constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Block Hybrid Method for the Solution of General Second Order Ordinary Differe...QUESTJOURNAL
ABSTRACT: We consider the construction of block hybrid method for the solution of general second order ODEs. Derivation of the method was based on the use of hermite polynomial as basis function. The main method and its additional equations are obtained from the same continuous formulation via interpolation and collocation procedures. The method is then applied in block form as simultaneous numerical integrator, this approach eliminates requirement for starting values, and it also reduces computational effort. The stability properties of the method is discussed and the stability region shown. Two numerical experiments were given to illustrate the accuracy and efficiency of the new method.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Block Hybrid Method for the Solution of General Second Order Ordinary Differe...QUESTJOURNAL
ABSTRACT: We consider the construction of block hybrid method for the solution of general second order ODEs. Derivation of the method was based on the use of hermite polynomial as basis function. The main method and its additional equations are obtained from the same continuous formulation via interpolation and collocation procedures. The method is then applied in block form as simultaneous numerical integrator, this approach eliminates requirement for starting values, and it also reduces computational effort. The stability properties of the method is discussed and the stability region shown. Two numerical experiments were given to illustrate the accuracy and efficiency of the new method.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
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.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
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.
Разработка схемотехнических решений на базе микроконтроллеров, испытательных стендов, а также алгоритмов и исследовательских, демонстрационных программ, предназначенных для совершенствования запатентованой технологии создания полиморфных микроджойстиков.
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
In this paper, we introduced the modified differential transform which is a modified version of a two-dimensional differential transform method. First, the properties of the modified differential transform method (MDTM) are presented. After this, by using the idea modified differential transform method we will find an analytical-numerical solution of linear partial integro-differential equations (PIDE) with convolution kernel which occur naturally in various fields of science and engineering. In some cases, the exact solution may be achieved. The efficiency and reliability of this method are illustrated by some examples.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
In this paper, a class of uncertain chaotic and non-chaotic systems is firstly introduced and the state observation problem of such systems is explored. Based on the time-domain approach with integral and differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20219.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20219/exponential-state-observer-design-for-a-class-of-uncertain-chaotic-and-non-chaotic-systems/yeong-jeu-sun
MATHEMATICAL MODELING OF COMPLEX REDUNDANT SYSTEM UNDER HEAD-OF-LINE REPAIREditor IJMTER
Suppose a composite system consisting of two subsystems designated as ‘P’ and
‘Q’ connected in series. Subsystem ‘P’ consists of N non-identical units in series, while the
subsystem ‘Q’ consists of three identical components in parallel redundancy.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
On the construction and comparison of an explicit iterative
1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
On the Construction and Comparison of an Explicit Iterative
Algorithm with Nonstandard Finite Difference Schemes
Sania Qureshi1* Zaib-un-Nisa Memon 2 Asif Ali Shaikh 3 Muhammad Saleem Chandio4
1.
Lecturer in Mathematics, Department of Basic Sciences and Related Studies, Mehran University of
Engineering and Technology, Jamshoro.
2. Lecturer in Mathematics, Department of Basic Sciences and Related Studies, Mehran University of
Engineering and Technology, Jamshoro.
3. Assistant Professor in Mathematics, Department of Basic Sciences and Related Studies, Mehran
University of Engineering and Technology, Jamshoro.
4. Professor of Mathematics, Institute of Mathematics and Computer Science, University of Sindh,
Jamshoro.
*Email of the corresponding author: sania.qureshi@faculty.muet.edu.pk
Abstract
An explicit iterative algorithm to solve both linear and nonlinear problems of ordinary differential equations with
initial conditions is formulated with main focus given on its comparison with some non-standard finite difference
schemes. Two first order linear initial value problems (IVPs) with periodic behavior are used to analyze the
performance of the proposed algorithm with respect to maximum absolute error and computational effort where
proposed algorithm performs better in both cases. The proposed algorithm efficiently follows the oscillatory
behavior of models like Lotka-Volterra predator-prey and mass-spring system (damped case) in comparison to
the nonstandard schemes. All necessary computations have been carried out through MATLAB version 8.1
(R2013a) in double precision arithmetic. Numerical results obtained by the proposed algorithm are found to be
computationally reliable and practical in comparison with two nonstandard finite difference schemes discussed
in literature.
Keywords:
absolute error.
Iterative algorithm, nonstandard finite difference scheme, Initial conditions, Maximum
1. Introduction
To solve models based on ordinary or partial differential equations is one of the major and challenging tasks
being faced by researchers belonging to various fields of science and engineering. It is seldom possible to
explicitly solve such models (especially nonlinear) in terms of elementary mathematical functions (Burden and
Faires, 2010; Ibijola and Ogunrinde, 2010; Soomro et al., 2013) and this is where computer simulation and
approximate methods play their dynamic role. Numerical techniques to solve Initial Value Problems (IVPs)
modelled by ordinary differential equations (ODEs) help us to analyze and investigate various features of
dynamical systems (Yin et al., 2013). A few of such dynamical and/or nonlinear systems based on a set of first
order ordinary differential equations include Lorenz model (Lorenz, 1963), Lotka – Volterra predator – prey
model (Lotka, 1910; Goel, 1971), Lane – Emden equation (Homer, 1870) and Van der Pol’s oscillator (Van der
Pol, 1927). These systems find number of applications in many scientific fields like, Meteorology, Ecology,
Biomathematics, oscillatory chemical reactions, chaos theory, seismology, and astrophysics; just to mention a
few, and are often considered to be model problems in order to test accuracy and efficacy of various newly
developed algorithms for solving IVPs as commented in (Soomro et al., 2013) .
In this paper, an attempt is made to improve an explicit numerical algorithm to solve a single ODE or a system
of ODEs subject to initial conditions as shown below:
y = f ( t , y ) ; y ( t0 ) = a }
(1.1)
78
2. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
y1 = f1 ( t , y1 , y2 ,
y2 = f 2 ( t , y1 , y2 ,
, yn )
yn = f n ( t , y1 , y2 ,
and
, yn )
, yn )
subject to
y1 ( t0 ) = a1 , y2 ( t0 ) = a 2 ,
ü
ï
ï
ï
ï
ý
ï
ï
ï
, y n ( t0 ) = a n ï
þ
(1.2 )
These models are said to be nonlinear if nonlinearity occurs in y ( t ) . There is no existence of a unique algorithm
to deal with every type of differential equation; therefore, these ODEs have been divided into classes with
respect to their order, type, and linearity (Zill, 2011). Many scholars have resorted to either develop new iterative
algorithms or improve the efficiency of existing ones in terms of their stability, convergence and number of
function evaluations. (Fatunla, 1976; Ibijola, 1997; Wazwaz, 2000; Ramos, 2007; Yang and X, 2008; Sunday
and Odekunle, 2012; Nik and Soleymani, 2013) are among many of those who have introduced nonstandard
finite difference schemes to solve first order IVPs and (Wambecq, 1976; Chandio and Memon, 2010; Rabiei and
Ismail, 2011; Anuar et al., 2011; Rabiei et al., 2012) are among those who have improved the efficiency of
existing standard algorithms.
The purpose of the coming section is to construct an improved iterative algorithm to solve problems of type
(1.1) and (1.2 ) . Various existing methods of past could not beat the introduced algorithm for one or the other
reason as shown by number of examples chosen for comparison. Limitations of the algorithm are also discussed.
2. Materials and Methods
2.1. Problem Description
Consider a well-posed1 initial value problem
dy ( t )
ü
= f (t, y (t )) ï
dt
ý
y ( t0 ) = y0 , t Î [ a, b ]ï
þ
( 2.1)
Approximation yi to the theoretical solution y ( ti ) , in a discrete fashion, will be produced at values called
nodes, in a closed interval
[ a, b ] ×
ti +1 = t0 + ( i + 1) Dt for each i = 0,1, 2,
The nodes are equally spaced throughout the interval
[ a, b ] ×
Also
, n , where Dt = ti +1 - ti = ( b - a ) / n is called the fixed step size.
2.2 Construction of an Explicit Iterative Algorithm
Consider a continuously differentiable function y = f ( t ) shown in fig. 1. Suppose the red line PL1 be the
(
)
(1)
tangent to the curve y = f ( t ) at P ( t0 , y0 ) and the green line QL2 be the line through Q t1 , y1 having the
slope
(
(1)
f t1 , y1
1
) , where y( ) = y + Dt f (t , y ) .
1
1
0
0
0
One and only one solution exists to the problem.
79
3. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
L2
y
L1
S
y = f (t )
P
R
Q
L3
t0 + Dt / 2
t0
0
t1 = t0 + Dt
t
Figure 1. Graphical representation of the proposed iterative algorithm.
(
( f (t , y ) + f (t , y )) 2 that is average
)
(1)
Now blue line QL3 is the line passing through Q t1 , y1 with slope
(
0
)
0
1
(1)
1
(1)
of two slopes f ( t0 , y0 ) and f t1 , y1 .
Coordinates of the point R with abscissa t0 + Dt / 2 and slope
( f (t , y ) + f (t , y )) 2 will be
0
0
(
æ
æ f t , y + f t , y (1)
1 1
ç
Dt
Dt ç ( 0 0 )
R ç t0 + , y0 + ç
2
2 ç
2
ç
è
è
Equation of the line (brown)
(1)
1
1
) ö÷ ö÷÷
÷
÷÷
øø
PS passing through P ( t0 , y0 ) with slope at R will be
y - y0 = ( t - t0 )
(
æ
æ f t , y + f t , y (1)
1 1
ç
Dt
Dt ç ( 0 0 )
f ç t0 + , y0 + ç
2
2 ç
2
ç
è
è
) ö÷ ö÷÷
÷
÷÷
øø
At t = t1 , we obtain
y1 = y0 + ( t1 - t0 )
(
æ
æ f t , y + f t , y (1)
1 1
ç
Dt
Dt ç ( 0 0 )
f ç t0 + , y0 + ç
2
2 ç
2
ç
è
è
(
æ
æ f t , y + f t , y (1)
1 1
ç
Dt
Dt ç ( 0 0 )
= y0 + Dt f ç t0 + , y0 + ç
2
2 ç
2
ç
è
è
) ö÷ ö÷÷
÷
÷÷
øø
) ö÷ ö÷÷
÷
÷÷
øø
Finally, this arrangement results in a point S that will better approximate the exact point lying on the curve
y = f (t ) .
Continuing in the same way, we will get the following proposed iterative algorithm:
æ
Dt
Dt æ f ( ti , yi ) + f ( ti +1 , yi + Dt f ( ti , yi ) ) ö ö
÷÷
yi +1 = yi + Dt f ç ti + , yi + ç
ç
÷÷
2
2 ç
2
è
øø
è
80
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for i = 0,1, 2,
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,n .
3. Numerical Examples and Discussion
In this section, we proceed to implement the proposed algorithm ( 2.2 ) on variety of problems of the type (1.1)
and (1.2 ) taken from the literature. All the computations and graphical displays have been carried out using
MATLAB version 8.1 (R2013a) in double precision arithmetic. Numerical results of proposed algorithm have
been compared with respect to exact solution, absolute errors, CPU time, step size taken and numerical solutions
obtained by existing iterative methods.
Though, linear differential equations can be solved using analytical methods but some of them consume large
amount of computer time and memory. For example, the following IVP:
y = sin 5t - 0.4 y ü
ï
ý
y (0) = 5
ï
þ
Problem: 01
possesses an exact solution in explicit form
y (t ) =
1 é
3270e-2t /5 - 125cos5t + 10sin 5t ù
û
629 ë
But it takes the CPU time of about 7.5432 seconds or if solved by hand involves complicated integration. On the
other hand, the CPU time elapsed is approximately 0.001611 seconds when solved using the proposed algorithm
showing advantage of implementing numerical algorithms. Further, integration steps (IS) taken by Euler and
proposed algorithm are 40 and 20 respectively; even though, the former could not beat the latter as depicted in
figure 2.
6
5
4
Exact
Proposed
Euler
y(t)
3
2
1
0
-1
-1
0
1
2
3
time
4
5
6
7
Figure 2. Comparison of Euler (IS = 40) and proposed algorithm (IS = 20) with respect to exact solution on the
interval [0 6].
Another initial value problem of considerable attention is the one given below:
Problem: 02
y ¢ = y cos ( t ) ü
ï
ý
y ( 0) = 1
ï
þ
sin ( t )
Being a linear first order IVP, its analytical solution, y ( t ) = e
, is a periodic function with period 2p in t.
For the purpose of comparison, a nonstandard second order and A – stable method by Ramos (2007) is selected,
with the iterative algorithm given as:
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yi +1 = yi +
2Dt é f ( ti , yi ) ù
ë
û
2
2 f ( ti , yi ) - Dtf ¢ ( ti , yi )
where f ¢ ( ti , yi ) = ft + f y f at ( ti , yi ) ×
3
y(t)
2
1
0
Proposed
Ramos
Exact
-1
0
2
4
6
time
8
10
Figure 3. Comparison of proposed method with a method of Ramos with respect to exact solution for step size
0.2.
It is observed from fig. 3 that the method of Ramos shows notable fluctuations at ridges of the solution curve
whereas proposed algorithm follows the periodic behavior of the solution curve even at considerably large step
size ( Dt = 0.2 ) . It should also be mentioned that Ramos method gives second order accuracy and maintains A –
stability specifically for singular, non-singular and singularly-perturbed problems. It must be admitted at this
stage that the methods like Heun’s and Ralston’s beat the proposed algorithm when it comes to periodicity.
Though, the proposed algorithm does not require very small step sizes in order that the error does not propagate
dramatically, for certain problems it will be necessary to take small step sizes and this occurs with problems
having solutions of oscillatory behavior. In the regions where curvature is considerably larger, the proposed
algorithm needs smaller step sizes to follow the solution as shown by figure 4 for the IVP given in problem 02.
Nevertheless, there are better options to solve such problems. For further details, see (Fang, 2009) and (Yang &
X., 2008).
3
step size = 0.5
exact solution
2.5
y(t)
2
1.5
step size = 0.25
1
0.5
step size = 1
0
-0.5
0
1
2
3
4
5
time
6
7
8
9
10
Figure 4. Proposed method follows the oscillatory behavior with reducing step size.
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Computation of errors and CPU time of different algorithms with various integration steps are given in table 1. It
can be observed that maximum absolute error produced by Ramos method is considerably higher than the one
produced by proposed algorithm with the same integration steps and so is the case with the CPU time noted for
both the methods.
Table 1. Comparison of Proposed Algorithm with Ramo's Method for y¢ = y cos ( t ) ; y ( 0 ) = 1 with respect to
errors and Computer time.
Integration
Steps
Method
25
50
100
200
400
25
50
100
200
400
Ramos
Proposed
Algorithm
Error at
( t = 10 )
Maximum Absolute Error
0.141939963560321
0.096591768133376
0.045508916402834
0.003532000426836
0.020619199968314
0.035530962934161
0.005305961565138
0.000849565979959
0.000152269593188
0.000030552080064
CPU Time
(seconds)
0.872205397973849
0.366086264684939
0.239923661433473
0.015513069156405
0.100016509934113
0.059822450713906
0.008011144164723
0.002314692012949
6.192312521497989e-04
1.599858792880049e-04
0.002321
0.002841
0.004234
0.007780
0.018664
0.000709
0.001571
0.005447
0.010940
0.012100
Another feature depicted by figure 5 is continuous oscillations in error plot of Ramos method with increasing
integration steps whereas error plot of proposed algorithm does not exhibit such behavior.
maximum absolute errors
10
10
10
10
10
0
-1
-2
Ramos
Proposed
-3
-4
0
50
100
150
200
250
300
integration steps
350
400
450
500
Figure 5. Error plots obtained by Ramos and Proposed Algorithm
The dynamical equations of a rationalized biological model of two challenging populations (shown by figure 6);
called Lotka – Volterra predator – prey system, are given by two coupled nonlinear first order ordinary
differential equations:
Problem: 03
y1 = a y1 ( t ) - b y1 ( t ) y2 ( t ) ü
ï
ý
y2 = -g y2 ( t ) + d y1 ( t ) y2 ( t ) ï
þ
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where y1 ( t ) and y2 ( t ) are number of prey (for example, rabbit) and predators (for example, fox) respectively
at any given time t, y1 and y2 represent growth rates of the two species respectively over time, and a , b , g ,
and d are factors concerning the interaction of the two species.
Figure 6. Predator – Prey interaction, source: www.personal.psu.ed
The system cannot exactly be solved except for two equilibria at ( 0, 0 ) and (g d , a b ) as detailed in Zill
(2009). Nonetheless, the system is capable of being analyzed through numerical algorithms. Figure 7 displays
numerical solution and phase portrait of the system obtained by Euler’s, Wambecq’s 2 and proposed algorithm
using step size of 0.1 with initial conditions y1 ( 0 ) = 2 and y2 ( 0 ) = 1 taking the parameters’ values
a = 1.2, b = 0.6, g = 0.8, d = 0.3 from Chapra (2012).
state variables vs time
15
10
5
0
0
10
Prey
Predator
5
10
20
30
0
0
40
(a) Euler's Time Plot
state variables vs time
10
20
30
0
0
40
(c) Wambecq's Time Plot
state variables vs time
6
8
10
2
4
6
8
(d) Wambecq's Phase Plane Portrait
5
5
0
0
4
5
5
0
0
2
(b) Euler's Phase Plane Portrait
10
20
30
0
0
40
(e) Proposed Time Plot
1
2
3
4
5
6
(f) Proposed Phase Plane Portrait
Figure 7. Numerical solution of predator – prey system using Euler’s, Wambecq’s and Proposed algorithm with
400 integration steps.
Although, same step size has been used for all three algorithms but fig. 7a shows oscillations with increasing
amplitudes and the same is apparent by the phase plane portrait (fig. 7b). This results in using much smaller step
sizes required by the Euler’s method. Likewise, Wambecq’s method have sharp corners at peaks of the
oscillations shown in fig. 7c and the same is confirmed by the phase portrait in fig. 7d. On the other hand,
2
yi +1 = yi + Dt ék12 ( 2k1 - k2 )ù ; where k1 = f ( ti , yi ) , k2 = f ( ti + 0.5Dt , yi + 0.5Dtk1 )
ê
ú
ë
û
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proposed algorithm produced better results with the same step size. Over time, a cyclical behavior develops in
fig. 7e; as expected. Also, the phase portrait (fig. 7f) reveals repetition of the process by a closed trajectory.
Finally, second order linear (autonomous) ordinary differential equation called harmonic oscillator (mass-spring
system) has been investigated through proposed algorithm and Wambecq’s method against its solution obtained
by ode45 solver offered by MATLAB keeping relative error tolerance as small as 1e-06. For the sake of
simplicity, we consider the case of damped harmonic oscillator ( b > 0 ) with mass of the object equals 1
(Blanchard et al., 2012):
y ( t ) + by ( t ) + ky ( t ) = 0 ü
ï
subject to
ý
ï
y ( 0) = a , y ( 0) = b
þ
Problem: 04
where y ( t ) and y ( t ) are position and velocity of the spring respectively, ( k > 0 ) is called spring constant and
b is known as damping constant.
The system has been solved numerically through three iterative algorithms with same step size as shown in
figure 8 choosing damping constant relatively small ( b = 0.17 ) and spring constant k = 1 with initial conditions:
g mp
y ( 0) = 3.14, y ( 0 ) = 0 . Solution to the system, as expected, goes up and down with decreasing amplitude and
slowly comes to rest as time goes on; and proposed algorithm favorably agrees with that of ode45 but
Wambecq’s shows amplitudes less than those obtained by rest of the two iterative algorithms; however, it also
attains resting position with time. This type of behavior is confirmed by the phase plane portrait shown in figure
9 where solutions are spiraling in to the point where mass is at rest and there is no velocity, called equilibrium
point. It can be observed from figure 9 that Wambecq came across various disruptions while reaching to
equilibrium point.
4
position
2
0
-2
-4
0
Proposed
ode45
Wambecq
10
20
30
40
50
time
Figure 8. Behavior of solution of damped harmonic oscillator investigated by three different iterative algorithms.
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9. Mathematical Theory and Modeling
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4
Wambecq
ode45
Proposed
3
position
2
1
0
-1
-2
-3
-3
-2
-1
0
velocity
1
2
3
Figure 9. Phase plane portrait of damped harmonic oscillator generated by three different iterative algorithms.
4. Conclusion
In this work, formulation and a comparative study of the proposed algorithm is carried out with two nonstandard
finite difference schemes. Numerical results obtained support the efficiency of the proposed algorithm in
comparison to nonstandard schemes even at larger step sizes. The algorithm is also found to have lower
computational cost with greater accuracy in results both for linear and nonlinear problems considered.
5. Future Work
In future, much of work would be based upon standard error analysis of the algorithm proposed. Convergence of
the proposed algorithm would be discussed in detail. Its stability region would also be drawn and a comparison
with standard and few more non-standard methods would be under consideration.
6. Acknowledgement
The authors would like to express their sincere thanks to anonymous referees of the journal for their thoughtful
reading of the work presented.
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