An Optimization on Quadrature Formulas and Numerical Solutions of Ordinary Differential Equations

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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Numerical Solutions of Initial Value Ordinary Differential Equations Using Finite Difference Method

OALib ◽

10.4236/oalib.1101614 ◽

2015 ◽

Vol 02 (06) ◽

pp. 1-7

Author(s):

Negesse Yizengaw

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Difference Method ◽

Numerical Solutions ◽

Initial Value

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Numerical Solution for Boundary Layer Flow of a Dusty Micropolar Fluid Due to a Stretching Sheet with Constant Wall Temperature

Journal of Advanced Research in Fluid Mechanics and Thermal Sciences ◽

10.37934/arfmts.87.1.3040 ◽

2021 ◽

Vol 87 (1) ◽

pp. 30-40

Author(s):

Anisah Dasman ◽

Abdul Rahman Mohd Kasim ◽

Iskandar Waini ◽

Najiyah Safwa Khashi’ie

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Wall Temperature ◽

Micropolar Fluid ◽

Stretching Sheet ◽

Numerical Solutions ◽

Numerical Study ◽

Particle Interaction ◽

Dust Particles ◽

Constant Wall Temperature

This paper aims to present the numerical study of a dusty micropolar fluid due to a stretching sheet with constant wall temperature. Using the suitable similarity transformation, the governing partial differential equations for two-phase flows of the fluid and the dust particles are reduced to the form of ordinary differential equations. The ordinary differential equations are then numerically analysed using the bvp4c function in the Matlab software. The validity of present numerical results was checked by comparing them with the previous study. The results graphically show the numerical solutions of velocity, temperature and microrotation distributions for several values of the material parameter K, fluid-particle interaction parameter and Prandtl number for both fluid and dust phase. The effect of microrotation is investigated and analysed as well. It is found that the distributions are significantly influenced by the investigated parameters for both phases.

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Modelling the Molecular Transportation of Subcutaneously Injected Salubrinal

Biomedical Engineering and Computational Biology ◽

10.4137/becb.s7050 ◽

2011 ◽

Vol 3 ◽

pp. BECB.S7050

Author(s):

Andy Chen ◽

Ping Zhang ◽

Zhiyao Duan ◽

Guofeng Wang ◽

Hiroki Yokota

Keyword(s):

Diffusion Coefficient ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Numerical Solutions ◽

Subcutaneous Tissue ◽

Subcutaneous Administration ◽

Cellular Damage ◽

Whole Body ◽

Chemical Agent ◽

Permeability Factor

For the subcutaneous administration of a chemical agent (salubrinal), we constructed a mathematical model of molecule transportation and subsequently evaluated the kinetics of diffusion, convection, and molecular turnover. Salubrinal is a potential therapeutic agent that can reduce cellular damage and death. The understanding of its temporal profiles in local tissue as well as in a whole body is important to develop a proper strategy for its administration. Here, the diffusion and convection kinetics was formulated using partial and ordinary differential equations in one- and three-dimensional (semi-spherical) coordinates. Several key parameters including an injection velocity, a diffusion coefficient, thickness of subcutaneous tissue, and a permeability factor at the tissue-blood boundary were estimated from experimental data in rats. With reference to analytical solutions in a simplified model without convection, numerical solutions revealed that the diffusion coefficient and thickness of subcutaneous tissue determined the timing of the peak concentration in the plasma, and its magnitude was dictated by the permeability factor. Furthermore, the initial velocity, induced by needle injection, elevated an immediate transport of salubrinal at t < 1h. The described analysis with a combination of partial and ordinary differential equations contributes to the prediction of local and systemic effects and the understanding of the transportation mechanism of salubrinal and other agents.

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YANG–MILLS INSTANTONS ON SEVEN-DIMENSIONAL MANIFOLD OF G2 HOLONOMY

Modern Physics Letters A ◽

10.1142/s0217732399002728 ◽

1999 ◽

Vol 14 (37) ◽

pp. 2595-2604 ◽

Cited By ~ 1

Author(s):

SAYURI MIYAGI

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Numerical Solutions ◽

Spherically Symmetric ◽

Dimensional Manifold ◽

Yang Mills

We investigate Yang–Mills instantons on a seven-dimensional manifold of G2 holonomy. By proposing a spherically symmetric ansatz for the Yang–Mills connection, we have ordinary differential equations as the reduced instanton equation, and give some explicit and numerical solutions.

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