In this article, a new method of analysis of the differential equations on the sphere using Leapfrog
method is presented. To illustrate the effectiveness of the Leapfrog method, an example of the differential
equations on the sphere has been considered and the solutions were obtained using methods taken taken from
the literature [17] and Leapfrog method. The obtained discrete solutions are compared with the exact solutions
of the differential equations on the sphere. Solution graphs for the differential equations on the sphere have
been presented in the graphical form to show the efficiency of this Leapfrog method. This Leapfrog method can
be easily implemented in a digital computer and the solution can be obtained for any length of time
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
In this study, nonlinear singularly perturbed problems with nonlocal condition are evaluated by finite
difference method. The exact solution 푢(푥) has boundary layers at 푥 = 0 and 푥 = 1. We present some
properties of the exact solution of the multi-point boundary value problem (1)-(3). According to the perturbation
parameter, by the method of integral identities with the use exponential basis functions and interpolating
quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference
scheme on Bakhvalov mesh is established. The error analysis for the difference scheme is performed. 휀 −
uniform convergence for approximate solution in the discrete maximum norm is provided, which is the firstorder
(O(h)). This theoretical process is applied on the sample. By Thomas Algorithm, it has been shown to be
consistent with the theoretical results of numerical results. The results were embodied in table and graphs. The
relationship between the approximate solution with the exact solution are obtained by Maple 10 computer
program.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
In this study, nonlinear singularly perturbed problems with nonlocal condition are evaluated by finite
difference method. The exact solution 푢(푥) has boundary layers at 푥 = 0 and 푥 = 1. We present some
properties of the exact solution of the multi-point boundary value problem (1)-(3). According to the perturbation
parameter, by the method of integral identities with the use exponential basis functions and interpolating
quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference
scheme on Bakhvalov mesh is established. The error analysis for the difference scheme is performed. 휀 −
uniform convergence for approximate solution in the discrete maximum norm is provided, which is the firstorder
(O(h)). This theoretical process is applied on the sample. By Thomas Algorithm, it has been shown to be
consistent with the theoretical results of numerical results. The results were embodied in table and graphs. The
relationship between the approximate solution with the exact solution are obtained by Maple 10 computer
program.
Comparative Analysis of Different Numerical Methods of Solving First Order Di...ijtsrd
A mathematical equation which relates various function with its derivatives is known as a differential equation.. It is a well known and popular field of mathematics because of its vast application area to the real world problems such as mechanical vibrations, theory of heat transfer, electric circuits etc. In this paper, we compare some methods of solving differential equations in numerical analysis with classical method and see the accuracy level of the same. Which will helpful to the readers to understand the importance and usefulness of these methods. Chandrajeet Singh Yadav"Comparative Analysis of Different Numerical Methods of Solving First Order Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd13045.pdf http://www.ijtsrd.com/mathemetics/applied-mathematics/13045/comparative-analysis-of-different-numerical-methods-of-solving-first-order-differential-equation/chandrajeet-singh-yadav
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
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Sinc collocation linked with finite differences for Korteweg-de Vries Fraction...IJECEIAES
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Comparative Analysis of Different Numerical Methods for the Solution of Initi...YogeshIJTSRD
A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well known and popular concept in mathematics because of its massive application in real world problems. Differential equations are one of the most important mathematical tools used in modeling problems in Physics, Biology, Economics, Chemistry, Engineering and medical Sciences. Differential equation can describe many situations viz exponential growth and de cay, the population growth of species, the change in investment return over time. We can solve differential equations using classical as well as numerical methods, In this paper we compare numerical methods of solving initial valued first order ordinary differential equations namely Euler method, Improved Euler method, Runge Kutta method and their accuracy level. We use here Scilab Software to obtain direct solution for these methods. Vibahvari Tukaram Dhokrat "Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-5 , August 2021, URL: https://www.ijtsrd.com/papers/ijtsrd45066.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/45066/comparative-analysis-of-different-numerical-methods-for-the-solution-of-initial-value-problems-in-first-order-ordinary-differential-equations/vibahvari-tukaram-dhokrat
Derivation and Application of Multistep Methods to a Class of First-order Ord...AI Publications
Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion method, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations. For the numerical integration technique, an interpolating polynomial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multi-step methods require more computational effort than the single-step methods.
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Continuous models of economy adopted to either define evolutions of economic
systems or investigate economic dynamics, amongst its other areas of applications, are
known to be related to differential equations. The considered model in this article takes
the form of a second order non-linear ordinary differential equation (ODE) which is
conventional solved by reducing to the system of first order. This approach is
computationally tasking, unlike block methods which bypasses reduction by directly
solving the model. A new approach is introduced named Modified Taylor Series
Approach (MTSA). Hence, the resultant MTSA-derived block method is implemented to
solve the second order non-linear model of economy under consideration.
Finding the Extreme Values with some Application of Derivativesijtsrd
There are many different way of mathematics rules. Among them, we express finding the extreme values for the optimization problems that changes in the particle life with the derivatives. The derivative is the exact rate at which one quantity changes with respect to another. And them, we can compute the profit and loss of a process that a company or a system. Variety of optimization problems are solved by using derivatives. There were use derivatives to find the extreme values of functions, to determine and analyze the shape of graphs and to find numerically where a function equals zero. Kyi Sint | Kay Thi Win "Finding the Extreme Values with some Application of Derivatives" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29347.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/29347/finding-the-extreme-values-with-some-application-of-derivatives/kyi-sint
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Numerical simulation on laminar free convection flow and heat transfer over a...eSAT Journals
Abstract
In the present numerical study, laminar free-convection flow and heat transfer over an isothermal vertical plate is presented. By
means of similarity transformation, the original nonlinear coupled partial differential equations of flow are transformed to a pair
of simultaneous nonlinear ordinary differential equations. Then, they are reduced to first order system. Finally, NewtonRaphson
method and adaptive Runge-Kutta method are used for their integration. The computer codes are developed for this numerical
analysis in Matlab environment. Velocity and temperature profiles for various Prandtl number are illustrated graphically. Flow
and heat transfer parameters are derived as functions of Prandtl number alone. The results of the present simulation are then
compared with experimental data published in literature and find a good agreement.
Keywords: Free Convection, Heat Transfer, Matlab, Numerical Simulation, Vertical Plate.
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Does Goods and Services Tax (GST) Leads to Indian Economic Development?iosrjce
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Childhood Factors that influence success in later lifeiosrjce
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Customer’s Acceptance of Internet Banking in Dubaiiosrjce
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Student`S Approach towards Social Network Sitesiosrjce
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Broadcast Management in Nigeria: The systems approach as an imperativeiosrjce
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A Study Factors Influence on Organisation Citizenship Behaviour in Corporate ...iosrjce
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Consumers’ Behaviour on Sony Xperia: A Case Study on Bangladeshiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Design of a Balanced Scorecard on Nonprofit Organizations (Study on Yayasan P...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Public Sector Reforms and Outsourcing Services in Nigeria: An Empirical Evalu...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Media Innovations and its Impact on Brand awareness & Considerationiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer experience in supermarkets and hypermarkets – A comparative studyiosrjce
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Secretarial Performance and the Gender Question (A Study of Selected Tertiary...iosrjce
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Implementation of Quality Management principles at Zimbabwe Open University (...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Organizational Conflicts Management In Selected Organizaions In Lagos State, ...iosrjce
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Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
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Exposé invité Journées Nationales du GDR GPL 2024
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Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
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A Study on Differential Equations on the Sphere Using Leapfrog Method
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. II (Nov. - Dec. 2015), PP 49-52
www.iosrjournals.org
DOI: 10.9790/5728-11624952 www.iosrjournals.org 49 | Page
A Study on Differential Equations on the Sphere Using Leapfrog
Method
S.Karunanithi1
, S. Chakravarthy1
, S. Sekar2
1
(Department of Mathematics, Government Thirumagal Mills College, Gudiyattam – 632 602, India)
2
(Department of Mathematics, Government Arts College (Autonomous), Salem – 636 007, India)
Abstract: In this article, a new method of analysis of the differential equations on the sphere using Leapfrog
method is presented. To illustrate the effectiveness of the Leapfrog method, an example of the differential
equations on the sphere has been considered and the solutions were obtained using methods taken taken from
the literature [17] and Leapfrog method. The obtained discrete solutions are compared with the exact solutions
of the differential equations on the sphere. Solution graphs for the differential equations on the sphere have
been presented in the graphical form to show the efficiency of this Leapfrog method. This Leapfrog method can
be easily implemented in a digital computer and the solution can be obtained for any length of time.
Keywords: Differential equations, system of differential equations, Differential equations on the sphere,
Single-term Haar wavelet series, Leapfrog Method.
I. Introduction
Mathematical modeling aims to describe the different aspects of the real world, their interaction, and
their dynamics through mathematics. It constitutes the third pillar of science and engineering, achieving the
fulfillment of the two more traditional disciplines, which are theoretical analysis and experimentation.
Nowadays, mathematical modeling has a key role also in fields such as the environment and industry, while its
potential contribution in many other areas is becoming more and more evident. One of the reasons for this
growing success is definitely due to the impetuous progress of scientific computation; this discipline allows the
translation of a mathematical model, which can be explicitly solved only occasionally, into algorithms that can
be treated and solved by ever more powerful computers.
Since 1960 numerical analysis, the discipline that allows mathematical equations (algebraic, functional,
differential, and integrals) to be solved through algorithms, had a leading role in solving problems linked to
mathematical modeling derived from engineering and applied sciences. Following this success, new disciplines
started to use mathematical modeling, namely information and communication technology, bioengineering,
financial engineering, and so on. As a matter of fact, mathematical models offer new possibilities to manage the
increasing complexity of technology, which is at the basis of modern industrial production. They can explore
new solutions in a very short time period, thus allowing the speed of innovation cycles to be increased. This
ensures a potential advantage to industries, which can save time and money in the development and validation
phases. We can state therefore that mathematical modeling and scientific computation are gradually and
relentlessly expanding in manifold fields, becoming a unique tool for qualitative and quantitative analysis. In the
following paragraphs we will discuss the role of mathematical modeling and of scientific computation in applied
sciences; their importance in simulating, analyzing, and decision making; and their contribution to technological
progress. We will show some results and underline the perspectives in different fields such as industry,
environment, life sciences, and sports.
The goal of this article is to construct a numerical method for addressing the differential equations on
the sphere problem by an application of the Leapfrog method which was studied by Sekar and team of his
researchers [2-15]. Recently, Sekar et al. [17] discussed the differential equations on the sphere problem using
STHW. In this paper, the same differential equations on the sphere problem was considered (discussed by Sekar
et al. [17]) but present a different approach using the Leapfrog method with more accuracy for the differential
equations on the sphere problem.
II. Leapfrog Method
The most familiar and elementary method for approximating solutions of an initial value problem is
Euler’s Method. Euler’s Method approximates the derivative in the form of d
Ryytyytfy ,,, 00 by
a finite difference quotient htyhtyty / . We shall usually discretize the independent variable in
equal increments:
01 ,...,1,0, tnhtt nn .
2. A Study on Differential Equations on the Sphere Using Leapfrog Method
DOI: 10.9790/5728-116XXXXX www.iosrjournals.org 50 | Page
Henceforth we focus on the scalar case, N = 1. Rearranging the difference quotient gives us the
corresponding approximate values of the dependent variable:
01 ,...,1,0,, tnythfyy nnnn
To obtain the leapfrog method, we discretize nt as in 01 ,...,1,0, tnhtt nn , but we double the time
interval, h, and write the midpoint approximation
2
h
tyhtyhty in the form
htyhtyhty /2
and then discretize it as follows:
011 ,...,1,0,,2 tnythfyy nnnn
The leapfrog method is a linear m = 2-step method, with 2,1,1,0 0110 bbaa and 01 b . It
uses slopes evaluated at odd values of n to advance the values at points at even values of n, and vice versa,
reminiscent of the children’s game of the same name. For the same reason, there are multiple solutions of the
leapfrog method with the same initial value 0yy . This situation suggests a potential instability present in
multistep methods, which must be addressed when we analyze them—two values, 0y and 1y , are required to
initialize solutions of 011 ,...,1,0,,2 tnythfyy nnnn uniquely, but the analytical problem
d
Ryytyytfy ,,, 00 only provides one. Also for this reason, one-step methods are used to initialize
multistep methods.
III. Differential Equations on the Sphere
This section features differential equations on the sphere of ordinary differential equations to be solved
numerically. The results from the Leapfrog method are compared with the results from some of the classical
integrators such as STHW method, which is mentioned in Section 2, and the build-in integrator ODE45 from
Matlab. For more information on ODE45, please refer to Shampine [16]. With the exception of ODE45, the
experiments performed in this chapter will involve integrators based on fixed step size h and fixed order.
Differential equation on the sphere was designed to be solved by the Leapfrog method and STHW
method, and it is another example featured in Hairer and Wanner [1].
01.0)cos(4.0
1.00
)cos(4.00
tt
tt
tt
dt
dy
,tytAty .
1
0
0
0
y (1)
The solution ty R3
in Eq. (1) is a vector of unit length, evolving on the unit sphere. That is, the
solution vector y(t) rotates around the spherical surface with respect to its origin, and naturally, the
homogeneous manifold in this case is the unit sphere. Then, the general presentation of the differential equation,
with which we want to solve for the solution y(t) is given by,
,tQtAtQ ,0 IdQ
where Q(t) is the corresponding 3 3 matrix in the differential equation. This is exactly the form of the
differential equation. To perform the numerical experiments, let us suppose that we want to integrate Eq. (1)
with the given initial condition y(0), over the time interval t = [0, 64] with a constant stepsize of h = 1/20, and
order p = 4 for the 4 integrators Leapfrog and STHW methods. Once again, we also integrate the same problem
with Matlab's ODE45 for comparison.
In the two graphs in Figure 1, we plotted the three dimensional solutions y(t) over the integration
period, on the unit sphere for the four methods under consideration, using the same stepsize h. It is apparent that
solutions from the RK integrator eventually drift away from the surface of the sphere, while solutions from the
other three integrators appear to adhere to the same surface. Again assuming that ODE45 produces the most
accurate solutions y(t) when solving the same initial value problem, then we can take a look at the difference of
solutions between ODE45's and the other four integrators. Such results are plotted in the bottom four graphs in
Figure 1. Note that the difference between ODE45's and the method such as Leapfrog method are significantly
smaller, than the differences between ODE45's and STHW method.
One of the advantage of using the Leapfrog method for solving this type of differential equations on the
sphere is that the Leapfrog method action produces special orthogonal matrices, which, when multiplied to the
vector ty R3
, preserves its length in the rotation, so that solutions y(t) will always remain on the sphere.
This is not true if we use classical integrators such as STHW method. We can see this by observing the variation
in the length of the solution vectors y(t) with respect to the initial length, at each time step for all four integrators
tested, in Figure 2.
3. A Study on Differential Equations on the Sphere Using Leapfrog Method
DOI: 10.9790/5728-116XXXXX www.iosrjournals.org 51 | Page
Figure 1: From solving Equation (1), the above 2 graphs are the solutions from each integrator plotted on the
surface of a unit sphere. In Figure 2 graphs are the difference between solutions from ODE45 and each of the 2
integrators. y0 = [0, 0, 1], t0 = 0, tf = 64, and h = 1/20.
Figure 2: From solving Equation (1), the variation in the length of solution vectors against time t is plotted for
each integrator. y0 = [0, 0, 1], t0 = 0, tf = 64, and h = 1/20.
IV. Conclusion
The obtained discrete solution of the numerical examples shows the efficiency of the Leapfrog for
finding the solution of differential equations on the sphere. From the Figures 1 and 2, we can observe that the
solution graph is come closer in Leapfrog method when compared to other methods from ODE45 taken from the
literature []. Hence, the Leapfrog method is more suitable for studying the differential equations on the sphere.
Acknowledgements
The authors gratefully acknowledge the Dr. K. Balamurugan, Professor, Department of English,
Bharathiyar College of Engineering & Technology, Thiruvettakudy, Karaikal – 609 609, for his support to
improve the language of this article.
4. A Study on Differential Equations on the Sphere Using Leapfrog Method
DOI: 10.9790/5728-116XXXXX www.iosrjournals.org 52 | Page
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