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.
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.
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
Optimum Solution of Quadratic Programming Problem: By Wolfe’s Modified Simple...IJLT EMAS
In this paper, an alternative approach to the Wolfe’s
method for Quadratic Programming is suggested. Here we
proposed a new approach based on the iterative procedure for
the solution of a Quadratic Programming Problem by Wolfe’s
modified simplex method. The method sometimes involves less or
at the most an equal number of iteration as compared to
computational procedure for solving NLPP. We observed that
the rule of selecting pivot vector at initial stage and thereby for
some NLPP it takes more number of iteration to achieve
optimality. Here at the initial step we choose the pivot vector on
the basis of new rules described below. This powerful technique
is better understood by resolving a cycling problem.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
Quadratic Programming : KKT conditions with inequality constraintsMrinmoy Majumder
In the case of Quadratic Programming for optimization, the objective function is a quadratic function. One of the techniques for solving quadratic optimization problems is KKT Conditions which is explained with an example in this tutorial.
A method for solving quadratic programming problems having linearly factoriz...IJMER
A new method namely, objective separable method based on simplex method is proposed for
finding an optimal solution to a quadratic programming problem in which the objective function can be
factorized into two linear functions. The solution procedure of the proposed method is illustrated with the
numerical example.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
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.
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
Optimum Solution of Quadratic Programming Problem: By Wolfe’s Modified Simple...IJLT EMAS
In this paper, an alternative approach to the Wolfe’s
method for Quadratic Programming is suggested. Here we
proposed a new approach based on the iterative procedure for
the solution of a Quadratic Programming Problem by Wolfe’s
modified simplex method. The method sometimes involves less or
at the most an equal number of iteration as compared to
computational procedure for solving NLPP. We observed that
the rule of selecting pivot vector at initial stage and thereby for
some NLPP it takes more number of iteration to achieve
optimality. Here at the initial step we choose the pivot vector on
the basis of new rules described below. This powerful technique
is better understood by resolving a cycling problem.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
Quadratic Programming : KKT conditions with inequality constraintsMrinmoy Majumder
In the case of Quadratic Programming for optimization, the objective function is a quadratic function. One of the techniques for solving quadratic optimization problems is KKT Conditions which is explained with an example in this tutorial.
A method for solving quadratic programming problems having linearly factoriz...IJMER
A new method namely, objective separable method based on simplex method is proposed for
finding an optimal solution to a quadratic programming problem in which the objective function can be
factorized into two linear functions. The solution procedure of the proposed method is illustrated with the
numerical example.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
New approach for wolfe’s modified simplex method to solve quadratic programmi...eSAT Journals
Abstract
In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
Keywords: Quadratic programming problem, New approach, Modified simplex method, and Optimal solution.
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.
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.
Oscillation Criteria for First Order Nonlinear Neutral Delay Difference Equat...inventionjournals
We discuss the oscillatory behavior of all solutions of first order nonlinear neutral delay difference equations with variable coefficients of the form where are sequences of positive real numbers, and are sequences of nonnegative real numbers, and are positive integers. Our proved results extend and develop some of the well-known results in the literature. Examples are inserted to demonstrate the confirmation of our new results
This article continues the study of concrete algebra-like structures in our polyadic approach, when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but "quantized"; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered.
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.
Finite Element Analysis of the Aeroelasticity Plates Under Thermal and Aerody...Mohammad Tawfik
At the flight condition of panel-flutter (usually supersonic flight conditions), the phenomenon is associated with elevated temperatures, produced from the aerodynamic heating through the boundary layer friction and the presence of shock waves. This heating adds to the complexity of the problem by introducing panel stiffness reduction and thermal loading, which might also be associated with post-buckling deflection. In the following, a literature review of panel-flutter analysis and control topics is presented.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
A report on the finite element analysis of a beam under thermal loading. Nonlinear deflections and solution procedures covered.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
FEM: Introduction and Weighted Residual MethodsMohammad Tawfik
What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods
New approach for wolfe’s modified simplex method to solve quadratic programmi...eSAT Journals
Abstract
In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
Keywords: Quadratic programming problem, New approach, Modified simplex method, and Optimal solution.
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.
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.
Oscillation Criteria for First Order Nonlinear Neutral Delay Difference Equat...inventionjournals
We discuss the oscillatory behavior of all solutions of first order nonlinear neutral delay difference equations with variable coefficients of the form where are sequences of positive real numbers, and are sequences of nonnegative real numbers, and are positive integers. Our proved results extend and develop some of the well-known results in the literature. Examples are inserted to demonstrate the confirmation of our new results
This article continues the study of concrete algebra-like structures in our polyadic approach, when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but "quantized"; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered.
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.
Finite Element Analysis of the Aeroelasticity Plates Under Thermal and Aerody...Mohammad Tawfik
At the flight condition of panel-flutter (usually supersonic flight conditions), the phenomenon is associated with elevated temperatures, produced from the aerodynamic heating through the boundary layer friction and the presence of shock waves. This heating adds to the complexity of the problem by introducing panel stiffness reduction and thermal loading, which might also be associated with post-buckling deflection. In the following, a literature review of panel-flutter analysis and control topics is presented.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
A report on the finite element analysis of a beam under thermal loading. Nonlinear deflections and solution procedures covered.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
FEM: Introduction and Weighted Residual MethodsMohammad Tawfik
What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods
Existence results for fractional q-differential equations with integral and m...IJRTEMJOURNAL
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
In this study, nonlinear singularly perturbed problems with nonlocal condition are evaluated by finite
difference method. The exact solution 푢(푥) has boundary layers at 푥 = 0 and 푥 = 1. We present some
properties of the exact solution of the multi-point boundary value problem (1)-(3). According to the perturbation
parameter, by the method of integral identities with the use exponential basis functions and interpolating
quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference
scheme on Bakhvalov mesh is established. The error analysis for the difference scheme is performed. 휀 −
uniform convergence for approximate solution in the discrete maximum norm is provided, which is the firstorder
(O(h)). This theoretical process is applied on the sample. By Thomas Algorithm, it has been shown to be
consistent with the theoretical results of numerical results. The results were embodied in table and graphs. The
relationship between the approximate solution with the exact solution are obtained by Maple 10 computer
program.
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.
The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
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.
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.
The study is concerned with a different perspective which the numerical solution of the singularly
perturbed nonlinear boundary value problem with integral boundary condition using finite difference method on
Bakhvalov mesh. So, we show some properties of the exact solution. We establish uniformly convergent finite
difference scheme on Bakhvalov mesh. The error analysis for the difference scheme is performed. The numerical
experiment implies that the method is the first order convergent in the discrete maximum norm, independently of
휀- singular perturbation parameter with effective and efficient iterative algorithm. The numerical results are
shown in table and graphs.
Local Model Checking Algorithm Based on Mu-calculus with Partial OrdersTELKOMNIKA JOURNAL
The propositionalμ-calculus can be divided into two categories, global model checking algorithm
and local model checking algorithm. Both of them aim at reducing time complexity and space complexity
effectively. This paper analyzes the computing process of alternating fixpoint nested in detail and designs
an efficient local model checking algorithm based on the propositional μ-calculus by a group of partial
ordered relation, and its time complexity is O(d2(dn)d/2+2) (d is the depth of fixpoint nesting, n is the
maximum of number of nodes), space complexity is O(d(dn)d/2). As far as we know, up till now, the best
local model checking algorithm whose index of time complexity is d. In this paper, the index for time
complexity of this algorithm is reduced from d to d/2. It is more efficient than algorithms of previous
research.
An Examination of Effectuation Dimension as Financing Practice of Small and M...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Does Goods and Services Tax (GST) Leads to Indian Economic Development?iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Childhood Factors that influence success in later lifeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Emotional Intelligence and Work Performance Relationship: A Study on Sales Pe...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer’s Acceptance of Internet Banking in Dubaiiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study of Employee Satisfaction relating to Job Security & Working Hours amo...iosrjce
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Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 2015), PP 01-11
www.iosrjournals.org
DOI: 10.9790/5728-11640111 www.iosrjournals.org 1 | Page
Numerical Solutions of Second Order Boundary Value Problems
by Galerkin Residual Method on Using Legendre Polynomials
M. B. Hossain1*
, M. J. Hossain2
, M. M. Rahaman3
, M. M. H. Sikdar4
M.A.Rahaman5
1, 3
Department of Mathematics, 2,5
Department of CIT, 4
Department of Statistics Patuakhali Science and
Technology University, Dumki, Patuakhali-8602
Abstract: In this paper, an analysis is presented to find the numerical solutions of the second order linear and
nonlinear differential equations with Robin, Neumann, Cauchy and Dirichlet boundary conditions. We use the
Legendre piecewise polynomials to the approximate solutions of second order boundary value problems. Here
the Legendre polynomials over the interval [0,1] are chosen as trial functions to satisfy the corresponding
homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition
to that the given differential equation over arbitrary finite domain [a,b] and the boundary conditions are
converted into its equivalent form over the interval [0,1]. Numerical examples are considered to verify the
effectiveness of the derivations. The numerical solutions in this study are compared with the exact solutions and
also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.
Keywords: Galerkin Method, Linear and Nonlinear VBP, Legendre polynomials
I. Introduction
In order to find out the numerical solutions of many linear and nonlinear problems in science and
engineering, namely second order differential equations, we have seen that there are many methods to solve
analytically but a few methods for solving numerically with various types of boundary conditions. In the
literature of numerical analysis solving a second order boundary value problem of differential equations, many
authors have attempted to obtain higher accuracy rapidly by using numerical methods. Among various
numerical techniques, finite difference method has been widely used but it takes more computational costs to get
higher accuracy. In this method, a large number of parameters are required and it can not be used to evaluate the
value of the desired points between two grid points. For this reason, Galerkin weighted residual method is
widely used to find the approximate solutions to any point in the domain of the problem.
Continuous or piecewise polynomials are incredibly useful as mathematical tools since they are
precisely defined and can be differentiated and integrated easily. They can be approximated any function to any
accuracy desired [1], spline functions have been studied extensively in [2-9]. Solving boundary value problems
only with Dirichlet boundary conditions has been attempted in [4] while Bernstein polynomials [10, 11] have
been used to solve the two point boundary value problems very recently by the authors Bhatti and Bracken [1]
rigorously by the Galerkin method. But it is limited to the second order boundary value problems with Dirichlet
boundary conditions and to first order nonlinear differential equation. On the other hand, Ramadan et al. [2] has
studied linear boundary value problems with Neumann boundary conditions using quadratic cubic polynomial
splines and nonpolynomial splines. We have also found that the linear boundary value problems with Robin
boundary conditions have been solved using finite difference method [12] and Sinc-Collocation method [13],
respectively. Thus except [9], little concentration has been given to solve the second order nonlinear boundary
value problems with dirichlet, Neumann and Robin boundary conditions. Therefore, the aim of this paper is to
present the Galerkin weighted residual method to solve both linear and nonlinear second order boundary value
problems with all types of boundary conditions. But none has attempted, to the knowledge of the present
authors, using these polynomials to solve the second order boundary value problems. Thus in this paper, we
have given our attention to solve some linear and nonlinear boundary value problems numerically with different
types of boundary conditions though it is originated in [1].
However, in this paper, we have solved second order differential equations with various types of
boundary conditions numerically by the technique of very well-known Galerkin method [15] and Legendre
piecewise polynomials [14] are used as trial function in the basis. Individual formulas for each boundary value
problem consisting of Dirichlet, Neumann, Robin and Cauchy boundary conditions are derived respectively.
Numerical examples of both linear and nonlinear boundary value problems are considered to verify the
effectiveness of the derived formulas and are also compared with the exact solutions. All derivations are
performed by MATLAB programming language.
2. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 2 | Page
II. Legendre Polynomials
The solution of the Legendre’s equation is called the Legendre polynomial of degree n and is denoted
by )(xpn .
Then
N
r
rn
n
r
n x
rnrnr
rn
xp
0
2
)!2()!(!2
)!22(
)1()(
where
2
n
N for n even
and
2
1
n
N for n odd
The first few Legendre polynomials are
xxp )(1
)13(
2
1
)( 2
2 xxp
)35(
2
1
)( 3
3 xxxp
)33035(
8
1
)( 24
4 xxxp
)157063(
8
1
)( 35
5 xxxxp
)5105315231(
16
1
)( 246
6 xxxxp
)35315693429(
16
1 357
7 xxxxp etc
Graphs of first few Legendre polynomials
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
p1 p2 p3 p4 p5 p6 p7
Shifted Legendre polynomials
Here the "shifting" function (in fact, it is an affine transformation) is chosen such that it bijectively
maps the interval [0, 1] to the interval [−1, 1], implying that the polynomials are
An explicit expression for the shifted Legendre polynomials is given by orthogonal on [0, 1]:
3. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 3 | Page
n
k
kn
x
k
kn
k
n
xp
0
)()1()(~
The analogue of Rodrigues' formula for the shifted Legendre polynomials is
n
n
n
xx
dx
d
n
xp )(
!
1
)(~ 2
To satisfy the condition 1,0)1()0( npp nn , we modified the shifted Legendre polynomials
given above in the following form
)1()1()(
!
1
)( 2
xxx
dx
d
n
xp nn
n
n
n .
Since Legendre polynomials have special properties at 0x and 1x : 0)0( np and
1,0)1( npn respectively, so that they can be used as set of basis function to satisfy the corresponding
homogeneous form of the Dirichlet boundary conditions to derive the matrix formulation of second order BVP
over the interval [0,1].
III. Formulation Of Second Order Bvp
We consider the general second order linear BVP [15]:
bxaxruxq
dx
du
xp
dx
d
),()()( (1a)
210110 )()(,)()( cbubucauau (1b)
where )(,)( xqxp and r are specified continuous functions and 211010 ,,,,, cc are specified
numbers. Since our aim is to use the Legendre polynomials as trial functions which are derived over the interval
[0,1], so the BVP (1) is to be converted to an equivalent problem on [0,1] by replacing x by ,)( axab and
thus we have:
10),()(~)(~
xxruxq
dx
du
xp
dx
d
(2a)
2
1
01
1
0 )1()1(,)0()0( cu
ab
ucu
ab
u
(2b)
where ))(()(~,))(()(~,))((
)(
1
)(~
2
axabrxraxabqxqaxabp
ab
xp
Using Legendre polynomials, )(xpi we assume an approximate solution in a form,
n
i
ii nxpaxu
1
1,)()(~ (3)
Now the Galerkin weighted residual equations corresponding to the differential equation (1a) is given by
njdxxpxruxq
dx
ud
xp
dx
d
j ,,2,1,0)()(~~)(~
~
)(~1
0
(4)
After minor simplification, from (2) we can obtain
i
n
i
jiji
ji
ji
a
pppabpppab
dxxpxpxq
dx
dp
dx
dp
xp
0
1
0 1
0
1
0 )0()0()0(~)()1()1()1(~)(
)()()(~)(~
1
0 1
1
1
2 )0()0(~)()1()1(~)(
)()(~
jj
j
ppabcppabc
dxxpxr (5)
Or, equivalently in matrix form
4. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 4 | Page
n
i
jiji njFaK
1
, ,,3,2,1, (6a)
where
1
0
1
01
0
,
)0()0()0(~)()1()1()1(~)(
)()()(~)(~
jiji
ji
ji
ji
pppabpppab
dxxpxpxq
dx
dp
dx
dp
xpK
(6b)
nj
ppabcppabc
dxxpxrF
jj
jj ,,2,1,
)0()0(~)()1()1(~)(
)()(~1
0 1
1
1
2
(6c)
Solving the system (6a), we find the values of the parameters ),,2,1( niai and then substituting
these parameters into eqn. (3), we get the approximate solution of the boundary value problem (2). If we replace
x by
ab
ax
in )(~ xu , then we get the desired approximate solution of the boundary value problem (1).
Now we discuss the different types of boundary value problems using various types of boundary conditions as
follows:
Case 1: The matrix formulation with the Robin boundary conditions ,0,0,0( 010
)01 , are already discussed in equation (6).
Case 2: The matrix formulation of the differential equation (1a) with the Dirichlet boundary conditions
)0,0,0,0.,.( 1010 ei is given by
n
i
jiji njFaK
1
, ,,2,1, (7a)
where
1
0
, ,,2,1,,)()()(~)(~ njidxxpxpxq
dx
dp
dx
dp
xpK ji
ji
ji (7b)
1
0
0
0 ,2,1,)()()(~)(~)()(~ njdxxpxxq
dx
dp
dx
d
xpxpxrF j
j
jj
(7c)
Case 3: The approximate solution of the differential equation (1a) consisting of Neumann boundary conditions
)0,0,0,0.,.( 1010 ei is given by
n
i
jiji njFaK
1
, ,,2,1, (8a)
where
1
0
, ,,2,1,,)()()(~)(~ njidxxpxpxq
dx
dp
dx
dp
xpK ji
ji
ji (8b)
nj
ppabcppabc
dxxpxrF
jj
jj ,,2,1,
)0()0(~)()1()1(~)(
)()(~1
0 1
1
1
2
(8c)
Case 4(i): The approximate solution of the differential equation (1a) consisting of Cauchy boundary conditions
)0,0.,.( 11 ei is given by
n
i
jiji njFaK
1
, ,,2,1, (9a)
where
1
0 1
0
, ,,2,1,,
)0()0()0(~
)()()(~)(~ nji
ppp
dxxpxpxq
dx
dp
dx
dp
xpK
ji
ji
ji
ji
(9b)
1
0 1
00
1
1
0
0
)0()0()0(~)0()0(~
)()()(~)(~)()(~
jj
j
j
jj
ppppc
dxxpxxq
dx
dp
dx
d
xpxpxrF (9c)
Case 4(ii): The matrix formulation with the Cauchy boundary conditions )0,0.,.( 11 ei
is given by
5. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 5 | Page
n
i
jiji njFaK
1
, ,,2,1, (10a)
where
1
0 1
0
, ,,2,1,,
)1()1()1(~
)()()(~)(~ nji
ppp
dxxpxpxq
dx
dp
dx
dp
xpK
ji
ji
ji
ji
(10b)
1
0 1
00
1
2
0
0 ,2,1,
)1()1()1(~)1()1(~
)()()(~)(~)()(~ nj
ppppc
dxxpxxq
dx
dp
dx
d
xpxpxrF
jj
j
j
jj
(10c)
Similar calculation for nonlinear boundary value problems using the Legendre polynomials can be
derived which will be discussed through numerical examples in next portion.
IV. Numerical Examples
In this section, we explain four linear and two nonlinear boundary value problems which are available
in the existing literatures, considering four types of boundary conditions to verify the effectiveness of the
present formulations described in the previous sections. The convergence of each linear boundary value problem
is calculated by
)()(1 xuxuE nn
where )(xun represents the approximate solution by the proposed method using n -th degree
polynomial approximation. The convergence of nonlinear boundary value problem is assumed when the
absolute error of two consecutive iterations is recorded below the convergence criterion such that
NN
uu
~1~
where N is the Newton’s iteration number and varies from 10-8
.
Example1. First we consider the boundary value problem with Robin boundary conditions [15]:
xxu
dx
ud
2
,cos2
2
2
(11a)
4)(4)(,1
2
3
2
uuuu (11b)
The exact solution is xxu cos)( .
The boundary value problem (11) over [0, 1] is equivalent to the BVP
10,
22
cos2
2
1
2
2
2
xxu
dx
ud
4)1(4)1(
2
,1)0(3)0(
2
uuuu
Using the formula derived in Case-1, equation (6) and using different number of Legendre
polynomials, the approximate solutions are shown in Table 1. It is observe that the accuracy is found nearly the
order 10-5
, 10-6
, 10-6
, 10-7
on using 6, 7, 8, and 9 Legendre polynomials respectively.
6. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 6 | Page
Table 1: Exact, approximate solutions and absolute differences for the example 1
Example2. We consider the boundary value problem with Dirichlet boundary conditions [1]:
100,2
2
2
xexu
dx
ud x
(12a)
0)10(,0)0( uu (12b)
The exact solution is:
x
e
ec
exexxe xxx
sin
2
10cos121
2
10cot
2cos21
2
1
10
2
The boundary value problem (12) is similar to the VBP
10,100
100
1 102
2
2
xexu
dx
ud x
(13a)
0)1()0( uu (13b)
Using the formula calculated in Case-2, equation (7), the approximate solutions are summarized in
Table 2. It is shown that the accuracy up to 3, 5, 6 and 8 significant digits are obtained for 8, 10, 12 and 15
Legendre polynomials respectively.
Table 2: Exact, approximate solutions and absolute differences for the example 2
x Exact
Approximate Error Approximate Error
6 Legendre polynomials 7 Legendre polynomials
π/2
11π/20
3π/5
13π/20
7π/10
3π/4
4π/5
17π/20
9π/10
19π/20
π
0.0000000000
-0.1564344650
-0.3090169944
-0.4539904997
-0.5877852523
-0.7071067812
-0.8090169944
-0.8910065242
-0.9510565163
-0.9876883406
-1.0000000000
0.0000000000
-0.1563952198
-0.3090826905
-0.4540545634
-0.5877674749
-0.7070318602
-0.8089692169
-0.8910363629
-0.9511166917
-0.9876784226
-1.0000000000
0.0000000000
3.9245266397E-005
6.5696082539E-005
6.4063642331E-005
1.7777435319E-005
7.4920958162E-005
4.7777521913E-005
2.9838679610E-005
6.0175450790E-005
9.9179536835E-006
0.0000000000
0.0000000000
-0.1564361741
-0.3090293644
-0.4539871936
-0.5877722257
-0.7071038639
-0.8090274888
-0.8910135046
-0.9510489959
-0.9876846996
-1.0000000000
0.0000000000
1.7090241794E-006
1.2370074109E-005
3.3061242330E-006
1.3026583116E-005
2.9172939944E-006
1.0494427349E-005
6.9803926110E-006
7.5203691469E-006
3.6409813292E-006
0.0000000000
8 Legendre polynomials 9 Legendre polynomials
π/2
11π/20
3π/5
13π/20
7π/10
3π/4
4π/5
17π/20
9π/10
19π/20
π
0.0000000000
-0.1564344650
-0.3090169944
-0.4539904997
-0.5877852523
-0.7071067812
-0.8090169944
-0.8910065242
-0.9510565163
-0.9876883406
-1.0000000000
0.0000000000
-0.1564359285
-0.3090176234
-0.4539882914
-0.5877849061
-0.7071088461
-0.8090175407
-0.8910047081
-0.9510563902
-0.9876895985
-1.0000000000
0.0000000000
1.4634359017E-006
6.2907488652E-007
2.2082959843E-006
3.4624056477E-007
2.0649209902E-006
5.4633655033E-007
1.8160907849E-006
1.2606701094E-007
1.2579438687E-006
0.0000000000
0.0000000000
-0.1564348025
-0.3090167598
-0.4539903134
-0.5877856122
-0.7071068440
-0.8090166411
-0.8910065885
-0.9510567651
-0.9876880856
-1.0000000000
0.0000000000
3.3747539643E-007
2.3457270337E-007
1.8630984755E-007
3.5985972813E-007
6.2781989829E-008
3.5332033022E-007
6.4299355618E-008
2.4875544480E-007
2.5495029321E-007
0.0000000000
x Exact Approximate Error Approximate Error
8 Legendre Polynomials 10 Legendre Polynomials
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000000000
1.1187780396
1.5229010422
1.0028335888
-0.0316812662
-0.7648884663
-0.6362448424
0.1621981240
0.8543002645
0.7816320450
0.0000000000
0.0000000000
1.1170936907
1.5275119317
0.9991649652
-0.0335749590
-0.7601267725
-0.6381149360
0.1584875515
0.8589605755
0.7799092631
0.0000000000
0.0000000000E+000
1.6843488272E-003
4.6108895155E-003
3.6686235984E-003
1.8936928166E-003
4.7616938417E-003
1.8700936538E-003
3.7105725450E-003
4.6603109949E-003
1.7227819094E-003
0.0000000000E+000
0.0000000000
1.1187696958
1.5229254770
1.0028621730
-0.0317936975
-0.7647484694
-0.6363187413
0.1621711868
0.8543785180
0.7815831186
0.0000000000
0.0000000000E+000
8.3437592517E-006
2.4434769731E-005
2.8584174599E-005
1.1243135195E-004
1.3999696859E-004
7.3898882488E-005
2.6937180385E-005
7.8253508555E-005
4.8926372264E-005
0.0000000000E+000
12 Legendre Polynomials 15 Legendre Polynomials
7. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 7 | Page
Example3. In this case we consider the boundary value problem with Neumann boundary conditions [2]:
10,1
2
2
xu
dx
ud
(14a)
1sin
1cos1
)1(,
1sin
1cos1
)0(
uu (14b)
whose exact solution is, 1sin
1sin
1cos1
cos)(
xxxu .
Applying the formula derived in Case-3,equation (8), the approximate solutions, given in Table 3, are
obtained on using 5, 7, 8 and 10 Legendre polynomials with the remarkable accuracy nearly the order of 10-11
,
10-13
, 10-13
and 10-17
. On the other hand, Ramadan et al. [6] has found nearly the accuracy of order 10-6
and 10-6
,
and 10-8
on using quadratic and cubic polynomial splines, and nonpolynomial spline respectively with h=1/128
where h= (b-a)/N, a and b are the endpoints of the domain and N is number of subdivision of intervals [a,b].
Table 3: Exact, approximate solutions and absolute differences for the example 3
Example4. We consider the Cauchy (mixed) boundary value problem [4]:
20,0)25(3
2
2
xu
dx
ud
(15a)
with mixed boundary conditions
)(0)2(,)(150)0( NeumannuDirichletu (15b)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000000000
1.1187780396
1.5229010422
1.0028335888
-0.0316812662
-0.7648884663
-0.6362448424
0.1621981240
0.8543002645
0.7816320450
0.0000000000
0.0000000000
1.1187868241
1.5228940750
1.0028413857
-0.0316917502
-0.7648764993
-0.6362555845
0.1622062945
0.8542931007
0.7816403137
0.0000000000
0.0000000000E+000
8.7845183310E-006
6.9672726579E-006
7.7969029351E-006
1.0484051623E-005
1.1967000683E-005
1.0742144593E-005
8.1704803209E-006
7.1637940648E-006
8.2686651681E-006
0.0000000000E+000
0.0000000000
1.1187780208
1.5229010732
1.0028335785
-0.0316812686
-0.7648884575
-0.6362448562
0.1621981449
0.8543002369
0.7816320520
0.0000000000
0.0000000000E+000
1.8750267117E-008
3.0956165631E-008
1.0297521058E-008
2.4126875567E-009
8.7905381863E-009
1.3803168386E-008
2.0844985821E-008
2.7584562634E-008
6.9470443842E-009
0.0000000000E+000
x Exact
Approximate Error Approximate Error
5 Legendre Polynomials 7 Legendre Polynomials
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000000000
0.0495434094
0.0886001279
0.1167799138
0.1338012040
0.1394939273
0.1338012040
0.1167799138
0.0886001279
0.0495434094
0.0000000000
0.0000000000
0.0495434085
0.0886001278
0.1167799150
0.1338012039
0.1394939260
0.1338012039
0.1167799150
0.0886001278
0.0495434085
0.0000000000
0.0000000000E+000
8.1779601840E-010
8.6266188637E-011
1.2220623125E-009
9.0318336143E-011
1.2807547800E-009
9.0318585944E-011
1.2220623263E-009
8.6266216393E-011
8.1779603922E-010
0.0000000000E+000
0.0000000000
0.0495434094
0.0886001279
0.1167799138
0.1338012040
0.1394939273
0.1338012040
0.1167799138
0.0886001279
0.0495434094
0.0000000000
0.0000000000E+000
5.3554383150E-013
7.7055029024E-013
6.1556315600E-013
2.8521629503E-013
8.7904683532E-013
2.8493873927E-013
6.1554927822E-013
7.7057804582E-013
5.3552995372E-013
0.0000000000E+000
8 Legendre Polynomials 10 Legendre Polynomials
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000000000
0.0495434094
0.0886001279
0.1167799138
0.1338012040
0.1394939273
0.1338012040
0.1167799138
0.0886001279
0.0495434094
0.0000000000
0.0000000000
0.0495434094
0.0886001279
0.1167799138
0.1338012040
0.1394939273
0.1338012040
0.1167799138
0.0886001279
0.0495434094
0.0000000000
0.0000000000E+000
5.3554383150E-013
7.7055029024E-013
6.1556315600E-013
2.8521629503E-013
8.7904683532E-013
2.8493873927E-013
6.1554927822E-013
7.7057804582E-013
5.3552995372E-013
0.0000000000E+000
0.0000000000
0.0495434094
0.0886001279
0.1167799138
0.1338012040
0.1394939273
0.1338012040
0.1167799138
0.0886001279
0.0495434094
0.0000000000
0.0000000000E+000
2.7755575616E-016
8.3266726847E-017
8.3266726847E-017
3.3306690739E-016
3.3306690739E-016
8.3266726847E-017
8.3266726847E-017
9.7144514655E-017
2.7061686225E-016
0.0000000000E+000
8. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 8 | Page
whose exact solution is
34
3
34
3
3
1
125
1
125
12525)(
e
e
e
e
exu
xx
x
Using the formula illustrated in Case-4(ii), equation (10), and using different number of Legendre
polynomials, the approximate solutions are shown in Table 4. It is observe that the accuracy is found nearly the
order 10-5
, 10-7
, 10-9
and 10-14
on using 8, 10, 12, and 15 Legendre polynomials respectively.
Table 4: Exact, approximate solutions and absolute differences for the example 4
We now also apply the procedure described in section 3, formulation of second order linear BVP, to
find the numerical solutions of one nonlinear second order boundary value problem.
Example5. We consider a nonlinear BVP with Dirichlet boundary conditions [16]
31,
4
1
4
8
1 3
2
2
xx
dx
du
u
dx
ud
(16a)
17)1( u and
3
43
)3( u (16b)
The exact solution of the problem is given by
x
xxu
16
)( 2
x Exact
Approximate Error Approximate Error
8 Legendre polynomials 10 Legendre polynomials
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
150.0000000000
130.1632369724
113.4892661882
99.4766167249
87.7038570984
77.8169206793
69.5184571131
62.5588894853
56.7289082771
51.8531763663
47.7850557499
44.4021973948
41.6028615824
39.3028580779
37.4330141035
35.9370939611
34.7701077404
33.8969582431
33.2913854326
32.9351766612
32.8176189233
150.0000000000
130.1632361028
113.4892227060
99.4766067840
87.7038954397
77.8169675873
69.5184703563
62.5588587219
56.7288570188
51.8531411416
47.7850603820
44.4022380045
41.6029098061
39.3028800573
37.4329943677
35.9370491681
34.7700776491
33.8969721439
33.2914230933
32.9351748738
32.8176189248
0.0000000000E+000
8.6964047341E-007
4.3482237473E-005
9.9408884751E-006
3.8341331972E-005
4.6908003981E-005
1.3243194047E-005
3.0763410692E-005
5.1258324511E-005
3.5224774393E-005
4.6321808611E-006
4.0609714048E-005
4.8223707843E-005
2.1979341483E-005
1.9735765207E-005
4.4793006822E-005
3.0091356948E-005
1.3 900810046E-005
3.7660648488E-005
1.7874420521E-006
0.0000000000E+000
150.0000000000
130.1632367772
113.4892661238
99.4766170265
87.7038572221
77.8169204267
69.5184568118
62.5588894927
56.7289085840
51.8531766501
47.7850557239
44.4021970899
41.6028613102
39.3028581183
37.4330144044
35.9370941670
34.7701075952
33.8969579798
33.2913855119
32.9351768223
32.8176189233
0.0000000000E+000
1.9518458316E-007
6.4393191224E-008
3.0160821041E-007
1.2375100766E-007
2.5263204861E-007
3.0132471807E-007
7.3751351692E-009
3.0687399288E-007
2.8373816008E-007
2.5985485763E-008
3.0494597070E-007
2.7216093912E-007
4.0339642737E-008
3.0088499159E-007
2.0580901605E-007
1.4517870284E-007
2.6333222536E-007
7.9248636098E-008
1.6105848744E-007
0.0000000000E+000
12 Legendre polynomials 15 Legendre polynomials
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
150.0000000000
130.1632369724
113.4892661882
99.4766167249
87.7038570984
77.8169206793
69.5184571131
62.5588894853
56.7289082771
51.8531763663
47.7850557499
44.4021973948
41.6028615824
39.3028580779
37.4330141035
35.9370939611
34.7701077404
33.8969582431
33.2913854326
32.9351766612
32.8176189233
150.0000000000
130.1632369713
113.4892661894
99.4766167252
87.7038570969
77.8169206790
69.5184571146
62.5588894862
56.7289082761
51.8531763648
47.7850557500
44.4021973964
41.6028615833
39.3028580769
37.4330141021
35.9370939616
34.7701077418
33.8969582428
33.2913854316
32.9351766622
32.8176189233
0.0000000000E+000
1.1265512967E-009
1.1613110473E-009
2.8325075618E-010
1.5066063952E-009
2.9929481116E-010
1.4935892523E-009
9.1058893759E-010
1.0389911154E-009
1.5080843241E-009
1.0601297618E-010
1.5544259213E-009
8.4782669774E-010
1.0288943031E-009
1.3430536683E-009
4.4136072574E-010
1.3676100252E-009
3.7826453081E-010
9.9556984878E-010
1.0126584016E-009
0.0000000000E+000
150.0000000000
130.1632369724
113.4892661882
99.4766167249
87.7038570984
77.8169206793
69.5184571131
62.5588894853
56.7289082771
51.8531763663
47.7850557499
44.4021973948
41.6028615824
39.3028580779
37.4330141035
35.9370939611
34.7701077404
33.8969582431
33.2913854326
32.9351766612
32.8176189233
0.0000000000E+000
2.8421709430E-014
1.4210854715E-014
2.5579538487E-013
2.7000623959E-013
5.6843418861E-014
3.4106051316E-013
1.4210854715E-014
3.6237679524E-013
2.8421709430E-014
3.6948222260E-013
7.1054273576E-014
2.9132252166E-013
8.5265128291E-014
2.9842794902E-013
8.5265128291E-014
1.8474111130E-013
3.6948222260E-013
4.2632564146E-014
1.0658141036E-013
0.0000000000E+000
9. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 9 | Page
To implement the Legendre polynomials, first we convert the BVP (16) to an equivalent BVP on the
interval [0,1] by replacing x by 12 x such that
10,)12(16
4
1 3
2
2
xx
dx
du
u
dx
ud
(17a)
17)0( u and
3
43
)1( u (17b)
Suppose that the approximate solution of the boundary value problem (17) applying the Legendre
polynomials is given by
n
i
ii nxpaxxu
1
0 1,)()()(~ (18)
Where
3
8
17)(0
x
x is specified by the dirichlet boundary conditions at 0x and 1x and
0)1()0( ii pp for each .,,2,1 ni
The weighted residual equations of (17a) corresponding to the approximation solution (18), given by
1
0
3
2
2
,,2,1,0)()12(16
~
~
4
1~
nkdxxpx
dx
ud
u
dx
ud
k (19)
Exploiting integration by parts with minor simplifications, we have
n
i
i
n
j
k
j
ijkik
iki adxp
dx
dp
papp
dx
d
p
dx
dp
dx
dp
dx
dp
1
1
0 1
0
0
4
1
4
1
1
0
0
0
03
,,2,1,
4
1
1216 nkdxp
dx
d
dx
dp
dx
d
px k
k
k
(20)
The above equation (20) is equivalent to the matrix form
GACD )( (21a)
where the elements of the matrix A, C, D and G are kikii dca ,, ,, and kg respectively, given by
1
0
0
0,
4
1
dxpp
dx
d
p
dx
dp
dx
dp
dx
dp
d kik
iki
ki
(21b)
n
j
k
j
ijki dxp
dx
dp
pac
1
1
0
,
4
1
(21c)
1
0
0
0
03
,,2,1,
4
1
1216 nkdxp
dx
d
dx
dp
dx
d
pxg k
k
kk
(21d)
The initial values of these coefficients ia are obtained by applying the Galerkin method to the BVP
neglecting the nonlinear term in (17a). Therefore, to find the initial coefficients, we will solve the system
GDA (22a)
where the elements of the matrices are given by
1
0
, dx
dx
dp
dx
dp
d ki
ki (22b)
1
0
03
,,2,1,1216 nkdx
dx
dp
dx
d
pxg k
kk
(22c)
Once the initial values of the parameters ia are obtained from equation (22a), they are substituted into
equation (21a) to obtain new estimates for the values of ia . This iteration process continues until the converged
values of the unknowns are obtained. Putting the final values of coefficients into equation (18), we obtain an
10. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 10 | Page
approximate solution of the BVP (17), and if we replace x by
2
1x
in this solution we will get the
approximate solution of the given BVP (16).
Using first 10 and 15 Legendre polynomials with 10 iterations, the absolute differences between exact and the
approximate solutions are given in Table 5. It is observed that the accuracy is found of the order nearly 10-6
and
10-9
on using 10 and 15 Legendre polynomials respectively.
Table 5: Exact, approximate solutions and absolute differences of example 5 using 10 iterations
V. Conclusions
In this paper, the formulation of one dimensional linear and nonlinear second order boundary value
problems have been discussed in details by the Galerkin weighted residual method applying Legendre
polynomials as the basis functions in the approximation. The proposed method is applied to solve some
numerical examples both linear and nonlinear. The computed results are compared with the exact solutions and
we have found a good agreement with the exact solution. All the mathematical formulations and numerical
computations have been done by MATLAB-10 code.
References
[1]. M. I. Bhatti and P. Bracken, “Solutions of Differential Equations in a Bernstein Polynomial Basis,” Journal of Computational and
Applied Mathematics, Vol. 205, No.1, 2007, pp.272-280. doi:10.1016/j.cam.2006.05.002.
[2]. M. A. Ramadan, I. F. Lashien and W. K. Zahra, “Polynomial and Nonpolynomial Spline Approaches to the Numerical Solution of
Second Order Boundary Value Problem,” Applied Mathematics and Computation, Vol.184, No. 2, 2007, pp.476-
484.doi:10.1016/j.amc.2006.06.053.
[3]. R. A. Usmani and M. Sakai, “A Connection between Quatric Spline and Numerov Solution of a Boundary value Problem,”
International Journal of Computer Mathematics, Vol. 26, No. 3, 1989, pp. 263-273. doi:10.1080/00207168908803700.
[4]. Arshad Khan, “Parametric Cubic Spline Solution of Two Point Boundary Value Problems,” Applied Mathematics and
Computation, Vol. 154, No. 1, 2004, pp.175-182. doi:10.1016/S009-3003(03)00701-X.
[5]. E. A. Al-Said, “Cubic Spline Method for Solving Two Point Boundary Value Problems,” Korean Journal of Computational and
Applied Mathematics, Vol. 5, 1998, pp. 759-770.
[6]. E. A. Al-Said, “Quadratic Spline Solution of Two Point Boundary Value Problems,” Journal of Natural Geometry, Vol. 12, 1997,
pp.125-134.
[7]. D. J. Fyfe, “The Use of Cubic splines in the Solution of Two Point Boundary Value Problems,” The Computer Journal, Vol. 12, No.
2, 1969, pp. 188-192. doi:10.1093/comjnl/12.2.188
[8]. A.K. Khalifa and J. C. Eilbeck, “Collocation with Quadratic and Cubic Splines,” The IMA Journal of Numerical Analysis, Vol. 2,
No. 1, 1982, pp. 111-121. doi:10.1093/imanum/2.1.111
[9]. G. Mullenheim, “Solving Two-Point Boundary Value Problems with Spline Functions,” The IMA Journal of Numerical Analysis,
Vol. 12, No. 4, 1992, pp. 503-518. doi:10.1093/imanum/12.4.503
[10]. J. Reinkenhof, “Differentiation and Integration Using Bernstein’s Polynomials,” International Journal for Numerical Methods in
Engineering, Vol. 11, No. 10, 1977, pp. 1627-1630. doi:10.1002/nme.1620111012
[11]. E. Kreyszig, “Bernstein Polynomials and Numerical Integration,” International Journal for Numerical Methods in Engineering, Vol.
14, No. 2, 1979, pp. 292-295.doi:10.1002/nme.1620140213
[12]. R. A. Usmani, “Bounds for the Solution of a Second Order Differential Equation with Mixed Boundary Conditions,” Journal of
Engineering Mathematics, Vol. 9, No. 2 1975, pp. 159-164. doi:10.1007/BF01535397
[13]. B. Bialecki, “Sinc-Collocation methods for Two Point Boundary Value Problems,” The IMA Journal of Numerical Analysis, Vol.
11, No. 3, 1991, pp. 357-375, doi:10.1093/imanum/11.3.357
x Exact Approximate Error Approximate Error
10 Legendre polynomials 15 Legendre polynomials
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
17.0000000000
15.7554545455
14.7733333333
13.9976923077
13.3885714286
12.9166666667
12.5600000000
12.3017647059
12.1288888889
12.0310526316
12.0000000000
12.0290476190
12.1127272727
12.2465217391
12.4266666667
12.6500000000
12.9138461538
13.2159259259
13.5542857143
13.9272413793
14.3333333333
17.0000000000
15.7554513780
14.7733354460
13.9976948749
13.3885686469
12.9166634176
12.5600012904
12.3017686614
12.1288904019
12.0310499612
11.9999964073
12.0290472659
12.1127304365
12.2465245358
12.4266656697
12.6499967192
12.9138454470
13.2159287029
13.5542862145
13.9272392744
14.3333333333
0.000000E+000
3.167443E-006
2.112697E-006
2.567226E-006
2.781699E-006
3.249050E-006
1.290388E-006
3.955476E-006
1.512994E-006
2.670420E-006
3.592663E-006
3.531724E-007
3.163751E-006
2.796678E-006
9.969229E-007
3.280754E-006
7.068234E-007
2.776953E-006
5.002600E-007
2.104884E-006
0.000000E-000
17.0000000000
15.7554545479
14.7733333301
13.9976923079
13.3885714325
12.9166666616
12.5599999987
12.3017647109
12.1288888877
12.0310526261
12.0000000012
12.0290476234
12.1127272702
12.2465217349
12.4266666696
12.6500000021
12.9138461496
13.2159259274
13.5542857152
13.9272413784
14.3333333333
0.000000E+000
2.436138E-009
3.184004E-009
2.160068E-010
3.968140E-009
5.087697E-009
1.277906E-009
5.053868E-009
1.141489E-009
5.513813E-009
1.181718E-009
4.306418E-009
2.555799E-009
4.242077E-009
2.950340E-009
2.129884E-009
4.234208E-009
1.455382E-009
8.964705E-010
9.028049E-010
0.000000E-000
11. Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method…
DOI: 10.9790/5728-11640111 www.iosrjournals.org 11 | Page
[14]. N. Saran, S. D. Sharma and T. N. Trivedi, “Special Functions,” Seventh Edition, Pragati Prakashan, 2000.
[15]. P. E. Lewis and J. P. Ward, “The Finite element Method, Principles and Applications,” Addison-Wesley, Boston, 1991.
[16]. R. L. Burden and J. D. Faires, “Numerical Analysis,” Books/Cole Publishing Co. Pacific Grove, 1992.
[17]. M. K. Jain, “Numerical Solution of Differential Equations,” 2nd
Edition, New Age International, New Delhi, 2000.
[18]. J. Stephen Chapman, “MATLAB Programming for Engineers,” Third Edition, Thomson Learning, 2004.
[19]. C. Steven Chapra, “Applied Numerical Methods with MATLAB for Engineers and Scientists,” Second Edition, Tata McGraw-Hill,
2007.