scholarly journals Cube Arithmetic: Improving Euler Method for Ordinary Differential Equation using Cube Mean

2018 ◽  
Vol 11 (3) ◽  
pp. 1109
Author(s):  
Nooraida Samsudin ◽  
Nurhafizah Moziyana Mohd Yusop ◽  
Syahrul Fahmy ◽  
Anis Shahida Niza Binti Mokhtar
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Simple Procedure ◽  
Numerical Procedure ◽  
Euler Method ◽  
Step Size ◽  
Easy Method ◽  
Euler Methods ◽  
Improved Accuracy

The Euler method is a first-order numerical procedure for solving Ordinary Differential Equation (ODEs) problems. It is an effective and easy method to solve initial value problems. Although Euler provides simple procedure for solving ODEs, there have been issues such as complexity, time of processing and accuracy that compelled the use of other, more complex, methods. Improvements to the Euler method have attracted much attention resulting in numerous modified Euler methods. This paper proposes Cube Arithmetic, a modified Euler method with improved accuracy. The efficiency of Cube Arithmetic was compared with Euler Arithmetic and tested using SCILAB against exact solutions. Results indicate that not only Cube Arithmetic provided solutions that are similar to exact solutions at small step size, but also at higher step size, hence producing more accurate results.

scholarly journals The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2113
Author(s):  
Alla A. Yurova ◽  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Airy Function ◽  
The Real ◽  
The Cauchy Problem

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

EXPLICIT AND EXACT SOLUTIONS TO THE mKdV-SINE-GORDON EQUATION

2008 ◽  
Vol 22 (15) ◽  
pp. 1471-1485 ◽  
Author(s):  
YUANXI XIE
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Variable Separation ◽  
Sine Gordon Equation ◽  
Simple Manner

By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation, rich types of explicit and exact solutions of the mKdV-sine-Gordon equation are presented in a simple manner.

A Numerical Procedure for Obtaining the Static and Pull-In Deflection and Voltage of Capacitive Microcantilever Beams

2006 ◽  
Author(s):  
Seyed Babak Ghaemi Oskouei ◽  
Aria Alasty
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Numerical Procedure ◽  
Nonlinear Ordinary Differential Equation ◽  
Static Deflection ◽  
Element Analysis ◽  
Fringing Field ◽  
Good Agreement

A numerical procedure is proposed for obtaining the static deflection, pull-in (PI) deflection and PI voltage of electrostatically excited capacitive microcantilever beams. The method is not time and memory consuming as Finite Element Analysis (FEA). Nonlinear ordinary differential equation of the static deflection of the beam is derived, w/wo considering the fringing field effects. The nondimensional parameters upon which PI voltage is dependent are then found. Thereafter, using the parameters and the numerical method, three closed form equations for pull-in voltage are developed. The results are in good agreement with others in literature.

scholarly journals Cube Polygon: A New Modified Euler Method to Improve Accuracy of Ordinary Differential Equation (ODE)

2020 ◽  
Vol 1532 ◽  
pp. 012020
Author(s):  
S. Nooraida ◽  
M.M.Y. Nurhafizah ◽  
M.S. Anis ◽  
A.M.M. Fahmi ◽  
A.W.F. Syarul ◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Euler Method ◽  
Improve Accuracy

EXPLICIT EXACT SOLUTIONS TO THE (2+1) DIMENSIONAL SINH-POISSON EQUATION

2010 ◽  
Vol 24 (17) ◽  
pp. 3395-3409
Author(s):  
YUANXI XIE ◽  
SHIYU PENG
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Poisson Equation ◽  
Variable Separation ◽  
Simple Manner

By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation, many explicit exact solutions of the (2+1) dimensional sinh-Poisson equation are presented in a simple manner.

NUMERICAL PROCEDURE FOR SOLVING THE STRIP PROBLEM BY THE SPECTRAL EQUATION

2011 ◽  
Vol 19 (03) ◽  
pp. 269-290 ◽  
Author(s):  
A. V. SHANIN ◽  
V. YU. VALYAEV
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Scattered Field ◽  
A Priori ◽  
Numerical Procedure ◽  
Current Paper ◽  
Method Of Images ◽  
A New Technique ◽  
Spectral Equation ◽  
Branched Surfaces

Recently, a new technique has been proposed by one of the authors for solving diffraction problems belonging to certain class. The class includes 2D problems that can be reduced to propagation problems on branched surfaces by applying the method of images. One of the simplest problems belonging to this class is diffraction by an infinitely thin ideal strip. Within the framework of the new method, a "spectral equation" is derived, which is an ordinary differential equation for the components of the directivity of the scattered field. The coefficients of the spectral equation are known up to several numerical parameters, and these are found by a complicated numerical procedure from the a-priori known monodromy data for the equation. In the current paper the numerical procedure is described and its accuracy and efficiency are analyzed.

The Kerr metric and stationary axisymmetric gravitational fields

1978 ◽  
Vol 358 (1695) ◽  
pp. 405-420 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Event Horizon ◽  
Kerr Metric ◽  
Symmetric Pair ◽  
General Solutions ◽  
Gravitational Fields ◽  
First Order ◽  
Metric Functions

A treatment of Einstein’s equations governing vacuum gravitational fields which are stationary and axisymmetric is shown to divide itself into three parts: a part essentially concerned with a choice of gauge (which can be chosen to ensure the occurrence of an event horizon exactly as in the Kerr metric); a part concerned with two of the basic metric functions which in two combinations satisfy a complex equation (Ernst’s equation) and in one combination satisfies a symmetric pair of real equations; and a third part which completes the solution in terms of a single ordinary differential equation of the first order. The treatment along these lines reveals many of the inner relations which characterize the general solutions, provides a derivation of the Kerr metric which is direct and verifiable at all stages, and opens an avenue towards the generation of explicit classes of exact solutions (an example of which is given).

Analysis of a Nonlinear Model of Heat Transfer Through a Rectangular Fin: Exact Solutions and Their Multiplicity

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Mohammad Danish ◽  
Shashi Kumar ◽  
Surendra Kumar
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Substitution Method ◽  
Porous Catalyst ◽  
Exact Analytical Solutions ◽  
The Heat Transfer Coefficient

Abstract Exact analytical solutions for the temperature profile and the efficiency of a nonlinear rectangular fin model have been obtained in the forms of well-known algebraic/non-algebraic functions. In the considered nonlinear fin model, the thermal conductivity and the heat transfer coefficient have been assumed to vary as distinct power-law functions of temperature thereby yielding a nonlinear BVP in a 2nd order ODE (ordinary differential equation). These exact solutions have been obtained by employing the derivative substitution method which not only include the solutions of previously studied simplified cases of the same problem but also the solutions of a similar problem of reaction-diffusion process occurring in a porous catalyst slab. These exact solutions have been successfully validated against their numerical counterparts. Besides, effects of various parameters on the obtained solutions have been studied, and the conditions for their existence, uniqueness/multiplicity and stability/instability are analyzed and discussed in detail.

Sign in / Sign up

Export Citation Format

Share Document