Numerical Solutions of Second Order Boundary Value Problems by Galerkin Resid...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Entanglement Behavior of 2D Quantum ModelsShu Tanaka
I gave an oral presentation at YITP Workshop on Quantum Information Physics (YQIP2014).
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
This presentation is based on the following papers.
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
Preprint version are available via
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
京都大学基礎物理学研究所で開催されたワークショップ、"YITP Workshop on Quantum Information Physics (YQIP2014)"で口頭講演を行いました。
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
本発表は以下の論文に基づいています。
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
プレプリントバージョンは、以下のサイトから閲覧できます。
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
Numerical Solutions of Second Order Boundary Value Problems by Galerkin Resid...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Entanglement Behavior of 2D Quantum ModelsShu Tanaka
I gave an oral presentation at YITP Workshop on Quantum Information Physics (YQIP2014).
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
This presentation is based on the following papers.
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
Preprint version are available via
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
京都大学基礎物理学研究所で開催されたワークショップ、"YITP Workshop on Quantum Information Physics (YQIP2014)"で口頭講演を行いました。
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
本発表は以下の論文に基づいています。
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
プレプリントバージョンは、以下のサイトから閲覧できます。
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
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.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
A Characterization of the Zero-Inflated Logarithmic Series Distributioninventionjournals
In this paper, we introduce a characterization of the zero-inflated logarithmic series distributions through a linear differential equation of its probability generating function.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
A Characterization of the Zero-Inflated Logarithmic Series Distributioninventionjournals
In this paper, we introduce a characterization of the zero-inflated logarithmic series distributions through a linear differential equation of its probability generating function.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
A friendly introduction to differential equationsEducation
A friendly introduction to differential equations Authored by Mohammed K A Kaabar
In this book, there are five chapters: The Laplace Transform, Systems of Homogenous Linear Differential Equations (HLDE), Methods of First and Higher Orders Differential Equations, Extended Methods of First and Higher Orders Differential Equations, and Applications of Differential Equations. In addition, there are exercises at the end of each chapter above to let students practice additional sets of problems other than examples, and they can also check their solutions to some of these exercises by looking at "Answers to Odd-Numbered Exercises" section at the end of this book. This book is a very useful for college students who studied Calculus II, and other students who want to review some concepts of differential equations before studying courses such as partial differential equations, applied mathematics, and electric circuits II.
Product Tank Amsterdam Pulse UX Presentationicemobile
In her talk Anna shares learnings of frequently testing with users in agile teams. Also she will show that testing with users can be simple, effective & prove its value.
Anna is an interaction designer & UX researcher with 9 years of experience. She likes to challenge the ‘WHY‘ of user actions. Two years ago she started the UX Lab at IceMobile to give a voice to the user. Today the UX Lab validates all concepts & products with users. She created 'Pulse UX' a simple method of frequent user testing that lets agile teams develop better products faster. Pulse UX is now part of every sprint at IceMobile.
To download, head to -
http://solarreference.com/parabolic-trough-collectors-comparison/
A detailed comparison of different types of parabolic trough collectors on the basis of specifications, technology, material etc. If CSP is your arena, this is one presentation you just can't miss !!!
Source: NREL
For more quality resources visit us at http://solarreference.com
Procesos sin despilfarro y a coste mínimo: el método Lean Thinking José María Apellidos
¿Qué es y para que sirve el Lean Thinking?
La mejora de procesos en la empresa
Los 8 tipos de desperdicios (Mudas)
Aspectos tecnológicos y organizativos del Lean
Herramientas básicas del Lean Management
Ejemplo práctico de la aplicación de la metodología en Rotulowcost, con Pepe Llabrés
Caso práctico a realizar entre los asistentes
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.
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.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
Within the framework of the general theory of relativity (GR) the modeling of the central symmetrical
gravitational field is considered. The mapping of the geodesic motion of the Lemetr and Tolman basis on
their motion in the Minkowski space on the world lines is determined. The expression for the field intensity
and energy where these bases move is obtained. The advantage coordinate system is found, the coordinates
and the time of the system coincide with the Galilean coordinates and the time in the Minkowski space.
Within the framework of the general theory of relativity (GR) the modeling of the central symmetrical
gravitational field is considered. The mapping of the geodesic motion of the Lemetr and Tolman basis on
their motion in the Minkowski space on the world lines is determined. The expression for the field intensity
and energy where these bases move is obtained. The advantage coordinate system is found, the coordinates
and the time of the system coincide with the Galilean coordinates and the time in the Minkowski space.
Significance of Mathematical Analysis in Operational Methods [2014]SanjayKumar Patel
Dr Ajay Shukla from SVNIT came to Ahmedabad on 2nd August 2014,to deliver the lecture on Significance of Mathematical Analysis in Operational Methods ....
Lecture was held in St. Xavier's College,Ahmedabad under the Father Valles Lecture Series...
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
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.
Similar to hebyshev Polynomial Based Numerical Inverse Laplace Transform Solutions of Linear Volterra Integral and Integro-Differential Equations (20)
Attending a job Interview for B1 and B2 Englsih learnersErika906060
It is a sample of an interview for a business english class for pre-intermediate and intermediate english students with emphasis on the speking ability.
Taurus Zodiac Sign_ Personality Traits and Sign Dates.pptxmy Pandit
Explore the world of the Taurus zodiac sign. Learn about their stability, determination, and appreciation for beauty. Discover how Taureans' grounded nature and hardworking mindset define their unique personality.
Memorandum Of Association Constitution of Company.pptseri bangash
www.seribangash.com
A Memorandum of Association (MOA) is a legal document that outlines the fundamental principles and objectives upon which a company operates. It serves as the company's charter or constitution and defines the scope of its activities. Here's a detailed note on the MOA:
Contents of Memorandum of Association:
Name Clause: This clause states the name of the company, which should end with words like "Limited" or "Ltd." for a public limited company and "Private Limited" or "Pvt. Ltd." for a private limited company.
https://seribangash.com/article-of-association-is-legal-doc-of-company/
Registered Office Clause: It specifies the location where the company's registered office is situated. This office is where all official communications and notices are sent.
Objective Clause: This clause delineates the main objectives for which the company is formed. It's important to define these objectives clearly, as the company cannot undertake activities beyond those mentioned in this clause.
www.seribangash.com
Liability Clause: It outlines the extent of liability of the company's members. In the case of companies limited by shares, the liability of members is limited to the amount unpaid on their shares. For companies limited by guarantee, members' liability is limited to the amount they undertake to contribute if the company is wound up.
https://seribangash.com/promotors-is-person-conceived-formation-company/
Capital Clause: This clause specifies the authorized capital of the company, i.e., the maximum amount of share capital the company is authorized to issue. It also mentions the division of this capital into shares and their respective nominal value.
Association Clause: It simply states that the subscribers wish to form a company and agree to become members of it, in accordance with the terms of the MOA.
Importance of Memorandum of Association:
Legal Requirement: The MOA is a legal requirement for the formation of a company. It must be filed with the Registrar of Companies during the incorporation process.
Constitutional Document: It serves as the company's constitutional document, defining its scope, powers, and limitations.
Protection of Members: It protects the interests of the company's members by clearly defining the objectives and limiting their liability.
External Communication: It provides clarity to external parties, such as investors, creditors, and regulatory authorities, regarding the company's objectives and powers.
https://seribangash.com/difference-public-and-private-company-law/
Binding Authority: The company and its members are bound by the provisions of the MOA. Any action taken beyond its scope may be considered ultra vires (beyond the powers) of the company and therefore void.
Amendment of MOA:
While the MOA lays down the company's fundamental principles, it is not entirely immutable. It can be amended, but only under specific circumstances and in compliance with legal procedures. Amendments typically require shareholder
What is the TDS Return Filing Due Date for FY 2024-25.pdfseoforlegalpillers
It is crucial for the taxpayers to understand about the TDS Return Filing Due Date, so that they can fulfill your TDS obligations efficiently. Taxpayers can avoid penalties by sticking to the deadlines and by accurate filing of TDS. Timely filing of TDS will make sure about the availability of tax credits. You can also seek the professional guidance of experts like Legal Pillers for timely filing of the TDS Return.
Affordable Stationery Printing Services in Jaipur | Navpack n PrintNavpack & Print
Looking for professional printing services in Jaipur? Navpack n Print offers high-quality and affordable stationery printing for all your business needs. Stand out with custom stationery designs and fast turnaround times. Contact us today for a quote!
Personal Brand Statement:
As an Army veteran dedicated to lifelong learning, I bring a disciplined, strategic mindset to my pursuits. I am constantly expanding my knowledge to innovate and lead effectively. My journey is driven by a commitment to excellence, and to make a meaningful impact in the world.
The world of search engine optimization (SEO) is buzzing with discussions after Google confirmed that around 2,500 leaked internal documents related to its Search feature are indeed authentic. The revelation has sparked significant concerns within the SEO community. The leaked documents were initially reported by SEO experts Rand Fishkin and Mike King, igniting widespread analysis and discourse. For More Info:- https://news.arihantwebtech.com/search-disrupted-googles-leaked-documents-rock-the-seo-world/
Cracking the Workplace Discipline Code Main.pptxWorkforce Group
Cultivating and maintaining discipline within teams is a critical differentiator for successful organisations.
Forward-thinking leaders and business managers understand the impact that discipline has on organisational success. A disciplined workforce operates with clarity, focus, and a shared understanding of expectations, ultimately driving better results, optimising productivity, and facilitating seamless collaboration.
Although discipline is not a one-size-fits-all approach, it can help create a work environment that encourages personal growth and accountability rather than solely relying on punitive measures.
In this deck, you will learn the significance of workplace discipline for organisational success. You’ll also learn
• Four (4) workplace discipline methods you should consider
• The best and most practical approach to implementing workplace discipline.
• Three (3) key tips to maintain a disciplined workplace.
What are the main advantages of using HR recruiter services.pdfHumanResourceDimensi1
HR recruiter services offer top talents to companies according to their specific needs. They handle all recruitment tasks from job posting to onboarding and help companies concentrate on their business growth. With their expertise and years of experience, they streamline the hiring process and save time and resources for the company.
India Orthopedic Devices Market: Unlocking Growth Secrets, Trends and Develop...Kumar Satyam
According to TechSci Research report, “India Orthopedic Devices Market -Industry Size, Share, Trends, Competition Forecast & Opportunities, 2030”, the India Orthopedic Devices Market stood at USD 1,280.54 Million in 2024 and is anticipated to grow with a CAGR of 7.84% in the forecast period, 2026-2030F. The India Orthopedic Devices Market is being driven by several factors. The most prominent ones include an increase in the elderly population, who are more prone to orthopedic conditions such as osteoporosis and arthritis. Moreover, the rise in sports injuries and road accidents are also contributing to the demand for orthopedic devices. Advances in technology and the introduction of innovative implants and prosthetics have further propelled the market growth. Additionally, government initiatives aimed at improving healthcare infrastructure and the increasing prevalence of lifestyle diseases have led to an upward trend in orthopedic surgeries, thereby fueling the market demand for these devices.
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Enterprise Excellence is Inclusive Excellence.pdfKaiNexus
Enterprise excellence and inclusive excellence are closely linked, and real-world challenges have shown that both are essential to the success of any organization. To achieve enterprise excellence, organizations must focus on improving their operations and processes while creating an inclusive environment that engages everyone. In this interactive session, the facilitator will highlight commonly established business practices and how they limit our ability to engage everyone every day. More importantly, though, participants will likely gain increased awareness of what we can do differently to maximize enterprise excellence through deliberate inclusion.
What is Enterprise Excellence?
Enterprise Excellence is a holistic approach that's aimed at achieving world-class performance across all aspects of the organization.
What might I learn?
A way to engage all in creating Inclusive Excellence. Lessons from the US military and their parallels to the story of Harry Potter. How belt systems and CI teams can destroy inclusive practices. How leadership language invites people to the party. There are three things leaders can do to engage everyone every day: maximizing psychological safety to create environments where folks learn, contribute, and challenge the status quo.
Who might benefit? Anyone and everyone leading folks from the shop floor to top floor.
Dr. William Harvey is a seasoned Operations Leader with extensive experience in chemical processing, manufacturing, and operations management. At Michelman, he currently oversees multiple sites, leading teams in strategic planning and coaching/practicing continuous improvement. William is set to start his eighth year of teaching at the University of Cincinnati where he teaches marketing, finance, and management. William holds various certifications in change management, quality, leadership, operational excellence, team building, and DiSC, among others.
3.0 Project 2_ Developing My Brand Identity Kit.pptxtanyjahb
A personal brand exploration presentation summarizes an individual's unique qualities and goals, covering strengths, values, passions, and target audience. It helps individuals understand what makes them stand out, their desired image, and how they aim to achieve it.
As a business owner in Delaware, staying on top of your tax obligations is paramount, especially with the annual deadline for Delaware Franchise Tax looming on March 1. One such obligation is the annual Delaware Franchise Tax, which serves as a crucial requirement for maintaining your company’s legal standing within the state. While the prospect of handling tax matters may seem daunting, rest assured that the process can be straightforward with the right guidance. In this comprehensive guide, we’ll walk you through the steps of filing your Delaware Franchise Tax and provide insights to help you navigate the process effectively.
Filing Your Delaware Franchise Tax A Detailed Guide
hebyshev Polynomial Based Numerical Inverse Laplace Transform Solutions of Linear Volterra Integral and Integro-Differential Equations
1. American Research Journal of Mathematics Original Article
ISSN 2378-704X Volume 1, Issue1, Feb-2015
www.arjonline.org 22
Chebyshev Polynomial Based Numerical Inverse Laplace
Transform Solutions of Linear Volterra Integral and
Integro-Differential Equations
Vinod Mishra1
, Dimple Rani
Department of Mathematics, Sant Longowal Institute of Engg. & Tech, Longowal (Punjab)
Abstract: There are enormous occasions when the methods for finding solutions of integral and integro-
differential equations lead to failure because of difficulty in inverting Laplace transform by standard technique.
Numerically inverting Laplace transform is cost effective in comparison to rather complicated technique of com-
plex analysis. In the process of numerical inversion, an odd cosine series which is ultimately based on Chebyshev
polynomial has been used. The adequacy of method is illustrated through numerical examples of convolution type
linear Volterra integral equations of second kind which include weakly singular Abel's integral equation and Vol-
terra integro-differential equation.
Keywords: Volterra integral equation; Volterra Integro-differential equation; Numerical Inversion of Laplace
transform; Gaussian quadrature; Cosine series; Chebyshev polynomial.
I. INTRODUCTION
Linear Volterra Integral and Integro-Differential equations are extensively used in the various specialties of science
and engineering. These include mathematical physics, chemical kinetic, heat conduction, seismology, fluid dynam-
ics, biological models, population dynamics, metallurgy and semi-conductors [11, 12 &13].
The Volterra Integral equation with a convolution kernel is defined by
x
Txdttftxkxyxf
0
,0,)()()()( (1)
while Volterra Integro-Differential equation by
00 )(,)()()()(
0
uxudxxutxkxuxu
x
x
, (2)
There have been many extant methods for solving Volterra Integral and Integro-Differential equations with convolu-
tion kernels. Wazwaz[12] and Al-Hayani[18] discussed the Adomian polynomial based Laplace Transform methods
to solve these equations. Yang[13] proposed a method by which solution is expressed in power series and to im-
prove the convergence rate applied the Pade approximant. Babolian-Shamloo[7], Aznam-Hussin[17] and Mishra et
al.[9] used operational matrices of piecewise constant orthogonal function or Haar wavelet to solve these equations.
Homotopy perturbation method with finite difference technique was used by Raftari[10] to solve Volterra Inte-
gro-Differential equations. Zarebnia[14] solved these equations using Sinc function. A modified Taylor series
method has been applied to approximate the solution of linear Integro-Differential equations in [11].
1.1 Numerical Inverse Laplace Transform
The Laplace transform of function is defined by
.)()()(
0
dttfesFtfL st
(3)
1
Corresponding Author: vinodmishra.2011@rediffmail.com
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As usual, the equations are first converted into algebraic equations using Laplace transform. The inverse Laplace
transform is then applied and the numerical solution is finally expressed in terms of Chebyshev series.
Applying Laplace transforms on both sides of above equations and then using convolution property in (1) & (2),
reduce the equations in the form
).()( sFxfL
The inversion leads to
)()( 1
sFLxf
.
Comprehensive literature consists of a number of methods for numerically inverting Laplace transform suited for
problems in particular situations. For a detailed survey of various methods for Laplace transform inversion numer-
ically refers to Cohen[2], Davies-Martin[5] and Mishra[16]. Bellman et al. (1966)[1] have outlined a method which
they derive from the consideration of Gauss-Legendre's quadrature rule. Substituting 0,
t
ex in [1 & 2],
)(sF is transformed from the interval ,0 to 1,0 as
,)(
1
)(
1
0
1/
dxxgxsF s
Where
xfxg log
1
)(
.
)(sF can be numerically inverted using Legendre series of
0
2 )()(
k
kk xPxg by considering )(xg being
even function in ]1,1[ . This method has slow convergence as the coefficients k decrease slowly due to the sin-
gularity of )(xg at 0x [1 & 2]. Erdelyi (1943), Papoulis (1956) and Lanczos (1957) have proposed Legendre's
function to find the approximate value of function )(tf [2, 5, 6, & 8]. In [3] Dubner and Abate (1968) have ex-
pressed the )(tf in terms Fourier cosine transforms. Durbin (1974) [4] has proposed the trapezoidal rule by result-
ing approximation.
1.2 Chebyshev Series based Inversion
Chebyshev polynomial is due to Panfnuty Chebyshev (b. 1821), a Russian. It is basically a class of orthogonal poly-
nomials. In sixties remarkable development to the theory lead to computation of function approximations, integrals
and solution of differential equations using Chebyshev polynomials, termed as Chebyshev series expansion of a
function. The range of problems covered includes singular problem and network synthesis [15].
Here we propose a technique parallel to Papoulis[6] by making the substitution
.0,sin
x
e (4)
The interval (0, ∞) transformed into 2,0
and )(xf becomes
).(sinlog
1
gf
(5)
Now eq. (3) takes the form
.)(cossin)(
2
0
1
dgsF
s
(6)
By setting ,)12( ks k = 0, 1, 2…, we have
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.)(cossin))12((
2
0
2
dgkF
k
(7)
Here we assume that 0)0(2 fg
. In case this does not hold then arrange it by subtracting a suitable function
from )(g . The function )(g can be expanded in 2,0
as the odd cosine series
0
)12cos()(
k
k kg (8)
and is valid in the interval ., 22
Now we have to determine the coefficients k
.
22
cossin
2
2
iikii
k ee
i
ee
Making substitution
i
ex , we find that
1
2
)1(
2
)1(
1
2
1
1
2
)1(
2
)1(
2
1
12
2
)1(
2
2
)1(
1
2
1
1
2
)1(
2
)1(
2
1
12
2
2
21
2
1
1
2
0
2
2
11
2
111
2
1
cossin12
11
122
122
1122
12
12
12
12
2
22
k
k
k
k
xk
k
k
k
x
rk
k
rk
k
xr
k
r
k
x
k
k
k
k
x
kk
x
x
x
x
x
x
x
kkkk
rkrk
rk
rrrk
k
k
k
k
k
kkk
= cos
2
1
2
1)122cos(
2
1
2
1)12cos(
11
k
k
k
k
rk
r
k
r
k
k
kr
(9)
Substitution of (8) & (9) and using the result of the orthogonality
,
4
)12cos( 2
2/
0
dk
eq. (7) gives
))12(( kF =
krk
rk
kk
r
k
r
k
k
k
k
k
.
2
1
2
1
2
1
2
1
412
1 1
0
1
2
(10)
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))12((2
4 2
kFk
krk
rk
k
r
k
r
k
k
k
k
k
.
2
1
2
1
2
1
2
1
1
1 1
0
1
(11)
which is the linear system in 0 , 1 , 2 ….. k …. It can be conveniently put in the matrix form
,ABC (12)
where the matrix C and the coefficient matrix A can be obtained by putting the values of ,2,1,0k in LHS
and RHS of (11) respectively.
A
k
k
k
k
k
)1(
2
1
2
132
011
001
,
k
B
2
1
0
Thus k can be obtained by solving CAB 1
and hence )(g can be obtained from eq. (7).
In general, we compute first )1( N terms of eq.(8), that is, the finite series
.)12cos()(
0
N
k
kN kg (13)
As N , )()( ggN . From )(g we can determine )(xf
.)12cos()(
0
k
k kxf (14)
Defining
sin,
cos
cos
)(1 x
k
xUk , where )(xUk is the Chebyshev polynomial of second kind of degree k
Then
2/12
1cos x
e
And
.)(1)(
0
2
2/12
k
x
kk
x
eUexf
1.3 Numerical Examples
Example1
Consider the weakly singular Volterra integral equation of second kind [12&13]
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x
xdt
tx
tu
xxu
0
1,0,
)(
2)( (15)
Taking Laplace transform on both sides of eq. (15) and using convolution theorem, we obtain
).()( sF
ss
uL
(16)
The necessary condition for compliance of inverse )(xu of eq.(16) is that 0)0( u . Following initial value theo-
rem,
.0lim)(lim)0(
ss
sssFu
ss
Therefore, the solution )(xu can be obtained and k can be computed using relation (12).
Table1.1. Coefficients in the Expansion of )(xu
k k
0 0.81399311
1 -0.0446193
2 0.03972445
3 0.00011582
4 0.01165101
5 0.00230399
6 0.00569674
7 0.00212046
8 0.00350064
9 0.00166594
10 0.00164977
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Table1.2. Computed Pade approximant [4/4] and Approximation Solution for Example1
X Pade approximant [4/4] method Present method Absolute Error
0 0 5.14E-17 5.13800E-17
0.1 0.41411018 0.4141034 6.78000E-06
0.2 0.50848304 0.5087519 2.68860E-04
0.3 0.56452274 0.5637984 7.24340E-04
0.4 0.60364034 0.6033569 2.83440E-04
0.5 0.63323515 0.6338924 6.57250E-04
0.6 0.65675942 0.655994 7.65420E-04
0.7 0.67610075 0.6740862 2.01455E-03
0.8 0.69240064 0.6911992 1.20144E-03
0.9 0.70639982 0.7070156 6.15780E-04
1 0.71860476 0.7202293 1.62454E-03
Example2. Consider the integral equation
.sin)()(
0
xdttxutu
x
(17)
Laplace transform on both sides of eq. (17) gives
).(
1
1
)(
2
sF
s
uL
(18)
The condition 0)0( u is not satisfied
.1
1
1
lim)(lim)0(
2
s
sssFu
ss
As per provision of condition a possible function need to be subtracted from )(xu is 1. A function which takes the
value 1 at 0x is
x
e
. Therefore,
x
exuxU
)()( as 0)0( U . Since
1
1
)()(
s
sFsF . Thereby, the
solution )(xu can be obtained by computing the coefficients k the relation (12).
Table2.1. Coefficients in the Expansion of )(xu
k k
0 0.26369654
1 -0.0735987
2 -0.1482495
3 -0.0985091
4 -0.0769397
5 -0.0547004
6 -0.042268
7 -0.0310263
8 -0.0239681
9 -0.0176349
10 -0.0133825
11 -0.0095073
12 -0.0067188
13 -0.003956
14 -0.001397
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Table2.2. Computed Exact and Approximation Solution for Example2
x Exact solution Present method Absolute error
0 1 1 3.00000E-09
0.1 0.99750157 0.99722919 2.72375E-04
0.2 0.99002498 0.98950321 5.21765E-04
0.3 0.97762625 0.97849647 8.70222E-04
0.4 0.96039823 0.95925921 1.13902E-03
0.5 0.93846981 0.93944053 9.70723E-04
0.6 0.91200486 0.9125871 5.82237E-04
0.7 0.88120089 0.87896932 2.23157E-03
0.8 0.84628735 0.84616791 1.19441E-04
0.9 0.8075238 0.81065911 3.13531E-03
1 0.76519768 0.7670378 1.84012E-03
Example3. Consider the Volterra integral with a convolution kernel given by [13]
.sin)()cos()(
0
x
xdttutxxu (19)
As usual taking Laplace transform on both sides of eq.(19) yield
).(
1
1
)( 2
sF
ss
uL
(20)
The condition 0)0( u is satisfied as in Example 1, that is
.0
1
1
lim)(lim)0( 2
ss
sssFu
ss
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.75
0.8
0.85
0.9
0.95
1
x
u(x)
Fig.2. Comparison of exact and present approximate solution
Exact solution
Present method
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Therefore, the solution )(xu is feasible. The coefficients k computed (12) are shown below.
Table3.1.Coefficients in the Expansion of )(xu
k k
0 0.42441318
1 0.03264717
2 -0.09372896
3 -0.07000573
4 -0.06126397
5 -0.04672609
6 -0.0391798
7 -0.03125062
8 -0.02649286
9 -0.02180741
10 -0.01875585
11 -0.01580044
12 -0.01381433
13 -0.01199908
14 -0.01103245
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
u(x)
Fig.3. Comaprison of Pade approximant [4/4] and present approximate solution
Pade approximant [4/4] method
Present method
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Table 3.2. Computed Pade approximant [4/4] and Approximation Solution for Example3
X Pade approximant [4/4] Present method Absolute error
0 0 4.12E-16 4.12297E-16
0.1 0.095004082 0.09158784 3.41624E-03
0.2 0.180063957 0.183098185 3.03423E-03
0.3 0.255316973 0.256151895 8.34922E-04
0.4 0.320980415 0.321179014 1.98599E-04
0.5 0.377341791 0.372097134 5.24466E-03
0.6 0.42474943 0.433955997 9.20657E-03
0.7 0.463603433 0.461889867 1.71357E-03
0.8 0.494347029 0.483582381 1.07646E-02
0.9 0.517458374 0.518579699 1.12133E-03
1 0.533442822 0.546451691 1.30089E-02
Example4. Consider the first order linear Volterra Integro-differential equation of the form [8]
x
x
tx
dttuexuxu
0
)()()( )(
, 00 )( uxu . (21)
If we choose 0 , 0 , 1 , 1 , 00 x , .10 u
Taking Laplace transform on both sides of eq. (21) and using derivative property and convolution theorem of Lap-
lace transform,
).(
1
1
)( 2
sF
ss
s
uL
(22)
The condition 0)0( u is not satisfied as
.1
1
1
lim)(lim)0( 2
ss
s
sssFu
ss
In this case we choose a possible function that will be subtracted from )(xu to be 1. A function which takes the
value 1 at 0x is
x
e
. Therefore,
x
exuxU
)()( as .0)0( U Since ,
1
1
)()(
s
sFsF the solution
)(xu exists. The coefficients k can be calculated using the relation (12).
Table 4.1. Coefficients in the Expansion of )(xu
k k
0 0.212206589
1 -0.08161792
2 -0.12163703
3 -0.06349292
4 -0.04422927
5 -0.02637033
6 -0.01850358
7 -0.01134116
8 -0.00779918
9 -0.00443629
10 -0.00269506
11 -0.00095958
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12 -5.86E-05
Table4.2. Computed Exact and Approximation Solution for Example4
x Exact solution Present method Absolute error
0 1 1 0.00E+00
0.1 0.9951666 0.99489203 2.75E-04
0.2 0.9813308 0.98137139 4.06E-05
0.3 0.9594808 0.95941456 6.62E-05
0.4 0.930587 0.93094843 3.61E-04
0.5 0.8955945 0.89502027 5.74E-04
0.6 0.8554164 0.85534569 7.07E-05
0.7 0.8109282 0.81181337 8.85E-04
0.8 0.762963 0.76330526 3.42E-04
0.9 0.7123077 0.71129202 1.02E-03
1 0.6597002 0.65824129 1.46E-03
II. CONCLUSION
In this paper we have gone through a new insight into the use of Chebyshev polynomials. The series solutions in
terms of Chebyshev polynomials have been used as numerically inverting Laplace transform tool for finding solu-
tions of Volterra integral and integro-differential equations. The outcome of four test problems have been compared
with exact or Pade approximants and found to be numerically efficient.
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0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
u(x)
Fig.4. Comparison of exact solution and present approximate solution
Exact solution
Present method
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