions ◽  
New Information ◽  
Pade Approximations ◽  
Backward Differentiation Formulae ◽  
The Stability

At the Department of Computer Science a system has been developed for plotting regions of absolute stability for a large class of formulae and methods for solving systems of ordinary differential equations. This report is a pictorial guide through the stability regions of a number of well-known formulae thereby showing the capabilities of our programs, and hopefully also giving some new information about the methods. In an appendix we give coefficients for Adams, Nystrom, generalized Milne-Simpson and backward differentiation formulae up to order 12 (resp. 11) and coefficients for Pade approximations to the exponential up to degree 6.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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