This document presents a novel numerical approach for solving odd higher order boundary value problems (BVPs), specifically fifth, seventh, and ninth order linear and nonlinear BVPs. The method uses the Galerkin weighted residual method with Legendre polynomials as basis functions to satisfy the boundary conditions. Matrix formulations are derived for the fifth, seventh, and ninth order cases. Several examples are presented and the results are compared to existing methods to demonstrate the reliability and efficiency of the proposed method.
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.
Exact solutions for weakly singular Volterra integral equations by using an e...iosrjce
In this paper, an iterative method proposed by Daftardar-Gejji and Jafari namely (DJM) will be
presented to solve the weakly singular Volterra integral equation (WSVIE) of the second kind. This method is
able to solve large class of linear and nonlinear equations effectively, more accurately and easily. In this
iterative method the solution is obtained in the series form that converges to the exact solution if it exists. The
main contribution of the current paper is to obtain the exact solution rather than numerical solution as done by
some existing techniques. The results demonstrate that the method has many merits such as being derivativefree,
overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in
Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational
Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations
as in Homotopy Perturbation Method (HPM). The results reveal that the method is accurate and easy to
implement. The software used for the calculations in this study was MATHEMATICA®
10.0.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
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 pictorial method of visualizing curl & determinant operation utilized i...ijscmcj
Curl and Determinant operations are well known in the field of Electromagnetics and fluid dynamics. Famous Maxwell’s equations involve curl operation. Though this operation is widely used to represent rotation, we do not have sufficient literature explaining how this operation represents rotation.
Most of the books omit detailed discussion on the physical interpretation of the Curl and Determinan operation. In this paper we have attempted to develop a pictorial method to provide a logical proof showing the versatility of this operation to study the rotating vectors. We further show how this pictorial method simplifies the fundamental expression with regard to determinants.
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.
Exact solutions for weakly singular Volterra integral equations by using an e...iosrjce
In this paper, an iterative method proposed by Daftardar-Gejji and Jafari namely (DJM) will be
presented to solve the weakly singular Volterra integral equation (WSVIE) of the second kind. This method is
able to solve large class of linear and nonlinear equations effectively, more accurately and easily. In this
iterative method the solution is obtained in the series form that converges to the exact solution if it exists. The
main contribution of the current paper is to obtain the exact solution rather than numerical solution as done by
some existing techniques. The results demonstrate that the method has many merits such as being derivativefree,
overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in
Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational
Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations
as in Homotopy Perturbation Method (HPM). The results reveal that the method is accurate and easy to
implement. The software used for the calculations in this study was MATHEMATICA®
10.0.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
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 pictorial method of visualizing curl & determinant operation utilized i...ijscmcj
Curl and Determinant operations are well known in the field of Electromagnetics and fluid dynamics. Famous Maxwell’s equations involve curl operation. Though this operation is widely used to represent rotation, we do not have sufficient literature explaining how this operation represents rotation.
Most of the books omit detailed discussion on the physical interpretation of the Curl and Determinan operation. In this paper we have attempted to develop a pictorial method to provide a logical proof showing the versatility of this operation to study the rotating vectors. We further show how this pictorial method simplifies the fundamental expression with regard to determinants.
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
On solving fuzzy delay differential equationusing bezier curves IJECEIAES
In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem.
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.
Initial and Boundary Value Problems Involving the Inhomogeneous Weber Equatio...theijes
Initial and boundary value problems of the inhomogeneous Weber differential equation are treated in this work. General solutions are expressed in terms of the parametric Nield-Kuznetsov functions of the first and second kinds, and are computed when the forcing function is a constant or a variable function of the independent variable
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
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
A numerical solution for Sine-Gordon type system was done by the use of two finite difference schemes, the first is the explicit scheme and the second is the Crank-Nicholson scheme. A comparison between the two schemes showed that the explicit scheme is easier and has faster convergence than the Crank-Nicholson scheme which is more accurate. The MATLAB environment was used for the numerical computations.
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.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
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.
In this paper, an accurate numerical method is presented to find the numerical solution of the singularinitial value problems. The second-order singular initial value problem under consideration is transferredinto a first-order system of initial value problems, and then it can be solved by using the fifth-order RungeKutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methodsis confirmed by solving three model examples, and the obtained approximate solutions are compared withthe existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numericalmethod for solving the singular initial value problems.
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
On solving fuzzy delay differential equationusing bezier curves IJECEIAES
In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem.
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.
Initial and Boundary Value Problems Involving the Inhomogeneous Weber Equatio...theijes
Initial and boundary value problems of the inhomogeneous Weber differential equation are treated in this work. General solutions are expressed in terms of the parametric Nield-Kuznetsov functions of the first and second kinds, and are computed when the forcing function is a constant or a variable function of the independent variable
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
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
A numerical solution for Sine-Gordon type system was done by the use of two finite difference schemes, the first is the explicit scheme and the second is the Crank-Nicholson scheme. A comparison between the two schemes showed that the explicit scheme is easier and has faster convergence than the Crank-Nicholson scheme which is more accurate. The MATLAB environment was used for the numerical computations.
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.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
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.
In this paper, an accurate numerical method is presented to find the numerical solution of the singularinitial value problems. The second-order singular initial value problem under consideration is transferredinto a first-order system of initial value problems, and then it can be solved by using the fifth-order RungeKutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methodsis confirmed by solving three model examples, and the obtained approximate solutions are compared withthe existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numericalmethod for solving the singular initial value problems.
ACCURATE NUMERICAL METHOD FOR SINGULAR INITIAL-VALUE PROBLEMSaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular
initial value problems. The second-order singular initial value problem under consideration is transferred
into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge
Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods
is confirmed by solving three model examples, and the obtained approximate solutions are compared with
the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical
method for solving the singular initial value problems.
ACCURATE NUMERICAL METHOD FOR SINGULAR INITIAL-VALUE PROBLEMSaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular
initial value problems. The second-order singular initial value problem under consideration is transferred
into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge
Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods
is confirmed by solving three model examples, and the obtained approximate solutions are compared with
the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical
method for solving the singular initial value problems.
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.
The approximate bound state of the nonrelativistic Schrӧdinger equation was
obtained with the modified trigonometric scarf type potential in the framework of
asymptotic iteration method for any arbitrary angular momentum quantum number l
using a suitable approximate scheme to the centrifugal term. The effect of the screening
parameter and potential depth on the eigenvalue was studied numerically. Finally, the
scattering phase shift of the nonrelativistic Schrӧdinger equation with the potential
under consideration was calculated.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
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.
Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Co...IOSR Journals
Tan-cot method is applied to get exact soliton solutions of non-linear partial differential equations notably generalized Benjamin-Bona-Mahony, Zakharov-Kuznetsov Benjamin-Bona-Mahony, Kadomtsov-Petviashvilli Benjamin-Bona-Mahony and Korteweg-de Vries equations, which are important evolution equations with wide variety of physical applications. Elastic behavior and soliton fusion/fission is shown graphically and discussed physically as far as possible
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Similar to A novel numerical approach for odd higher order boundary value problems (20)
A novel numerical approach for odd higher order boundary value problems
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.5, 2014
1
A Novel Numerical Approach for Odd Higher Order Boundary
Value Problems
Md. Bellal Hossain1
and Md. Shafiqul Islam2
*
1
Department of Mathematics, Patuakhali Science and Technology University,
Dumki, Patuakhali-8602, Bangladesh. Email: bellal77pstu@yahoo.com
2
Department of Mathematics, University of Dhaka, Dhaka – 1000, Bangladesh.
*
Email of the corresponding author: mdshafiqul_mat@du.ac.bd
Abstract
In this paper, we investigate numerical solutions of odd higher order differential equations, particularly the fifth,
seventh and ninth order linear and nonlinear boundary value problems (BVPs) with two point boundary
conditions. We exploit Galerkin weighted residual method with Legendre polynomials as basis functions.
Special care has been taken to satisfy the corresponding homogeneous form of boundary conditions where the
essential types of boundary conditions are given. The method is formulated as a rigorous matrix form. Several
numerical examples, of both linear and nonlinear BVPs available in the literature, are presented to illustrate the
reliability and efficiency of the proposed method. The present method is quite efficient and yields better results
when compared with the existing methods.
Keywords: Galerkin method, fifth, seventh and ninth order linear and nonlinear BVPs, Legendre Polynomials.
1. Introduction
From the literature on the numerical solutions of BVPs, it is observed that the higher order differential equations
arise in some branches of applied mathematics, engineering and many other fields of advanced physical sciences.
Particularly, the solutions of fifth order BVPs arise in the mathematical modeling of viscoelastic flows in (Davis,
Karageorghis & Philips, 1988), the seventh order BVPs arise in modeling induction motors with two rotor
circuits in (Richards & Sarma, 1994), and the ninth order boundary value problems are known to arise in
hydrodynamic, hydro magnetic stability and applied sciences. The performance of the induction motor behavior
is modeled by a fifth order differential equation (Siddiqi, Akram & Iftikhar, 2012). This model is constructed
with two stator state variables, two rotor state variables and one shaft speed. Generally, two more variables must
be added to account for the effects of a second rotor circuit representing deep bars, a starting cage or rotor
distributed parameters (Siddiqi & Iftikhar, 2013) . For neglecting the computational burden of additional state
variables when additional rotor circuits are needed, the model is often bounded to the fifth order. The rotor
impedance is done under the assumption that the frequency of rotor currents depends on rotor speed. This
process is appropriate for the steady state response with sinusoidal voltage, but it does not hold up during the
transient conditions, when the rotor frequency is not a single value (Siddiqi, Akram & Iftikhar, 2012). So this
behavior is modeled in the seventh order differential equation . Recently, the BVPs of ninth order have been
developed due to their mathematical importance and the potential for applications in hydrodynamic, hydro
magnetic stability. Agarwal (1986) discussed extensively the existence and uniqueness theorem of solutions of
such BVPs in his book without any numerical examples. Caglar & Twizell (1999) used sixth degree B-spline
functions for the numerical solution of fifth order BVPs where their approach is divergent and unexpected
situation is found near the boundaries of the interval. The spline methods have been discussed for the solution of
other higher order BVPs. Siddiqi & Twizell (1997, 1998, 1996) applied twelfth, tenth, eighth and sixth degree
splines for solving linear BVPs of orders 12, 10, 8 and 6 successively. Kasi Viswanadham, Murali Krishna &
Prabhakara (2010) presented the numerical solution of fifth order BVPs by collocation method with sixth order
B-splines. Lamnii, Mraoui, Sbibih & Tijini (2008) derived sextic spline solution of fifth order BVPs. The
numerical solution of fifth order BVPs by the decomposition method was developed in (Wazwaz, 2001).
Recently, Erturk (2007) has investigated differential transformation method for solving nonlinear fifth order
BVPs. Siddiqi, Akram & Iftikhar (2012) presented the solutions of seventh order BVPs by the differential
transformation method and variational iteration technique respectively. Very recently the solution of seventh
order BVPs was developed in (Siddiqi & Iftikhar, 2013) using variation of parameters method. The modified
variational iteration method has been applied for solving tenth and ninth order BVPs in (Mohyud-Din &
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2
Yildirim, 2010). Nadjafi & Zahmatkesh (2010) also investigated the homotopy perturbation method for solving
higher order BVPs.
In the present paper, first we shall employ the Galerkin weighted residual method (Reddy, 1991) with Legendre
polynomials (Davis &Robinowitz, 2007) as basis functions for the numerical solution of fifth, seventh and ninth
order linear BVPs of the following form:
5 4 3 2
5 4 3 2 1 05 4 3 2
,
d u d u d u d u du
c c c c c c u r a x b
dxdx dx dx dx
(1a)
21100 )(,)(,)(,)(,)( AauBbuAauBbuAau (1b)
7 6 5 4 3 2
7 6 5 4 3 2 1 07 6 5 4 3 2
,
d u d u d u d u d u d u du
c c c c c c c c u r a x b
dxdx dx dx dx dx dx
(2a)
3221100 )(,)(,)(,)(,)(,)(,)( AauBbuAauBbuAauBbuAau (2b)
and
9 8 7 6 5 4 3 2
9 8 7 6 5 4 3 2 1 09 8 7 6 5 4 3 2
,
d u d u d u d u d u d u d u d u du
c c c c c c c c c c u r
dxdx dx dx dx dx dx dx dx
a x b
(3a)
0 0 1 1 2 2 3 3
( )
4
( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) ,
( )iv
u a A u b B u a A u b B u a A u b B u a A u b B
u a A
(3b)
where 4,3,2,1,0, iAi and 3,2,1,0, jBj are finite real constants and 9,1,0, ici and r are all continuous
functions defined on the interval [a, b].
However, in section 2 of this paper, we give a short description on Legendre polynomials. In section 3, the
formulations are presented for solving linear fifth, seventh and ninth order BVPs by the Galerkin weighted
residual method. Numerical examples and results for both linear and nonlinear BVPs are considered to verify the
proposed formulation and the solutions are compared with the existing methods in the literature in section 4.
2. Legendre Polynomials
The general form of the Legendre polynomials of degree n is defined by
1],)1[(
)!(2
)1(
)( 2
nx
dx
d
n
xP n
n
n
n
n
n
Now we modify the above Legendre polynomials as
1),1()1(
!
1
)( 2
nxxx
dx
d
n
xL nn
n
n
n
We write first few modified Legendre polynomials over the interval [0, 1]:
)1(2)(1 xxxL , 2
2 )1(6)( xxxL , )61510)(1(2)( 2
3 xxxxxL ,
5432
4 7021023011020)( xxxxxxL
65432
5 252882119077024030)( xxxxxxxL ,
765432
6 924369659224830210046242)( xxxxxxxxL
8765432
7 343215444286442818215750495681256)( xxxxxxxxxL
98765432
8 12870643501355641561561067224389010500133272)( xxxxxxxxxxL
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Vol.4, No.5, 2014
3
10
98765432
9
48620
26741063063083226067267234234210857020460207090)(
x
xxxxxxxxxxL
1110
98765432
10
1847561108536
2892890430287040154402438436966966244530372903080110)(
xx
xxxxxxxxxxL
Si
nce the modified Legendre polynomials have special properties at 0x and 1x : 0)0( nL and
1,0)1( nLn 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 in the Galerkin method
to solve a BVP over the interval [0, 1].
3. Matrix Formulation
In this section, we first derive the matrix formulation rigorously for fifth-order linear BVP and then we extend
our idea for solving seventh and ninth order linear BVPs. To solve the BVPs (1) by the Galerkin weighted
residual method we approximate )(xu as
1,)()()(~
1
,0
nxLaxxu
n
i
nii (4)
Here )(0 x is specified by the essential boundary conditions, )(, xL ni are the Legendre polynomials which must
satisfy the corresponding homogeneous boundary conditions such that 0)()( ,, bLaL nini for each
.,,3,2,1 ni
Using eqn. (4) into eqn. (1a), the Galerkin weighted residual equations are
0)(~
~~~~~
,012
2
23
3
34
4
45
5
5
dxxLruc
dx
ud
c
dx
ud
c
dx
ud
c
dx
ud
c
dx
ud
c nj
b
a
(5)
Integrating by parts the terms up to second derivative on the left hand side of (5), we get
b
a
b
a
nj
b
a
njnj dx
dx
ud
xLc
dx
d
dx
ud
xLcdxxL
dx
ud
c 4
4
,54
4
,5,5
5
5
~
)(
~
)()(
~
dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
b
a
nj
b
a
nj 3
3
,52
2
3
3
,5
~
)(
~
)(
[Since 0)()( ,, bLaL njnj ]
dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
b
a
nj
b
a
nj
b
a
nj 2
2
,53
3
2
2
,52
2
3
3
,5
~
)(
~
)(
~
)(
3 2 2 3 4
5 , 5 , 5 , 5 ,3 2 2 3 4
( ) ( ) ( ) ( )
b b b
j n j n j n j n
aa a
d d u d d u d du d du
c L x c L x c L x c L x dx
dx dx dxdx dx dx dx dx
(6)
In the same way of equation (6), we have
dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dxxL
dx
ud
c
b
a
nj
b
a
nj
b
a
njnj
b
a
~
)(
~
)(
~
)()(
~
,43
3
,42
2
2
2
,4,4
4
4
(7)
b
a
b
a
nj
b
a
njnj dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dxxL
dx
ud
c
~
)(
~
)()(
~
,32
2
,3,3
3
3 (8)
b
a
b
a
njnj dx
dx
ud
xLc
dx
d
dxxL
dx
ud
c
~
)()(
~
,2,2
2
2 (9)
Substituting eqns. (6) – (9) into eqn. (5) and using approximation for )(~ xu given in eqn. (4) and after imposing
the boundary conditions given in eqn. (1b) and rearranging the terms for the resulting equations we get a system
of equations in matrix form as
njFaD ji
n
i
ji ,,2,1,
1
,
(10a)
where
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4
4 3 2
, 5 , 4 , 3 , 2 , 1 , ,4 3 2
( ) ( ) ( ) ( ) ( ) [ ( )]
b
i j j n j n j n j n j n i n
a
d d d d d
D c L x c L x c L x c L x c L x L x
dx dxdx dx dx
3 3
0 , , 5 , , 5 , ,3 3
( ) ( ) ( ) [ ( )] ( ) [ ( )]i n j n j n i n j n i n
x b x a
d d d d
c L x L x dx c L x L x c L x L x
dx dxdx dx
2 2 2
5 , , 4 , ,2 2 2
( ) [ ( )] ( ) [ ( )]j n i n j n i n
x b x b
d d d d
c L x L x c L x L x
dxdx dx dx
(10b)
)()()()()( ,2,32
2
,43
3
,54
4
, xLc
dx
d
xLc
dx
d
xLc
dx
d
xLc
dx
d
xrLF njnjnj
b
a
njnjj
ax
nj
bx
njnjnj
dx
d
xLc
dx
d
dx
d
xLc
dx
d
dxxLc
dx
d
xLc
3
0
3
,53
0
3
,5,00
0
,1 )()()()(
2,52
2
2
0
2
,42
0
2
,52
2
)()()( AxLc
dx
d
dx
d
xLc
dx
d
dx
d
xLc
dx
d
ax
nj
bx
nj
bx
nj
2,41,53
3
1,53
3
)()()( AxLc
dx
d
AxLc
dx
d
BxLc
dx
d
ax
nj
ax
nj
bx
nj
1,31,42
2
1,42
2
)()()( BxLc
dx
d
AxLc
dx
d
BxLc
dx
d
bx
nj
ax
nj
bx
nj
1,3 )( AxLc
dx
d
ax
nj
(10c)
Solving the system (10a), we obtain the values of the parameters ia and then substituting these parameters into
eqn. (4), we get the approximate solution of the desired BVP (1).
Similarly, for seventh order linear BVP given in eqn. (2), and for ninth order linear BVP given in eqn. (3), we
can obtain equivalent system of equations in matrix form. For nonlinear BVP, we first compute the initial values
on neglecting the nonlinear terms and using the system (10). Then using the Newton’s iterative method we find
the numerical approximations for desired nonlinear BVP. This formulation is described through the numerical
examples in the next section.
4. Numerical Examples and Results
To test the applicability of the proposed method, we consider both linear and nonlinear problems which are
available in the literature. For all the examples, the solutions obtained by the proposed method are compared
with the exact solutions. All the calculations are performed by MATLAB 10. The convergence of linear BVP is
calculated by
)(~)(~
1 xuxuE nn
where )(~ xnu denotes the approximate solution using n-th polynomials and (depends on the problem) which
is less than 12
10 .
In addition, the convergence of nonlinear BVP is calculated by the absolute error of two
consecutive iterations such that
N
n
N
n uu ~~ 1
, where is less than 10
10
and N is the Newton’s iteration number.
Example 1: Consider the linear BVP of fifth order (Viswanadham, Krishna & Prabhakara, 2010):
11,sin2sin41cos2cos19 23
5
5
xxxxxxxxxu
dx
ud
(11a)
1sin81cos3)1(,1sin1cos4)1()1(,1cos)1()1( uuuuu (11b)
The analytic solution of the above system is .cos)12()( 2
xxxu
In Table 1, we list the maximum absolute errors for this problem to compare with the existing methods.
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Table 1: Maximum absolute errors for the example 1
x Exact Results 12 Legendre Polynomials
Approximate Abs. Error
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5403023059
0.1950778786
-0.2310939722
-0.6263214759
-0.9016612516
-1.0000000000
-0.9016612516
-0.6263214759
-0.2310939722
0.1950778786
0.5403023059
0.5403023059
0.1950778786
-0.2310939722
-0.6263214759
-0.9016612516
-1.0000000000
-0.9016612516
-0.6263214759
-0.2310939722
0.1950778786
0.5403023059
0.0000000E+000
4.7337134E-013
3.3925640E-013
6.9122486E-013
1.3414825E-012
1.5528689E-012
6.3637984E-013
6.4226402E-013
1.0645096E-012
6.7196249E-013
0.0000000E+000
On the contrary, the maximum absolute error has been obtained by Kasi Viswanadham, Murali Krishna &
Prabhakara ( 2010) is 4
105162.4
.
Now the exact and approximate solutions are depicted in Fig. 1 of example 1 for 12n .
Fig. 1: Graphical representation of exact and approximate solutions of example 1.
Example 2: Consider the linear BVP of seventh order [Siddiqi & Iftikhar, 2013]
10),35122( 2
7
7
xxxeu
dx
ud x
(12a)
3)0(,4)1(,0)0(,)1(,1)0(,0)1()0( ueuueuuuu . (12b)
The analytic solution of the above system is x
exxxu )1()( .
The maximum absolute errors for this problem are shown in Table 2 to compare with the existing methods.
On the other hand, the maximum absolute error has been found by Siddiqi & Iftikhar (2013) is .10482.7 10
In
Fig. 2, the exact and approximate solutions are given for example 2.
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Table 2: Maximum absolute errors for the example 2
x Exact Results 12 Legendre Polynomials
Approximate Abs. Error
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000000000
0.0994653826
0.1954244413
0.2834703496
0.3580379274
0.4121803177
0.4373085121
0.4228880686
0.3560865486
0.2213642800
0.0000000000
0.0000000000
0.0994653826
0.1954244413
0.2834703496
0.3580379274
0.4121803177
0.4373085121
0.4228880686
0.3560865486
0.2213642800
0.0000000000
6.2597190E-026
5.8911209E-014
1.0269563E-014
1.0608181E-013
6.8500761E-014
9.6977981E-014
7.1442852E-014
6.1228800E-014
5.0015547E-014
5.9674488E-015
0.0000000E+000
Fig. 2: Graphical representation of exact and approximate solutions of example 2.
Example 3: Consider the following ninth-order linear BVP (Nadjafi and Zahmatkesh (2010), Mohy-ud-Din &
Yildirim (2010), Wazwaz, 2000)
10,99
9
xeu
dx
ud x
(13a)
3)0(,3)1(,2)0(,2)1(,1)0(,)1(,0)0(,0)1(,1)0( )(
iv
ueuueuueuuuu (13b)
Table 3: Maximum absolute errors for the example 3
x Exact Results 12 Legendre Polunomials
Approximate Abs. Error
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0000000000
0.9946538263
0.9771222065
0.9449011653
0.8950948186
0.8243606354
0.7288475202
0.6041258122
0.4451081857
0.2459603111
0.0000000000
1.0000000000
0.9946538263
0.9771222065
0.9449011653
0.8950948186
0.8243606354
0.7288475202
0.6041258122
0.4451081857
0.2459603111
0.0000000000
0.0000000E+000
1.4432899E-015
3.2196468E-015
4.5519144E-015
4.8849813E-015
3.1086245E-015
5.6621374E-015
4.4408921E-016
3.1641356E-015
4.4408921E-016
0.0000000E+000
The analytic solution of the above system is x
exxu )1()( . The numerical results for this problem are
summarized in Table 3. On the other hand, the accuracy is found nearly the order 10
10
by Mohy-ud-Din and
Yildirim (2010) and by Wazwaz (2000) and nearly the order 9
10
by Nadjafi and Zahmatkesh. In Fig. 3, the
exact and approximate solutions are given of example 3 for 12n .
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7
Fig. 3: Graphical representation of exact and approximate solutions of example 3.
Example 4: Consider the nonlinear BVP of fifth order (Erturk, 2007 and Wazwaz, 2001)
10,2
5
5
xeu
dx
ud x
(14a)
1)0(,)1(,1)0(,)1(,1)0( ueuueuu (14b)
The exact solution of this BVP is .)( x
exu
Consider the approximate solution of )(xu as
1,)()()(~
1
,0
nxLaxxu
n
i
nii (15)
Here )1(1)(0 exx is specified by the essential boundary conditions in (14b). Also 0)1()0( ,, nini LL
for each ni ,,2,1 . Using eqn. (15) into eqn. (14a), the Galerkin weighted residual equations are
1
0
,
2
5
5
,,2,1,0)(~
~
nkdxxLeu
dx
ud
nk
x
(16)
Integrating first term of (16) by parts, we obtain
dx
dx
ud
dx
xLd
dx
ud
dx
xLd
dx
ud
dx
xLd
dx
ud
dx
xdL
dxxL
dx
ud nknknknk
nk
~)(~)(~)(~)(
)(
~ 1
0
4
,
41
0
1
0
3
,
3
1
0
2
2
2
,
21
0
3
3
,
,5
5
(17)
Putting eqn. (17) into eqn. (16) and using approximation for )(~ xu given in eqn. (15) and after applying the
boundary conditions given in eqn. (14b) and rearranging the terms for the resulting equations, we obtain
dxexLxLxLaxLxLe
dx
xdL
dx
xLdn
i
n
j
x
nknjnijnkni
xnink
1
1
0 1
,,,,,0
,
4
,
4
))()()(()()(2
)()(
i
x
nink
x
nink
x
nink
a
dx
xLd
dx
xLd
dx
xLd
dx
xdL
dx
xLd
dx
xdL
1
2
,
2
2
,
2
0
3
,
3
,
1
3
,
3
, )()()()()()(
0
3
0
3
,
1
3
0
3
,
1
0
,
2
0
0
4
,
4
)()(
)(
)(
x
nk
x
nk
nk
xnk
dx
d
dx
xdL
dx
d
dx
xdL
dxxLe
dx
d
dx
xLd
0
3
,
3
1
3
,
3
0
2
,
2
1
2
0
2
2
,
2
)()()()(
x
nk
x
nk
x
nk
x
nk
dx
xLd
e
dx
xLd
dx
xLd
dx
d
dx
xLd
(18)
The above equation (18) is equivalent to matrix form as
GABD )( (19a)
where the elements of A, B, D, G are kikii dba ,, ,, and kg respectively, given by
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8
1
3
,
3
,
1
0
,,0
,
4
,
4
,
)()(
)()(2
)()(
x
nink
nkni
xnink
ki
dx
xLd
dx
xdL
dxxLxLe
dx
xdL
dx
xLd
d
1
2
,
2
2
,
2
0
3
,
3
, )()()()(
x
nink
x
nink
dx
xLd
dx
xLd
dx
xLd
dx
xdL
(19b)
n
j
x
nknjnijki dxexLxLxLab
1
1
0
,,,, ))()()(( (19c)
0
3
0
3
,
1
3
0
3
,
1
0
,
2
0
0
4
,
4
)()(
)(
)(
x
nk
x
nk
nk
xnk
k
dx
d
dx
xdL
dx
d
dx
xdL
dxxLe
dx
d
dx
xLd
g
0
3
,
3
1
3
,
3
0
2
,
2
1
2
0
2
2
,
2
)()()()(
x
nk
x
nk
x
nk
x
nk
dx
xLd
e
dx
xLd
dx
xLd
dx
d
dx
xLd
(19d)
The initial values of these coefficients ia are obtained by applying Galerkin method to the BVP neglecting the
nonlinear term in (14a). That is, to find initial coefficients we solve the system
GDA (20a)
whose matrices are constructed from
0
3
,
3
,
1
3
,
3
,
1
0
,
4
,
4
,
)()()()()()(
x
nink
x
ninknink
ki
dx
xLd
dx
xdL
dx
xLd
dx
xdL
dx
dx
xdL
dx
xLd
d
1
2
,
2
2
,
2
)()(
x
nink
dx
xLd
dx
xLd
(20b)
1
2
0
2
2
,
2
0
3
0
3
,
1
3
0
3
,
1
0
0
4
,
4
)()()()(
x
nk
x
nk
x
nknk
k
dx
d
dx
xLd
dx
d
dx
xdL
dx
d
dx
xdL
dx
dx
d
dx
xLd
g
0
3
,
3
1
3
,
3
0
2
,
2
)()()(
x
nk
x
nk
x
nk
dx
xLd
e
dx
xLd
dx
xLd
(20c)
Once the initial values of ia are obtained from eqn. (20a), they are substituted into eqn. (19a) to obtain new
estimates for the values of ia . This iteration process continues until the converged values of the unknown
parameters are obtained. Substituting the final values of the parameters into eqn. (15), we obtain an approximate
solution of the BVP (14). The maximum absolute errors for this problem are shown in Table 4 with 6 iterations.
On the other hand, the maximum absolute errors have been obtained by Wazwaz (2001) and Erturk (2007) are
8
101.4
and ,1052.1 10
respectively. We depict the exact and approximate solutions in Fig. 4 of example 4
for 12n .
Example 5: Consider the nonlinear BVP of seventh order (Siddiqi, Akram & Iftikhar, 2012)
10,2
7
7
xeu
dx
ud x
(21a)
1)0(,)1(,1)0(,)1(,1)0(,)1(,1)0( ueuueuueuu . (21b)
The exact solution of this BVP is x
exu )( . Following the proposed method illustrated in section 3 as well as in
example 4; the maximum absolute errors for this problem are summarized in Table 5.
9. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.5, 2014
9
Table 4: Maximum absolute errors of example 4 using 6 iterations
x Exact Results 12 Legendre Polynomials
Approximate Abs. Error
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0000000000
1.1051709181
1.2214027582
1.3498588076
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409285
2.4596031112
2.7182818285
1.0000000000
1.1051709181
1.2214027582
1.3498588076
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409285
2.4596031112
2.7182818285
0.0000000E+000
2.9918024E-012
3.4379395E-011
1.1878055E-012
2.3980112E-011
3.4213951E-012
3.6519182E-012
2.8570975E-012
1.4395748E-011
2.8618543E-011
1.2759373E-000
Fig. 4: Graphical representation of exact and approximate solutions of example 4.
Table 5: Maximum absolute errors of example 5 using 6 iterations
x Exact Results 12 Legendre Polynomials
Approximate Abs. Error
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0000000000
1.1051709181
1.2214027582
1.3498588076
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409285
2.4596031112
2.7182818285
1.0000000000
1.1051709181
1.2214027580
1.3498588075
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409282
2.4596031112
2.7182818285
0.0000000E+000
1.6084911E-011
1.3298251E-011
5.0746074E-011
7.1547213E-012
6.2061467E-011
2.9898306E-012
1.8429702E-013
6.5281114E-011
1.6604496E-011
0.0000000E+000
On the contrary, the maximum absolute error has been obtained by Siddiqi , Akram & Iftikhar (2012) is
10
10586.7
. We depict the exact and approximate solutions in Fig. 5 of example 5 for 12n .
10. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.5, 2014
10
Fig. 5: Graphical representation of exact and approximate solutions of example 5 for.
5. Conclusions
In this paper, Galerkin method has been used for finding the numerical solutions of fifth, seventh and ninth order
linear and nonlinear BVPs with Legendre polynomials as basis functions. The numerical examples available in
the literature have been considered to verify the proposed method. We see from the tables that the numerical
results obtained by our method are better than other existing methods. It may also notice that the numerical
solutions coincide with the exact solution even Legendre polynomials are used in the approximation which are
shown in Figs. [1-5]. The algorithm can be coded easily and may be used for solving any higher order BVP.
6. References
[1] Agarwal, R.P. (1986). Boundary Value Problems for Higher Order Differential Equations, World
Scientific, Singapore.
[2] Caglar H.N., Caglar S.H., & Twizell E.H. (1999). The numerical solution of fifth order boundary value
problems with sixth degree B-spline functions, Appl. Math. Lett., 12, 25-30.
[3] Davis P.J., & Robinowitz P. (2007). Methods of Numerical Integration, Dover Publications, Inc, 2nd
Edition.
[4] Davis A.R., Karageorghis A., & Phillips T.N. (1988). Spectral Galerkin methods for the primary two point
boundary value problem in modeling viscoelastic flows, International J. Numer. Methods Eng. 26, 647-662.
[5] Karageorghis A., Phillips T.N. & Davis A.R., (1988). Spectral Collocation methods for the primary two
point boundary value problem in modeling viscoelastic flows, International J. Numer. Methods Eng. 26,
805-813.
[6] Erturk V.S. (2007). Solving nonlinear fifth order boundary value problems by differential transformation
method, Selcuk J. Appl. Math., 8 (1), 45-49.
[7] Kasi Viswanadham K.N.S., Murali Krishna P., & Prabhakara Rao, C. (2010). Numerical solution of fifth
order boundary value problems by collocation method with sixth order B-splines, International Journal of
Applied Science and Engneering, 8 (2), 119-125.
[8] Lamnii A., Mraoui H., Sbibih D., & Tijini A. (2008). Sextic spline solution of fifth order boundary value
problems, Math. Computer Simul., 77, 237-246.
[9] Mohyud-Din S.T., & Yildirim A. (2010). Solutions of tenth and ninth order boundary value problems by
modified variational iteration method, Application Applied Math., 5 (1), 11-25.
[10] Nadjafi, J.S., & Zahmatkesh, S. (2010). Homotopy perturbation method (HPM) for solving higher order
boundary value problems, Applied Math. Comput. Sci., 1 (2), 199-224.
11. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.5, 2014
11
[11] Reddy, J.N. (1991). Applied Functional Analysis and Variational Methods in Engineering, Krieger
Publishing Co. Malabar, Florida.
[12] Richards, G. & Sarma, P.R.R. (1994). Reduced order models for induction motors with two rotor circuits,
IEEE Transactions Energy Conversion, 9 (4), 673-678.
[13] Siddiqi, S. S., & Twizell E.H. (1997). Spline solutions of linear twelfth order boundary value problems, J.
Comp. Appl. Maths. 78, 371-390.
[14] Siddiqi, S. S., Twizell, E.H. (1998). Spline solutions of linear tenth order boundary value problems, Intern.
J. Computer Math. 68 (3), 345-362.
[15] Siddiqi, S. S., & Twizell, E.H. (1996). Spline solutions of linear eighth order boundary value problems,
Comp. Meth. Appl. Mech. Eng. 131, 309-325.
[16] Siddiqi, S. S., & Twizell, E.H. (1996). Spline solutions of linear sixth order boundary value problems,
Intern. J. Computer Math. 60 (3), 295-304.
[17] Siddiqi, S. S., Ghazala Akram & Muzammal Iftikhar (2012). Solution of seventh order boundary value
problem by differential transformation method, World Applied Sciences Journal, 16 (11), 1521-1526.
[18] Siddiqi S. S., Ghazala Akram & Muzammal Iftikhar (2012). Solution of seventh order boundary value
problems by variational iteration technique, Applied Mathematical Sciences, 6, (94), 4663-4672.
[19] Siddiqi, S. S. & Muzammal Iftikhar (2013). Solution of seventh order boundary value problem by variation
of parameters method, Research Journal of Applied Sciences, Engineering and Technology, 5 (1), 176-179.
[20] Wazwaz, A.M. (2001). The numerical solution of fifth order boundary value problems by the decomposition
method, J. Comput. Appl. Math., 136, 259-270.
[21] Wazwaz A.M. (2000). Approximate solutions to boundary value problems of higher order by the modified
decomposition method, Comput. Math. Appl., 40, 679-691.
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