The Itô-Stratonovich formula for an operator of order four

2016 ◽  
pp. 165-169 ◽  
Author(s):  
Rémi Léandre
Keyword(s):  
Order Four

Effect of Viscous Dissipation on MHD Williamson Nanofluid Flow in a Porous Medium

2019 ◽  
Vol 8 (3) ◽  
pp. 5795-5802 ◽  
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Numerical Solution ◽  
Viscous Dissipation ◽  
Flow Problem ◽  
Numerical Study ◽  
Steady State Flow ◽  
Nanofluid Flow ◽  
Shooting Technique ◽  
Order Four

The main objective of this paper is to focus on a numerical study of viscous dissipation effect on the steady state flow of MHD Williamson nanofluid. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional PDEs (nonlinear) into coupled dimensionless ODEs. The numerical solution of these modeled ordinary differential equations (ODEs) is achieved by utilizing shooting technique together with Adams-Bashforth Moulton method of order four. Finally, the results of discussed for different parameters through graphs and tables.

scholarly journals Unsteady hybrid-nanofluid flow comprising ferrousoxide and CNTs through porous horizontal channel with dilating/squeezing walls

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Muhammad Bilal ◽  
Hamna Arshad ◽  
Muhammad Ramzan ◽  
Zahir Shah ◽  
Poom Kumam
Keyword(s):  
Carbon Nanotubes ◽  
Differential Equations ◽  
Numerical Approach ◽  
Hybrid Nanofluid ◽  
Multi Wall Carbon Nanotubes ◽  
Porous Walls ◽  
Industrial Cleaning ◽  
Energy Equations ◽  
Single Wall Nanotubes ◽  
Order Four

AbstractThe key objective of the present research is to examine the hybrid magnetohydrodynamics (MHD) nanofluid (Carbon-nanotubes and ferrous oxide–water) CNT–Fe3O4/H2 flow into a horizontal parallel channel with thermal radiation through squeezing and dilating porous walls. The parting motion is triggered by the porous walls of the channel. The fluid flow is time-dependent and laminar. The channel is asymmetric and the upper and lower walls are distinct in temperature and are porous. With the combination of nanoparticles of Fe3O4 and single and multi-wall carbon nanotubes, the hybrid nanofluid principle is exploited. By using the similarity transformation, the set of partial differential equations (PDEs) of this mathematical model, governed by momentum and energy equations, is reduced to corresponding ordinary differential equations (ODEs). A very simple numerical approach called the Runge–Kutta system of order four along with the shooting technique is used to achieve the solutions for regulating ODEs. MATLAB computing software is used to create temperature and velocity profile graphs for various emerging parameters. At the end of the manuscript, the main conclusions are summarized. Through different graphs, it is observed that hybrid-nanofluid has more prominent thermal enhancement than simple nanofluid. Further, the single-wall nanotubes have dominated impact on temperature than the multi-wall carbon nanotubes. From the calculations, it is also noted that Fe2O3–MWCNT–water has an average of 4.84% more rate of heat transfer than the Fe2O3–SWCNT–water. On the other hand, 8.27% more heat flow observed in Fe2O3–SWCNT–water than the simple nanofluid. Such study is very important in coolant circulation, inter-body fluid transportation, aerospace engineering, and industrial cleaning procedures, etc.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

Interpolants for Runge-Kutta pairs of order four and five

Computing ◽  
1990 ◽  
Vol 45 (4) ◽  
pp. 383-388
Author(s):  
M. Calvé ◽  
R. Vaillancourt
Keyword(s):  
Runge Kutta ◽  
Order Four

EXISTENCE THEOREMS OF POSITIVE SOLUTIONS FOR FOURTH-ORDER FOUR-POINT BOUNDARY VALUE PROBLEMS

2004 ◽  
Vol 02 (01) ◽  
pp. 71-85 ◽  
Author(s):  
YUJI LIU ◽  
WEIGAO GE
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence Theorems ◽  
Boundary Value ◽  
Growth Conditions ◽  
Point Boundary ◽  
Order Ordinary Differential Equation ◽  
Order Four

In this paper, we study four-point boundary value problems for a fourth-order ordinary differential equation of the form [Formula: see text] with one of the following boundary conditions: [Formula: see text] or [Formula: see text] Growth conditions on f which guarantee existence of at least three positive solutions for the problems (E)–(B1) and (E)–(B2) are imposed.

Quantum Synchronizable Codes From the Cyclotomy of Order Four

2019 ◽  
Vol 23 (1) ◽  
pp. 12-15 ◽  
Author(s):  
Lanqiang Li ◽  
Shixin Zhu ◽  
Li Liu
Keyword(s):  
Order Four
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