This document presents a new method for solving integral equations of Chandrasekhar type that arise in radiative transfer problems. The method approximates the Jacobian inverse as a diagonal matrix using variational techniques. This avoids explicitly computing or storing the Jacobian. The method formulates an optimization problem to minimize corrections to the diagonal matrix while satisfying a weak secant condition. Solving this yields a unique solution for updating the diagonal matrix approximation at each iteration. Numerical results on benchmark problems demonstrate the reliability and efficiency of the approach compared to other Newton-like methods. The fact that it solves the integral equations without derivative computation or matrix storage is a clear advantage.
Numerical Solution of Diffusion Equation by Finite Difference Methodiosrjce
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.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
This document discusses numerical methods for solving differential and partial differential equations. It begins by providing some historical context on the development of numerical analysis. It then discusses several common numerical methods including Lagrangian interpolation, finite difference methods, finite element methods, spectral methods, and finite volume methods. For each method, it provides a brief overview of the approach and discusses aspects like discretization, accuracy, computational cost, and common applications. Overall, the document serves as an introduction to various numerical techniques for approximating solutions to differential equations.
A BRIEF SURVEY OF METHODS FOR SOLVING NONLINEAR LEAST-SQUARES PROBLEMSShannon Green
This document provides a summary of methods for solving nonlinear least squares problems. It discusses structured quasi-Newton methods that take into account the special structure of the Hessian matrix for nonlinear least squares problems. It also discusses derivative-free methods based on the Gauss-Newton method and hybrid approaches. The document focuses on structured quasi-Newton methods, which combine Gauss-Newton and quasi-Newton ideas to make good use of the Hessian structure. It also discusses sizing techniques to help structured quasi-Newton methods handle small residual problems better.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
Numerical Solutions of Stiff Initial Value Problems Using Modified Extended B...IOSR Journals
This document presents numerical solutions for stiff initial value problems using a modified extended backward differentiation formula (MEBDF). The MEBDF method is developed based on linear multi-step methods. Three stages of the two-step MEBDF are constructed and used to solve a sample stiff initial value problem. The numerical solutions from each stage are compared to the exact solution and each other to determine which stage provides the most accurate solutions.
11.[36 49]solution of a subclass of lane emden differential equation by varia...Alexander Decker
This document describes applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate successive approximations to the solution. It introduces a polynomial initial approximation satisfying the boundary conditions. The Lagrange multiplier is determined using variational theory. Estimates are established to prove the iterative sequence converges uniformly to the exact solution. The method is illustrated on some example problems and shown to produce exact polynomial solutions, demonstrating the effectiveness of the variational iteration method.
This document discusses various numerical analysis methods for solving differential and partial differential equations. It begins with a brief history of numerical analysis, then discusses different interpolation methods like Lagrangian interpolation. It also covers finite difference methods, finite element methods, spectral methods, and the method of lines - explaining how each method discretizes equations. The document concludes by discussing multigrid methods, which use a hierarchy of grids to accelerate convergence in solving equations.
Numerical Solution of Diffusion Equation by Finite Difference Methodiosrjce
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.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
This document discusses numerical methods for solving differential and partial differential equations. It begins by providing some historical context on the development of numerical analysis. It then discusses several common numerical methods including Lagrangian interpolation, finite difference methods, finite element methods, spectral methods, and finite volume methods. For each method, it provides a brief overview of the approach and discusses aspects like discretization, accuracy, computational cost, and common applications. Overall, the document serves as an introduction to various numerical techniques for approximating solutions to differential equations.
A BRIEF SURVEY OF METHODS FOR SOLVING NONLINEAR LEAST-SQUARES PROBLEMSShannon Green
This document provides a summary of methods for solving nonlinear least squares problems. It discusses structured quasi-Newton methods that take into account the special structure of the Hessian matrix for nonlinear least squares problems. It also discusses derivative-free methods based on the Gauss-Newton method and hybrid approaches. The document focuses on structured quasi-Newton methods, which combine Gauss-Newton and quasi-Newton ideas to make good use of the Hessian structure. It also discusses sizing techniques to help structured quasi-Newton methods handle small residual problems better.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
Numerical Solutions of Stiff Initial Value Problems Using Modified Extended B...IOSR Journals
This document presents numerical solutions for stiff initial value problems using a modified extended backward differentiation formula (MEBDF). The MEBDF method is developed based on linear multi-step methods. Three stages of the two-step MEBDF are constructed and used to solve a sample stiff initial value problem. The numerical solutions from each stage are compared to the exact solution and each other to determine which stage provides the most accurate solutions.
11.[36 49]solution of a subclass of lane emden differential equation by varia...Alexander Decker
This document describes applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate successive approximations to the solution. It introduces a polynomial initial approximation satisfying the boundary conditions. The Lagrange multiplier is determined using variational theory. Estimates are established to prove the iterative sequence converges uniformly to the exact solution. The method is illustrated on some example problems and shown to produce exact polynomial solutions, demonstrating the effectiveness of the variational iteration method.
This document discusses various numerical analysis methods for solving differential and partial differential equations. It begins with a brief history of numerical analysis, then discusses different interpolation methods like Lagrangian interpolation. It also covers finite difference methods, finite element methods, spectral methods, and the method of lines - explaining how each method discretizes equations. The document concludes by discussing multigrid methods, which use a hierarchy of grids to accelerate convergence in solving equations.
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The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
This document is a literature review for a project on modeling fluid dynamics using spectral methods in MATLAB. It summarizes two key papers: (1) Balmforth et al.'s paper on modeling the dynamics of interfaces and layers in a stratified turbulent fluid, which derived coupled differential equations; and (2) Trefethen's book on spectral methods in MATLAB, which provided guidance on using Chebyshev polynomials and differentiation matrices. It also outlines the methodology used in Balmforth et al.'s paper and chapters from Trefethen's book on finite differences, Chebyshev points, and constructing Chebyshev differentiation matrices.
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
Numerical Solutions of Burgers' Equation Project ReportShikhar Agarwal
This report summarizes the use of finite element methods to numerically solve Burgers' equation. It introduces finite element methods and the Galerkin method for approximating solutions. MATLAB codes are presented to solve example boundary value problems and differential equations. The method of quasi-linearization is also described for solving Burgers' equation numerically. The report concludes that finite element methods can accurately predict numerical solutions that are close to exact solutions for problems where no closed-form solution exists.
This document provides an introduction and overview to Unit 13 of the course MST209 Mathematical methods and models. Unit 13 focuses on modelling systems using non-linear differential equations. It discusses two main examples - modeling the interaction between predator and prey populations using the Lotka-Volterra equations, and modeling the motion of a pendulum using differential equations. The unit emphasizes qualitative analysis and interpretation of solutions rather than explicit solutions. It introduces concepts like equilibrium solutions and linearizing near equilibria to understand behavior. Sections 1 and 2 develop these ideas for the Lotka-Volterra population model, while Section 3 applies similar techniques to the pendulum motion model.
An efficient improvement of the Newton method for solving nonconvex optimizat...Christine Maffla
This document summarizes a research article that proposes a new modification of the Newton method for solving non-convex optimization problems. The proposed method sets the step length αk at each iteration equal to 1. The convergence of the iterative algorithm is analyzed under suitable conditions. Some examples are provided to demonstrate the validity and applicability of the presented method, and it is compared to several other existing methods. The goal of the research is to develop an efficient improvement of the Newton method for solving non-convex optimization problems.
This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.
11.solution of a subclass of lane emden differential equation by variational ...Alexander Decker
This document discusses applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate iterative approximations to the solution. It is shown that under certain conditions, the iterative sequence converges to the exact solution of the Lane-Emden equation. The variational iteration method provides an efficient means of obtaining analytical solutions and has been successfully used to solve many types of nonlinear problems. The method is illustrated through examples and shown to produce polynomial solutions.
Solution of a subclass of lane emden differential equation by variational ite...Alexander Decker
This document discusses applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate iterative approximations to the solution. It is shown that under certain conditions, the iterative sequence converges to the exact solution of the Lane-Emden equation. The variational iteration method provides an efficient means of obtaining polynomial solutions without linearization, perturbation or discretization. Illustrative examples from literature are shown to produce exact polynomial solutions when treated with this method.
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.
This document provides an introduction to the basic concepts of computational fluid dynamics (CFD). It discusses the need for CFD due to the inability to analytically solve the governing equations for most engineering problems. The document then summarizes some common applications of CFD in industry, including simulating vehicle aerodynamics, mixing manifolds, and bio-medical flows. It also outlines the overall strategy of CFD in discretizing the continuous problem domain into a discrete grid before discussing specific discretization methods like the finite difference and finite volume methods.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
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.
IRJET- Wavelet based Galerkin Method for the Numerical Solution of One Dimens...IRJET Journal
This document presents a wavelet-based Galerkin method for numerically solving one-dimensional partial differential equations using Hermite wavelets. Hermite wavelets are used as the basis functions in the Galerkin method. The method is demonstrated on some test problems, and the numerical results obtained from the proposed method are compared to exact solutions and a finite difference method to evaluate the accuracy and efficiency of the proposed wavelet Galerkin approach.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
This document discusses solving quadratic equations by completing the square. It begins by stating the objectives and key understandings of quadratic equations and the completing the square method. It then outlines the steps to complete the square, which involves rewriting the equation as a perfect square trinomial. Several examples are shown using algebra tiles and calculations. Finally, applications of completing the square to real-life problems in physics, engineering, architecture and economics are described.
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
The document provides a 5-step process for requesting writing assistance from HelpWriting.net:
1. Create an account with a password and valid email.
2. Complete a 10-minute order form providing instructions, sources, deadline, and attaching a writing sample.
3. Review bids from writers and choose one based on qualifications, history, and feedback and place a deposit.
4. Review the completed paper and authorize final payment if pleased, with free revisions available.
5. Request multiple revisions to ensure satisfaction, with plagiarized work resulting in a full refund.
IELTS Academic Essay Writing Tips For A Better ScoreJeff Brooks
The document provides tips for getting a better score on the IELTS Academic Essay Writing exam. It outlines 5 steps: 1) Create an account; 2) Complete an order form providing instructions, sources, and deadline; 3) Review bids from writers and choose one; 4) Ensure the paper meets expectations and authorize payment; 5) Request revisions to ensure satisfaction and get a refund for plagiarized work. The document stresses choosing HelpWriting.net for high-quality, original content and customer satisfaction.
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This report summarizes the use of finite element methods to numerically solve Burgers' equation. It introduces finite element methods and the Galerkin method for approximating solutions. MATLAB codes are presented to solve example boundary value problems and differential equations. The method of quasi-linearization is also described for solving Burgers' equation numerically. The report concludes that finite element methods can accurately predict numerical solutions that are close to exact solutions for problems where no closed-form solution exists.
This document provides an introduction and overview to Unit 13 of the course MST209 Mathematical methods and models. Unit 13 focuses on modelling systems using non-linear differential equations. It discusses two main examples - modeling the interaction between predator and prey populations using the Lotka-Volterra equations, and modeling the motion of a pendulum using differential equations. The unit emphasizes qualitative analysis and interpretation of solutions rather than explicit solutions. It introduces concepts like equilibrium solutions and linearizing near equilibria to understand behavior. Sections 1 and 2 develop these ideas for the Lotka-Volterra population model, while Section 3 applies similar techniques to the pendulum motion model.
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This document summarizes a research article that proposes a new modification of the Newton method for solving non-convex optimization problems. The proposed method sets the step length αk at each iteration equal to 1. The convergence of the iterative algorithm is analyzed under suitable conditions. Some examples are provided to demonstrate the validity and applicability of the presented method, and it is compared to several other existing methods. The goal of the research is to develop an efficient improvement of the Newton method for solving non-convex optimization problems.
This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.
11.solution of a subclass of lane emden differential equation by variational ...Alexander Decker
This document discusses applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate iterative approximations to the solution. It is shown that under certain conditions, the iterative sequence converges to the exact solution of the Lane-Emden equation. The variational iteration method provides an efficient means of obtaining analytical solutions and has been successfully used to solve many types of nonlinear problems. The method is illustrated through examples and shown to produce polynomial solutions.
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This document discusses applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate iterative approximations to the solution. It is shown that under certain conditions, the iterative sequence converges to the exact solution of the Lane-Emden equation. The variational iteration method provides an efficient means of obtaining polynomial solutions without linearization, perturbation or discretization. Illustrative examples from literature are shown to produce exact polynomial solutions when treated with this method.
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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.
This document provides an introduction to the basic concepts of computational fluid dynamics (CFD). It discusses the need for CFD due to the inability to analytically solve the governing equations for most engineering problems. The document then summarizes some common applications of CFD in industry, including simulating vehicle aerodynamics, mixing manifolds, and bio-medical flows. It also outlines the overall strategy of CFD in discretizing the continuous problem domain into a discrete grid before discussing specific discretization methods like the finite difference and finite volume methods.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
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
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A Low Memory Solver For Integral Equations Of Chandrasekhar Type In The Radiative Transfer Problems
1. Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 467017, 12 pages
doi:10.1155/2011/467017
Research Article
A Low Memory Solver for Integral
Equations of Chandrasekhar Type in
the Radiative Transfer Problems
Mohammed Yusuf Waziri, Wah June Leong,
Malik Abu Hassan, and Mansor Monsi
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Malaysia
Correspondence should be addressed to Mohammed Yusuf Waziri, mywaziri@gmail.com
Received 4 February 2011; Revised 14 July 2011; Accepted 8 August 2011
Academic Editor: Alexei Mailybaev
Copyright q 2011 Mohammed Yusuf Waziri et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The problems of radiative transfer give rise to interesting integral equations that must be faced
with efficient numerical solver. Very often the integral equations are discretized to large-scale
nonlinear equations and solved by Newton’s-like methods. Generally, these kind of methods
require the computation and storage of the Jacobian matrix or its approximation. In this paper,
we present a new approach that was based on approximating the Jacobian inverse into a diagonal
matrix by means of variational technique. Numerical results on well-known benchmarks integral
equations involved in the radiative transfer authenticate the reliability and efficiency of the
approach. The fact that the proposed method can solve the integral equations without function
derivative and matrix storage can be considered as a clear advantage over some other variants of
Newton’s method.
1. Introduction
The study of Chandrasekhar’s integral equation involved in the radiative transfer prob-
lem has been a foremost subject of much investigations and was first formulated by
Chandrasekhar 1 in 1960. It arose originally in connection with scattering through a
homogeneous semi-infinite plane atmosphere and since it has been used to model diverse
forms of scattering via the H-function of Chandrasekhar 2, defined by
Hx 1 Hx
1
0
x
x t
ψtHtdt. 1.1
2. 2 Mathematical Problems in Engineering
Chandrasekhar H-function plays a crucial role in radiative transfer and transport theory 3,
4. Since then, there have been diverse solvers of 1.1. It is well known that the numerical
solution of Chandrasekhar integral equation is difficult to be obtained 5, and thus it is con-
venient to have a reliable and efficient solver. The problem of finding approximate solution of
such integral equations is still popular today, and various methods of solving these integral
equations have been established 5–7. The common approach for the approximate solution of
1.1 is at first discretizing 1.1 by a vector x ∈ Rn
, then replacing the integrals by quadrature
sums and the derivatives by difference quotients involving only the components of x ∈ Rn
see 8, e.g.. By doing so, 1.1 becomes a problem of finding the solution of system of n
nonlinear equations with n unknowns
Fx 0, 1.2
where F : Rn
→ Rn
is a nonlinear mapping. Often, the mapping F is assumed to satisfy the
following assumptions:
A1 there exists x∗
∈ Rn
s.t. Fx∗
0,
A2 F is a continuously differentiable mapping in an open neighborhood of x∗
,
A3 F′
x∗
is invertible.
The famous iterative method for solving 1.2 is the classical Newton’s method, where the
Newtonian iteration is given by
xk1 xk −
F′
xk
−1
Fxk, k 0, 1, 2, . . . . 1.3
The convergence rate for the Newton’s method is quadratic from any initial point x0 in the
neighborhood of x∗
9. However, an iteration of 1.3 turns to be expensive, because it re-
quires to compute and store the Jacobian matrix, as well as solving Newton’s system which
is a linear system in each iteration. The major difficulty of Newton’s type method is the ma-
trix storage requirements especially when handling large systems of nonlinear equations
5, 6, 9. There are quite a number of revised Newton’s type methods being introduced,
which include fixed Newton’s and quasi-Newton’s, to diminish the weakness of 1.3. Fixed
Newton method 10 for the determination of solution x∗
is given by
xk1 xk −
F′
x0
−1
Fxk, k 0, 1, 2, . . . . 1.4
The method avoids computation and storing the Jacobian in each iteration except at k 0.
However, it still requires solving the systems of n linear equations and may consume more
CPU time as the system’s dimension increases 10.
A quasi-Newton’s method is another variant of Newton’s type methods, and it
replaces the Jacobian or its inverse with an approximation which can be updated at each
iteration 11, and its updating scheme is given by
xk1 xk − B−1
k Fxk, 1.5
where the matrix Bk is the approximation of the Jacobian at xk. The main idea behind quasi-
Newton’s method is to eliminate the evaluation cost of the Jacobian matrix, in which if
3. Mathematical Problems in Engineering 3
function evaluations are very expensive, the cost of finding a solution by quasi-Newton’s
methods could be much smaller than some other Newton’s-like methods 7, 12, 13. Various
Jacobian approximations matrices such as the Broyden’s method 11, 14 are proposed.
However, the most critical part of such solvers is that they need the storage of full matrix
of the approximate Jacobian, which can be a very expensive task as the dimension of systems
increases 15. In this paper, we propose an alternative approximation to the Jacobian inverse
into a diagonal matrix by means of variational techniques. It is worth mentioning that the sug-
gested method can be applied to solve 1.2 without the cost of computing or storing the
true Jacobian. Hence, it can reduce computational cost, storage requirement, processing time
CPU time and also eliminates the need for solving n linear equations in each iteration. The
proposed method works efficiently, and the results so far are very encouraging. This paper
is arranged as follows; we present our proposed method in Section 2; numerical results are
reported in Section 3; finally conclusion is given in Section 4.
2. Chandrasekhar H-Equation
In this section, we present the detailed process of discretizing the Chandrasekhar-type inte-
gral equations in the radiative transfer problem. Chandrasekhar and Breen 16 compute H-
equation as the solution of the nonlinear integral equation
Hx − c
x
2
Hx
1
0
H
y
x y
dy 1, 2.1
where c ∈ 0, 1 and H : 0, 1 → R is an unknown continuous function. From 2.1, we
obtain
Hx
1 −
c
2
1
0
xH
y
x y
dy
1. 2.2
Let us partition 0, 1 into n subinterval, 0 x1 · · · xj j/n · · · 1. Denote Hk as
Hxk, then the evaluation of 2.1 at every xi yields the equation
Hi
1 −
c
2
1
0
xiH
y
xi y
dy
1, i 1, 2, . . . , n. 2.3
After multiplying both sides of 2.2 by 1 − c/2
1
0
xHy/x ydy
−1
and performing
some algebra, we arrive at 2.4, which is known as the Chandrasekhar H-equation 15
FHx Hx −
1 −
c
2
1
0
xH
y
dy
x y
−1
0. 2.4
4. 4 Mathematical Problems in Engineering
If 2.4 is discretized by using the midpoint quadrature formula
1
0
ftdt
1
n
n
j0
f
tj
, 2.5
for tj j − 0.5h, 0 ≤ j ≤ 1, i 2, . . . , n, h 1/n, c ∈ 0, 1, then we have the following:
Fi xi −
⎛
⎝1 −
c
2n
n
j1
tixj
ti tj
⎞
⎠
−1
. 2.6
Function 2.6 is called the discretized Chandrasekhar H-equation which can be solved by
some iterative methods.
Nevertheless, the most difficult part in solving 2.6 arises dramatically as c approach-
es 1, since its Jacobian is singular at c 1. Due to this disadvantage, we aim to derive a
method that hopefully will not be affected by this difficulty.
3. Derivation of the Method (LMSI)
Firstly, note that by the mean value theorem, we have
F′xkxk1 − xk Fxk1 − Fxk, 3.1
where F′xk
1
0
F′
xk θxk1 − xkdθ.
Let us denote Δxk xk1 − xk and ΔFk Fxk1 − Fxk, then 3.1 becomes
F′xkΔxk ΔFk. 3.2
Equation 3.2 is always regarded as the secant equation. Alternatively, we can rearrange 3.2
to obtain
Δxk
F′xk
−1
ΔFk. 3.3
Here, we propose to use a diagonal matrix, say D, to approximate F′xk
−1
, that is,
F′xk
−1
≈ Dk. 3.4
Let us consider an updating scheme for D, in which we should update D by adding a cor-
rection M which is also a diagonal matrix at every iteration
Dk1 Dk Mk. 3.5
5. Mathematical Problems in Engineering 5
In order to incorporate correct information of the Jacobian inverse into the updating matrix,
Dk1, we require that Dk1 satisfies the secant equation 3.2, that is,
Δxk Dk MkΔFk. 3.6
However, since it is difficult to have a diagonal matrix that satisfies the secant equation, in
particular, because Jacobian approximations are not usually done in element wise, we con-
sider the use of the weak secant condition 17 instead,
ΔFT
k Δxk ΔFT
k Dk MkΔFk. 3.7
To encourage good condition number as well as numerical stability in the approximation, we
attempt to control the growth error of the correction by minimizing its magnitude under some
norms here, we consider the Frobenuis norm, such that 3.7 holds. To this end, we con-
sider the following problem:
min
1
2
Dk1 − Dk2
F
s.t. ΔFT
k Dk1ΔFk ΔFT
k Δxk,
3.8
where · F is the Frobenuis norm. If we let Dk1 − Dk Mk diagβ1, β2, . . . , βn and ΔFk
ΔF
1
k
, ΔF
2
k
, . . . , ΔF
n
k
, the above problem can be expressed as follows:
min
1
2
β2
1 β2
2 · · · β2
n
s.t.
n
i1
ΔF
i2
k
βi − ΔFT
k Δxk ΔFT
k DkΔFk 0.
3.9
Since the objective function and the constraint are convex, we will have unique solution for
3.9. The solution can be obtained by considering the Lagrangian function of problem 3.9
L
βi, λ
1
2
β2
1 β2
2 · · · β2
n
λ
n
i1
ΔF
i2
k
βi − ΔFT
k Δxk ΔFT
k DkΔFk , 3.10
where λ is the corresponding Lagrangian multiplier.
Taking the partial derivatives of 3.10 with respect to each βi and λ, respectively, and
setting them equal to zero, we have
∂L
∂βi
βi λ
ΔF
i
k
2
0, i 0, 1, 2, . . . , n, 3.11
∂L
∂λ
n
i1
ΔF
i
k
2
βi − ΔFT
k Δxk ΔFT
k DkΔFk 0. 3.12
6. 6 Mathematical Problems in Engineering
Premultiplying both sides of 3.11 by ΔF
i2
k
and summing them all yield
n
i1
ΔF
i
k
2
βi λ
n
i1
ΔF
i
k
4
0. 3.13
It follows from 3.13 that
n
i1
ΔF
i
k
2
βi −λ
n
i1
ΔF
i
k
4
. 3.14
Invoking the constraint 3.12, we have
n
i1
ΔF
i
k
2
βi ΔFT
k Δxk − ΔFT
k DkΔFk. 3.15
Equating 3.14 with 3.15 gives
λ −
ΔFT
k
Δxk − ΔFT
k
DkΔFk
n
i1
ΔF
i
k
4
. 3.16
Substituting 3.16 into 3.14 and after some simplifications, we obtain
βi
ΔFT
k
Δxk − ΔFT
k
DkΔFk
n
i1
ΔF
i
k
4
ΔF
i
k
2
, i 1, 2, . . . , n. 3.17
Denoting Gk diagΔF
1
k
2
, ΔF
2
k
2
, . . . , ΔF
n
k
2
and
n
i1ΔF
i
k
4
TrG2
k
where Tr is
the trace operation, we obtain, therefore,
Mk
ΔFT
k
Δxk − ΔFT
k
DkΔFk
Tr
G2
k
Gk. 3.18
Finally, the proposed updating formula for the approximation of the Jacobian inverse is given
as follows:
Dk1 Dk
ΔFT
k
Δxk − ΔFT
k
DkΔFk
Tr
G2
k
Gk. 3.19
To safeguard possibly very small ΔFk and TrG2
k
, we require that ΔFk ≥ ǫ1 for some chosen
small ǫ1 0. Else, we will skip the update by setting Dk1 Dk.
Now, we can describe the algorithm for our proposed method LMSI as follows.
7. Mathematical Problems in Engineering 7
Algorithm LMSI
Steps are the following.
Step 1. Given x0 and D0, set k 0.
Step 2. Compute Fxk and xk1 xk − DkFxk.
Step 3. If Δxk2 Fxk2 ≤ 10−4
, stop. Else, go to Step 4.
Step 4. If ΔFk2 ≥ ǫ1 where ǫ1 10−4
, compute Dk1, if not, Dk1 Dk. Set k : k 1 and go
to Step 2.
4. Local Convergence Results
In this section, we will give some convergence properties of LMSI method. Before we proceed
further, we will make the following standard assumptions on nonlinear systems F.
Assumption 4.1. We have the following.
i F is differentiable in an open-convex set E in Rn
.
ii There exists x∗
∈ E such that Fx∗
0, and F′
x is continuous for all x.
iii F′
x satisfies Lipschitz condition of order one, that is, there exists a positive
constant µ such that
F′
x − F′
y
≤ µ
x − y
, 4.1
for all x, y ∈ Rn
.
iv There exists constants c1 ≤ c2 such that c1ω2
≤ ωT
F′
xω ≤ c2ω2
for all x ∈ E
and ω ∈ Rn
.
We will also need the following result which is a special case of a more general theorem of
15.
Theorem 4.2. Assume that Assumption 4.1 holds. If there exists KB 0, δ 0, and δ1 0, such
that for x0 ∈ Bδ and the matrix-valued function Bx satisfies I − BxF′
x∗
ρx δ1 for
all x ∈ Bδ, then the iteration
xk1 xk − BxkFxk 4.2
converges linearly to x∗
.
For the proof of Theorem 4.2, see 15.
Using Assumption 4.1 and Theorem 4.2, one has the following result.
Theorem 4.3. Assume that Assumption 4.1 holds. There exist β 0, δ 0, α 0, and γ 0, such
that if x0 ∈ E and D0 satisfies I − D0F′
x∗
F δ for all xk ∈ E, then for iteration
xk1 xk − DkFxk, 4.3
8. 8 Mathematical Problems in Engineering
Dk defined by 3.19,
I − DkF′
x∗
F
δk, 4.4
holds for some constant δk 0, k ≥ 0.
Proof. Since Dk1F Dk MkF, it follows that
Dk1F ≤ DkF MkF. 4.5
For k 0 and assuming D0 I, we have
M
i
0
ΔFT
0 Δx0 − ΔFT
0 D0ΔF0
TrG2
ΔF
i2
0
≤
ΔFT
0 Δx0 − ΔFT
0 D0ΔF0
Tr
G2
0
ΔF
max2
0 , 4.6
where ΔF
max
0
2
is the largest element among ΔF
i
0 , i 1, 2, . . . , n. After multiplying 4.6
by ΔF
max
0
2
/ΔF
max
0
2
and substituting TrG2
0
n
i1ΔF
i
0
4
, we have
M
i
0
≤
ΔFT
0 Δx0 − ΔFT
0 D0ΔF0
ΔF
max
0
2n
i1
ΔFi
4
ΔF
max
0
4
. 4.7
Since ΔF
max
0
4
/
n
i1ΔF
i
0
4
≤ 1, then 4.7 turns into
M
i
0
≤
ΔFT
0 F′
xΔF0 − ΔFT
0 D0ΔF0
ΔF
max
0
2
. 4.8
From Assumption 4.1iv and D0 I, 4.8 becomes
M
i
0
≤
|c − 1|
ΔFT
0 ΔF0
ΔF
max
0
2
, 4.9
where c max{|c1|, |c2|}.
Since ΔF
i
0
2
≤ ΔF
max
0
2
for i 1, . . . , n, it follows that
M
i
0
≤
|c − 1|n
ΔF
max
0
2
ΔF
max
0
2
. 4.10
Hence, we obtain
M0F ≤ n3/2
|c − 1|. 4.11
9. Mathematical Problems in Engineering 9
Table 1: Results of Chandrasekhar H-equation number of iteration/CPU time.
n NM FN BM LMSI
200 4/6.9134 7/5.1673 6/4.7112 5/0.0312
500 4/14.0945 7/11.0418 8/6.9813 5/0.0624
1000 4/64.5813 9/27.2974 8/9.7416 5/0.1716
c 0.9 2000 4/95.6180 9/118.2094 8/11.1690 5/1.6410
5000 — — — 5/2.3556
10000 — — — 5/2.8209
20000 — — — 5/3.7915
200 6/8.1834 8/6.4282 7/5.1946 6/0.0780
500 6/18.2457 8/14.6132 7/7.6137 6/0.0824
1000 6/83.0569 9/30.0542 8/11.0432 6/0.2184
c 0.99 2000 6/121.5309 9/153.0351 8/15.9724 4/1.3104
5000 — — — 4/1.4976
10000 — — — 5/1.7910
20000 — — — 5/2.0371
200 8/15.1792 — 9/5.0388 5/0.3312
500 8/36.0330 — 10/7.1412 5/0.6024
1000 8/134.9045 — 10/12.0644 6/0.7803
c 0.9999 2000 8/175.0521 — 10/14.9064 6/1.2537
5000 — — — 6/1.5132
10000 — — — 6/1.5808
20000 — — — 6/1.6301
Suppose that α n3/2
|c − 1|, then
M0F ≤ α. 4.12
From the fact that D0F
√
n, it follows that
D1F ≤ β, 4.13
where β
√
n α 0.
Therefore, if we assume that I − D0F′
x∗
F δ, then
I − D1F′
x∗
F
I − D0 M0F′
x∗
F
≤
I − D0F′
x∗
F
M0F′
x∗
F
≤
I − D0F′
x∗
F
M0F
F′
x∗
F
,
4.14
hence I − D1F′
x∗
F δ αφ δ1.
And hence, by induction, I − DkF′
x∗
F δk for all k.
10. 10 Mathematical Problems in Engineering
0
20
40
60
80
100
120
140
0 1000 2000 3000 4000 5000 6000
NM
FN
BM
LMSI
CPU
time
(seconds)
System’s dimensions
Figure 1: Comparison of NM, FN, BM, and LMSI methods when c 0.9, in terms of CPU time.
5. Numerical Results
In this section, we compare the performance of LMSI method with that of the Newton’s meth-
od NM, fixed Newton’s method FN, and Broyden’s method BM. We apply the algo-
rithms to the well-known benchmarks integral equations involved in radiative transfer. The
comparison is based upon the following criterion: number of iterations, CPU time in seconds,
and storage requirement. The computations are done in MATLAB 7.0 using double-precision
computer. The stopping criterion used is
Δxk Fxk ≤ 10−4
. 5.1
The starting point x0 is given by 1, 1, . . . 1T
.
The symbol “−” is used to indicate a failure due to the following:
1 The number of iteration is at least 200, but no point of xk satisfying 5.1 is obtained,
2 CPU time in second reaches 200,
3 insufficient memory to initial the run.
The numerical results of the methods when solving Chandrasekhar H-Equation in
different parameter are reported in Table 1. The first column of the table contains the param-
eter of problem. Generally, with our choice of c, the corresponding Jacobian is not diagonally
dominate; however, when c → 1, the Jacobian is nearly singular. From Table 1, it was shown
that only LMSI is able to solve problems where n 2000. This is due to the fact that LMSI
requires very low-storage requirement in building the approximation of the Jacobian inverse.
Indeed, the size of the updating matrix increases in On as the dimension of the system
increases, as opposed to NM, FN, and BM methods that increase in On2
.
Moreover, we observe that LMSI method has a 100% of success rate convergence to
the solution when compared with NM method having 57%, FN method with 39% and BM
with 71%, respectively. In addition, it is worth mentioning that the result of LMSI in solving
11. Mathematical Problems in Engineering 11
0
20
40
60
80
100
120
140
160
180
0 1000 2000 3000 4000 5000 6000
NM
FN
BM
LMSI
CPU
time
(seconds)
System’s dimensions
Figure 2: Comparison of NM, FN, BM, and LMSI methods when c 0.99, in terms of CPU time.
0
20
40
60
80
100
120
140
160
180
200
0 1000 2000 3000 4000 5000 6000
NM LMSI
BM
CPU
time
(seconds)
System’s dimensions
Figure 3: Comparison of NM, FN, BM, and LMSI methods when c 0.9999, in terms of CPU time.
problem 1 when c 0.9999 shows that the method could be a good solver even when the
Jacobian is nearly singular. Figures 1, 2, and 3 reveal that the CPU time of LMSI method in-
creases linearly as the dimension of the systems increases, whereas for NM, FN, and BM, the
rates grow exponentially. This also suggests that our solver is a good alternative when the
dimension of the problem is very high.
6. Conclusion
In this paper, we present a low memory solver for integral equation of Chandrasekhar type in
the radiative transfer problems. Our approach is based on approximating the Jacobian inverse
into a diagonal matrix. The fact that the LMSI method can solve the discretized integral
12. 12 Mathematical Problems in Engineering
equations without computing and storing the Jacobian makes clear the advantage over NM
and FN methods. It is also worth mentioning that the method is capable of significantly
reducing the execution time CPU time, as compared to NM, FN, and BM methods while
maintaining good accuracy of the numerical solution to some extend. Another fact that makes
the LMSI method appealing is that throughout the numerical experiments it never fails to
converge. Finally, we conclude that our method LMSI is a good alternative to Newton-type
methods for solving large-scale nonlinear equations with nearly singular Jacobian.
References
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