This document discusses regular perturbation theory through three chapters:
1. It defines key concepts like asymptotic sequences, asymptotic expansions, and order symbols used in perturbation theory.
2. It explains the fundamental ideas of regular and singular perturbations. Regular perturbations do not change the order of the problem when the perturbation parameter is set to zero, while singular perturbations do.
3. It provides examples of applying regular perturbation theory to solve algebraic equations. The technique involves expanding the solutions in powers of the perturbation parameter and solving the equations order-by-order. This allows approximating solutions even when exact solutions are not available.
This document discusses methods for solving numerical equations, including the bisection method, Newton-Raphson method, and method of false position. It provides definitions and step-by-step computations for each method. For the bisection method, it gives an example of finding the positive root of x3 - x = 1. For Newton-Raphson, it gives examples of finding the root of 2x3 - 3x - 6 = 0 and x3 = 6x - 4. The document serves to introduce numerical methods for solving equations.
Formal expansion method for solving an electrical circuit modelTELKOMNIKA JOURNAL
We investigate the validity of the formal expansion method for solving a second order ordinary differential equation raised from an electrical circuit problem. The formal expansion method approximates the exact solution using a series of solutions. An approximate formal expansion solution is a truncated version of this series. In this paper, we confirm using simulations that the approximate formal expansion solution is valid for a specific interval of domain of the free variable. The accuracy of the formal expansion approximation is guaranteed on the time-scale 1.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
1) Nonhomogeneous linear systems generalize the theory of single nth order linear equations. They can be written in the form x' = P(t)x + g(t).
2) The general solution is the sum of the general solution to the homogeneous system x' = P(t)x and a particular solution to the nonhomogeneous system.
3) If the matrix P is constant and diagonalizable, the system can be transformed to uncoupled equations y' = Dy + h(t) via diagonalization, whose solutions give the solution to the original system.
This document discusses partial differential equations (PDEs). It provides examples of how PDEs can be formed by eliminating constants or functions from relations involving multiple variables. It also discusses different types of first-order PDEs and methods for solving them. Several example problems are presented with step-by-step solutions showing how to derive and solve PDEs that model different physical situations. Standard forms and techniques for reducing PDEs to simpler forms are also outlined.
(1) This document discusses ordinary differential equations of first order and first degree. Examples of differential equations are given and defined.
(2) Methods for solving first order differential equations are discussed, including variable separable, homogeneous, and linear methods. Examples of solving differential equations using these methods are provided.
(3) The order and degree of differential equations are defined. The process of forming differential equations from given functions is demonstrated through several examples.
Differential Equations Lecture: Non-Homogeneous Linear Differential Equationsbullardcr
This document discusses the method of undetermined coefficients for solving nonhomogeneous second-order linear differential equations. It explains that the solution to a nonhomogeneous equation is the sum of the solution to the corresponding homogeneous equation and a particular solution to account for the nonhomogeneous term. It also outlines the steps to use the method of undetermined coefficients, which involves guessing a form for the particular solution based on the type of nonhomogeneous term and solving for the coefficients.
This document discusses methods for solving numerical equations, including the bisection method, Newton-Raphson method, and method of false position. It provides definitions and step-by-step computations for each method. For the bisection method, it gives an example of finding the positive root of x3 - x = 1. For Newton-Raphson, it gives examples of finding the root of 2x3 - 3x - 6 = 0 and x3 = 6x - 4. The document serves to introduce numerical methods for solving equations.
Formal expansion method for solving an electrical circuit modelTELKOMNIKA JOURNAL
We investigate the validity of the formal expansion method for solving a second order ordinary differential equation raised from an electrical circuit problem. The formal expansion method approximates the exact solution using a series of solutions. An approximate formal expansion solution is a truncated version of this series. In this paper, we confirm using simulations that the approximate formal expansion solution is valid for a specific interval of domain of the free variable. The accuracy of the formal expansion approximation is guaranteed on the time-scale 1.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
1) Nonhomogeneous linear systems generalize the theory of single nth order linear equations. They can be written in the form x' = P(t)x + g(t).
2) The general solution is the sum of the general solution to the homogeneous system x' = P(t)x and a particular solution to the nonhomogeneous system.
3) If the matrix P is constant and diagonalizable, the system can be transformed to uncoupled equations y' = Dy + h(t) via diagonalization, whose solutions give the solution to the original system.
This document discusses partial differential equations (PDEs). It provides examples of how PDEs can be formed by eliminating constants or functions from relations involving multiple variables. It also discusses different types of first-order PDEs and methods for solving them. Several example problems are presented with step-by-step solutions showing how to derive and solve PDEs that model different physical situations. Standard forms and techniques for reducing PDEs to simpler forms are also outlined.
(1) This document discusses ordinary differential equations of first order and first degree. Examples of differential equations are given and defined.
(2) Methods for solving first order differential equations are discussed, including variable separable, homogeneous, and linear methods. Examples of solving differential equations using these methods are provided.
(3) The order and degree of differential equations are defined. The process of forming differential equations from given functions is demonstrated through several examples.
Differential Equations Lecture: Non-Homogeneous Linear Differential Equationsbullardcr
This document discusses the method of undetermined coefficients for solving nonhomogeneous second-order linear differential equations. It explains that the solution to a nonhomogeneous equation is the sum of the solution to the corresponding homogeneous equation and a particular solution to account for the nonhomogeneous term. It also outlines the steps to use the method of undetermined coefficients, which involves guessing a form for the particular solution based on the type of nonhomogeneous term and solving for the coefficients.
This document discusses ordinary differential equations (ODEs). It defines ODEs and differentiates them from partial differential equations. ODEs can be classified by type, order, and linearity. Initial value problems involve solving an ODE with initial conditions specified at a point, while boundary value problems involve conditions at boundary points. The document provides examples of solving first- and second-order initial value problems. It also discusses the existence and uniqueness of solutions to initial value problems under certain continuity conditions on the functions defining the ODE.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
The document discusses numerical methods for solving ordinary differential equations (ODEs), including Taylor's series method and Picard's method. It provides examples of applying Taylor's series method to approximate solutions of first order ODEs at different values of x to 4-5 decimal places of accuracy. The examples given include solving ODEs with initial conditions and computing solutions at multiple x values by taking terms from the Taylor series expansion.
In this presentation we can get to know the meaning of basic discrete distribution for bivariate. There are also discussions regarding the topic along with marginal tables. Also there are certain illustrative example for the ease of understanding. Overall it is a great presentation for the junior engineers aiming in their course.
The document discusses solving systems of linear equations with two or three variables. There are three possible cases for the solution: 1) a unique solution, 2) infinitely many solutions (a dependent system), or 3) no solution. The document demonstrates solving systems using substitution and elimination methods, and provides examples of each case. Graphically, case 1 corresponds to intersecting lines or planes, case 2 to coinciding lines or intersecting planes, and case 3 to parallel lines or non-intersecting planes.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
The document discusses series solutions to second order linear differential equations near ordinary points. It provides an example of finding the series solution to the differential equation y'' + y = 0 near x0 = 0. The solution is found to be a cosine series which represents the cosine function, a fundamental solution. A second example finds the series solution to Airy's equation near x0 = 0, obtaining fundamental solutions related to Airy functions.
The document provides an introduction to partial differential equations (PDEs). Some key points:
- PDEs involve functions of two or more independent variables, and arise in physics/engineering problems.
- PDEs contain partial derivatives with respect to two or more independent variables. Examples of common PDEs are given, including the Laplace, wave, and heat equations.
- The order of a PDE is defined as the order of the highest derivative. Methods for solving PDEs through direct integration and using Lagrange's method are briefly outlined.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This document describes Picard's method for solving simultaneous first order differential equations numerically. It presents the iterative formula used in Picard's method and applies it to solve four example problems of simultaneous differential equations. The problems are solved over multiple iterations to obtain successive approximations of the solutions at increasing values of x, with the approximations being carried to three or four decimal places.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods. The Newton-Raphson method uses iterative approximations based on the Taylor series expansion and its derivative to find roots. The secant method approximates the derivative as the difference quotient. MATLAB functions are provided to implement both methods to find roots of sample non-linear equations.
This document contains a summary of a student group project on first order ordinary differential equations (ODEs). It defines key terms related to ODEs such as order, degree, general solutions, and singular solutions. It also categorizes common types of first order ODEs including separable, homogeneous, exact, and linear equations. Solution methods are described for each type. Additional topics covered include Bernoulli equations, orthogonal trajectories, and applications of ODEs in areas like radioactivity, electrical circuits, economics, and physics. The document is authored by six chemical engineering students at G.H. Patel College of Engineering and Technology.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
The document is an introduction to ordinary differential equations prepared by Ahmed Haider Ahmed. It defines key terms like differential equation, ordinary differential equation, partial differential equation, order, degree, and particular and general solutions. It then provides methods for solving various types of first order differential equations, including separable, homogeneous, exact, linear, and Bernoulli equations. Specific examples are given to illustrate each method.
1. The document discusses differential equations, which relate functions and their derivatives. First order equations relate a function to its first derivative, while second order equations relate it to its second derivative.
2. Differential equations are used in physics, such as Newton's second law relating force, mass and acceleration. They can have many solutions or no solutions. Simultaneous differential equations involve multiple dependent variables.
3. Examples of simultaneous differential equations include models of survival with AIDS, earthquake effects on buildings with multiple floors, and harvesting renewable resources like fish populations over time.
Dallas Technologies are providing best Training of ERP and CRM, Mainframes and AS400, C & C++/ Java & J2EE, .Net, Embedded Technologies, PLC & SCADA industrial automation training, Software Testing, Android App Development, Big Data, Salesforce, Web Services, Data Warehousing and Business Intelligence, (CCNA, A+, N+, Networking), Etc.
We gain knowledge of your business and help you apply technology quickly and intelligently to meet your goals. Our skilled team of local Solution Managers and Software Engineers work closely with each client to accurately assess their requirements and determine the appropriate business offering.
This document discusses regular perturbation theory and its application to solving algebraic equations. It begins by defining regular and singular perturbations. For regular perturbations, the order of the perturbed and unperturbed problems are the same when the perturbation parameter is set to zero. The document then shows how regular perturbation theory can be used to solve algebraic equations. Specifically, it demonstrates obtaining the series solutions for the quadratic equation x^2-1=ε and the cubic equation x^3-x+ε=0 by assuming power series solutions in ε and solving the equations order-by-order in ε. This yields convergent power series expansions for the roots as functions of the small perturbation parameter ε.
Dallas Technologies are providing best Training of ERP and CRM, Mainframes and AS400, C & C++/ Java & J2EE, .Net, Embedded Technologies, Software Testing, Android App Development, Web Services, Data Warehousing and Business Intelligence, (CCNA, A+, N+, Networking), Etc.
This document discusses ordinary differential equations (ODEs). It defines ODEs and differentiates them from partial differential equations. ODEs can be classified by type, order, and linearity. Initial value problems involve solving an ODE with initial conditions specified at a point, while boundary value problems involve conditions at boundary points. The document provides examples of solving first- and second-order initial value problems. It also discusses the existence and uniqueness of solutions to initial value problems under certain continuity conditions on the functions defining the ODE.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
The document discusses numerical methods for solving ordinary differential equations (ODEs), including Taylor's series method and Picard's method. It provides examples of applying Taylor's series method to approximate solutions of first order ODEs at different values of x to 4-5 decimal places of accuracy. The examples given include solving ODEs with initial conditions and computing solutions at multiple x values by taking terms from the Taylor series expansion.
In this presentation we can get to know the meaning of basic discrete distribution for bivariate. There are also discussions regarding the topic along with marginal tables. Also there are certain illustrative example for the ease of understanding. Overall it is a great presentation for the junior engineers aiming in their course.
The document discusses solving systems of linear equations with two or three variables. There are three possible cases for the solution: 1) a unique solution, 2) infinitely many solutions (a dependent system), or 3) no solution. The document demonstrates solving systems using substitution and elimination methods, and provides examples of each case. Graphically, case 1 corresponds to intersecting lines or planes, case 2 to coinciding lines or intersecting planes, and case 3 to parallel lines or non-intersecting planes.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
The document discusses series solutions to second order linear differential equations near ordinary points. It provides an example of finding the series solution to the differential equation y'' + y = 0 near x0 = 0. The solution is found to be a cosine series which represents the cosine function, a fundamental solution. A second example finds the series solution to Airy's equation near x0 = 0, obtaining fundamental solutions related to Airy functions.
The document provides an introduction to partial differential equations (PDEs). Some key points:
- PDEs involve functions of two or more independent variables, and arise in physics/engineering problems.
- PDEs contain partial derivatives with respect to two or more independent variables. Examples of common PDEs are given, including the Laplace, wave, and heat equations.
- The order of a PDE is defined as the order of the highest derivative. Methods for solving PDEs through direct integration and using Lagrange's method are briefly outlined.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This document describes Picard's method for solving simultaneous first order differential equations numerically. It presents the iterative formula used in Picard's method and applies it to solve four example problems of simultaneous differential equations. The problems are solved over multiple iterations to obtain successive approximations of the solutions at increasing values of x, with the approximations being carried to three or four decimal places.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods. The Newton-Raphson method uses iterative approximations based on the Taylor series expansion and its derivative to find roots. The secant method approximates the derivative as the difference quotient. MATLAB functions are provided to implement both methods to find roots of sample non-linear equations.
This document contains a summary of a student group project on first order ordinary differential equations (ODEs). It defines key terms related to ODEs such as order, degree, general solutions, and singular solutions. It also categorizes common types of first order ODEs including separable, homogeneous, exact, and linear equations. Solution methods are described for each type. Additional topics covered include Bernoulli equations, orthogonal trajectories, and applications of ODEs in areas like radioactivity, electrical circuits, economics, and physics. The document is authored by six chemical engineering students at G.H. Patel College of Engineering and Technology.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
The document is an introduction to ordinary differential equations prepared by Ahmed Haider Ahmed. It defines key terms like differential equation, ordinary differential equation, partial differential equation, order, degree, and particular and general solutions. It then provides methods for solving various types of first order differential equations, including separable, homogeneous, exact, linear, and Bernoulli equations. Specific examples are given to illustrate each method.
1. The document discusses differential equations, which relate functions and their derivatives. First order equations relate a function to its first derivative, while second order equations relate it to its second derivative.
2. Differential equations are used in physics, such as Newton's second law relating force, mass and acceleration. They can have many solutions or no solutions. Simultaneous differential equations involve multiple dependent variables.
3. Examples of simultaneous differential equations include models of survival with AIDS, earthquake effects on buildings with multiple floors, and harvesting renewable resources like fish populations over time.
Dallas Technologies are providing best Training of ERP and CRM, Mainframes and AS400, C & C++/ Java & J2EE, .Net, Embedded Technologies, PLC & SCADA industrial automation training, Software Testing, Android App Development, Big Data, Salesforce, Web Services, Data Warehousing and Business Intelligence, (CCNA, A+, N+, Networking), Etc.
We gain knowledge of your business and help you apply technology quickly and intelligently to meet your goals. Our skilled team of local Solution Managers and Software Engineers work closely with each client to accurately assess their requirements and determine the appropriate business offering.
This document discusses regular perturbation theory and its application to solving algebraic equations. It begins by defining regular and singular perturbations. For regular perturbations, the order of the perturbed and unperturbed problems are the same when the perturbation parameter is set to zero. The document then shows how regular perturbation theory can be used to solve algebraic equations. Specifically, it demonstrates obtaining the series solutions for the quadratic equation x^2-1=ε and the cubic equation x^3-x+ε=0 by assuming power series solutions in ε and solving the equations order-by-order in ε. This yields convergent power series expansions for the roots as functions of the small perturbation parameter ε.
Dallas Technologies are providing best Training of ERP and CRM, Mainframes and AS400, C & C++/ Java & J2EE, .Net, Embedded Technologies, Software Testing, Android App Development, Web Services, Data Warehousing and Business Intelligence, (CCNA, A+, N+, Networking), Etc.
El documento describe las actividades realizadas el 4 de marzo como parte del período de adaptación en el jardín de infantes. Las actividades incluyeron juegos de conteo con números del 1 al 10 en matemáticas, diálogos sobre intereses y necesidades para favorecer la lengua oral, y escuchar canciones infantiles para generar disfrute en música. También hubo tiempo para el aseo personal y la merienda.
Penas de muerte en el mundo presentaciónradioclaudi
El documento discute la pena de muerte, notando que Estados Unidos es el país más conocido por aplicarla, habiendo ejecutado a más de 1,400 personas. Aunque la pena de muerte ha existido desde tiempos antiguos, en la mayoría de países democráticos ya no se aplica, con excepción de Estados Unidos y Japón. Una encuesta de 2000 encontró un 52% de apoyo a nivel mundial, pero no está claro si ese apoyo proviene de personas bien informadas sobre el tema.
A girl named Morena Di Berardini who is 11 years old from Bahia Blanca introduces herself, her family which includes her parents Alex and Lia and her 4 month old brother Luca. She lists her hobbies as tissue dancing, listening to music, and studying English and her abilities as running, walking, swimming and riding a bike. She has 3 pets named Peter, Wendy and Roma and most importantly loves her family and friends.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...ijfls
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point
theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for
the approximate controllability of the system.
United Arab Emirates, A developed CountryHira Sohaib
There are two types of countries: developed and undeveloped. A developed country, as defined by Kofi Annan, allows all citizens to enjoy free, healthy, and safe lives. Developed countries have happy, educated citizens where governments work to better people's lives, as seen in present-day Dubai in the UAE. The document expresses a hope that all nations can achieve development to improve life worldwide.
Voorstelling van de federatie ICS en Fedelec en hun samenwerking, door Dirk Peytier, op de kick-off van de Ecodesign Roadshow, 24 september, Anderlecht.
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
Math 1102-ch-3-lecture note Fourier Series.pdfhabtamu292245
1. The document discusses Fourier series and orthogonal functions. It defines orthogonal functions and provides examples of orthogonal function sets, such as cosine and sine functions.
2. The chief advantage of orthogonal functions is that they allow functions to be represented as generalized Fourier series expansions. The orthogonality of the functions helps determine the Fourier coefficients in a simple way using integrals.
3. Euler's formulae give the expressions for calculating the Fourier coefficients a0, an, and bn of a periodic function f(x) from its values over one period using integrals of f(x) multiplied by cosine and sine terms.
This document presents several theorems regarding the existence, uniqueness, and extendibility of solutions to ordinary differential equation systems. It begins by introducing Peano's theorem on existence of solutions and Picard-Lindelof theorem on uniqueness of solutions locally. It then discusses extension theorems which aim to extend local solutions to open connected domains. The main results section discusses using extension theorems to show existence of solutions on the entire positive real line by extending solutions from finite intervals to infinity.
The document summarizes key concepts of quantum mechanics from chapters 3 and 4 of McQuarrie, including:
1) The Schrodinger equation and its solutions in 1D and 3D.
2) Solving the time-independent Schrodinger equation involves finding the general solution, applying boundary conditions, and normalizing the wavefunction.
3) Wavefunctions represent probability distributions, and the probability of finding a particle in a region is calculated by integrating the wavefunction.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
This document discusses conjugate gradient methods for minimizing quadratic functions. It begins by introducing quadratic functions and noting that conjugate gradient methods can minimize them without needing the full Hessian matrix, unlike Newton's method. It then defines what it means for a set of vectors to be conjugate with respect to a positive definite matrix A. Vectors are conjugate if their inner products with respect to A are all zero. The document proves that a set of conjugate vectors forms a basis and describes a simple conjugate gradient algorithm that finds the minimum in n iterations using n conjugate search directions.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
This document provides an overview of Floquet theory and periodic differential equations. It begins by introducing periodic differential equations and framing their solution in terms of spectral theory and eigenvalue problems. It then covers key results of Floquet theory, including that any periodic differential equation has a non-trivial solution that is periodic up to a constant multiplier. The document defines Hill's equation and shows how general periodic differential equations can be transformed into this form. It concludes by examining cases for the discriminant of Hill's equation that determine the nature of the solutions.
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).pptDediTriLaksono1
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3. 1
PREFACE
The book “A study on regular perturbation problems” is intended for the PG
students in kerala university. In this book all the topic have been deal within a
simple and lucid manner. A sufficiently large number of problems have been
solved by studying this book , the student is expected to understand the concept of
regular perturbation, the fundamental ideas of perturbation. To do more problems
involving the regular perturbation and fundamental ideas of perturbation.
Suggestion for the further improvement of this book will be highly
appreciated.
Shareena . P.R.
4. 2
CHAPTER 1
Defintion 1.1
Pertubation theory is the study of the effect of small distrurbance if the effect
are small, the distrurbance or perturbations are said to be regular, otherwise they
are said to be singular
Defintion 1.2
Asympotic Sequence
A set of function {∅n( )}n = 0,1,2…is an asymptotic as
→ 0 , ℎ > 0, ∅nti( ) = 0 (∅n( )) as → 0, that is each subsequent
term gets smaller.
Examples
(a) {1, , 2
, 3
,…}
(b){1, /
, /
,…}
(c) {1, , log , , 2
log ,…}
Defintion 1.3
Asymptotic expansion
5. 3
Define an asymptotic series,
y= yo+ y1 + 2
y2+…,
where y1, y1, y2, … are sufficiently smooth functions.
The standard asymptotic sequence is {1, , 2, 3
…} as 0 and fn(x)
represents the members of asymptotic sequence then fn+1( ) = 0(fn
( )) as xa
that is →
( )
( )
= 0. The general expression for asymptotic expansion of
function fn( ) is the series of terms
f(x)=∑ ( ) + → 0, ℎ
= ( ( )) → 0 lim → RN=0.
Definition 1.4
The expression f(x) =∑ fn( ) + RN ,where f(x; ) depends on an
independent variable x and small parameter . The coefficient of the gauge
function fn( ) are functions of x and the remainder term after N terms is a function
of both x and is RN = O (fn+1( )) is said to be uniform asymptotic expansion, if
RN Cfn+1( ), where c is the constant.
Example
f(x, )= = 1+ sinx+ 2
(sinx)2
+…as 0.
The remainder term RN = 1+ sinx+ 2
(sinx)2
+…-∑ ( ) ,
where → ( ) = (sinx)N
+1
6. 4
Defintion 1.5
The expression f(x) = ∑ an fn ( ) + RN, where f(x; ) depends on an
independent variable x and small parameter ∈ is said to be non – uniform
asymptotic expansion, if there is no constants exists but the relation RN≤ Cfn+1 ( )
satisfied is known as non-uniform asymptotic expansion
Example
f(x, )= = 1+ x+ 2
(x)2
+…as 0.
The remainder term RN = 1+ x+ 2
(x)2
+…- n
(x)n
,
Where → ( ) =(x)N
+1. There is no fixed constant C exists such that
RN≤ N+1.
7. 5
CHAPTER 2
The Fundamental Ideas of Perturbation
2.1 Definition Of Regular And Singular Perturbations
Definition 2.1.1.
The problem which does not contain any small parameter is known as unperturbed
problem.
Example 2.1.1
(a) x2
+ 3+ 1 = 0.
(b)
+2 + y = 2x2
– 8x + 4, y(o) = 3, (0) = 3.
Definition 2.1.2. The problem which contains a small parameter is known as
perturbed problem.
Example 2.1.2.
(a) x2
— x+ = 0.
(b) +y = y2
, y(0) = 1.
8. 6
Depending upon the nature of perturbation, a perturbed problem can be divided
into two categories. They are,
1. Regularly perturbed
2. Singularly perturbed
Definition 2.1.3. The perturbation problem is said to be regular in nature, when the
order (degree) of the perturbed and the un-perturbed problem are same, when we
set = 0. Generally, the parameter presented at lower order terms. The following is
an example of regularly perturbed problem.
Example 2.1.3
(a) x2
-1 = x
(b) )
+ y = y2
, y(0) = 1, (0) = -1
Definition 2.1.4. The perturbed problem is said to be singularly perturbed, when
the order (degree) of the problem is reduced when we set = 0. Generally, the
parameter presented at higher order terms and the lower order terms starts to
dominate. Sometime the above statement is considered as the definition of
singularly perturbation problem. The following is an example of singularly
perturbed problem.
Example 2.1.4
(a) x2
-3x+8=0
(b)
+ = 2 + 1, y(0) = 0, y(1) = 4
9. 7
2.2 The Fundamental Theorem Of Perturbation Theory
If A0, A1 +…+AN
N
+ O( N+1
) = 0 for sufficiently small and if the coefficients A0,
A1… are independent of , then
A0 = A1 = …= AN = 0
2.3 Order Symbols
The letters O and o are order symbols. They are used to describe the rate at
which the function approaches to limit value.
If a function f(x) approaches to a limit value at the same rate of another function
g(x) at x x 0, then we can write f(x) = O(g(x)) as x —> xo. The functions are
said to be of same order as x x 0 . We can write it as,
0
lim
xx
( )
( )
= C where C is
finite. We can say here “f is big – oh of g”. If the expression f(x) = o(g(x)) as
the x x 0 means
0
lim
xx
( )
( )
= 0. We can say here “ f is little – oh of g” x x 0 and
f(x) is smaller than g(x) as x x 0.
Example 2.3.1
(a) sin x = O(x) as x→ 0 since
0
lim
xx
= 1.
10. 8
(b) = o(x) as x→ ∞.
(c)Sin x2
= o(x) as x →0 because
0
lim
xx
= 0.
(c) 3x+x3
= O(x) as x →0 since
0
lim
xx
= 3.
(e) e-x
= o( ) as x→ ∞.
(f) sin(2x) = O(x) as x →0.
(g) x+e-x
= O(x) as x →∞.
“Big-oh" notation and "Little-oh" notation are generally called Landau
symbols. The expression f(x) ~ g(x) as x→x0 means
0
lim
xx
( )
( )
= 1 is called "f is
asymptotically equal or approximately equal to g".
11. 9
CHAPTER 3
Regular Perturbation problems
Very often, a mathematical problem cannot be solved exactly or, if the
exact solution is available, it exhibits such an intricate dependency in the
parameters that it is hard to use as such. It may be the case, however, that a
parameter can be identified, say , such that the solution is available and
reasonably simple for = 0. Then, one may wonder how this solution is altered for
non-zero but small . Perturbation theory gives a systematic answer to this
question.
3.1 Solution Of Algebraic Equations.
Example 3.1.1
Consider the quadratic equation
x2
-1 = (3.1)
The two roots of this equation are
x1 = + 1 + , = − 1 + (3.2)
For small , these roots are well approximated by the first few terms of their Taylor
series expansion (see figure 1)
= 1 + + + ( ), 2 = −1 + - + ( ). (3.3)
12. 10
Can we obtain (3.3) without prior knowledge of the exact solutions of (3.1)?.Yes,
using regular perturbation theory. The technique involves four steps.
Assume that the solution(s) of (3.1) can be Taylor expanded in varepsilon. Then
we have
x=X0+ X1 + 2
X2 + O( 3
) (3.4)
for X0, X1, X2 to be determined.
Substitute (3.4) into (3.1) written as x2
- 1 - X = 0, and expand the left hand side
of the resulting equation in power series of . Using
x2
= + 2 X0X1 + 2
( + 2X0X2) + O( 3
), (3.5)
x = X0 + 2
X1+O( 3
)
13. 11
Figure 1: The root x1 plotted as a function of (solid line), compared with the
approximations by truncation of the Taylor series at O (
2
) , x1 = 1+
(dotted line),
and O (
3
), x1 = 1+
+
(dashed line). Notice that even though the
approximations are a priori valid in the range <<1 only, the approximation
= 1+
+
is fairly good even up to = 2.
this gives
– 1+ (2X0X1 –X0)+
2
( +2X0X2 ─X1) + O(
3
)= 0 (3.6)
Equate to zero the successive terms of the series in the left hand
side of (3.6):
O (
0
) : ─1 = 0,
O (
1
) :2X0X1 – X0 = 0, (3.7)
O (
2
) : X2
1 + 2X0X1─X1 = 0,
O (
3
) : …
Successively solve the sequence of equations obtained in (3.7). Since X2
0-1=0 has
two roots, X0 = ±1, one obtains
X0 = 1, XI = , X2 = (3.8)
X0 = -1, XI = , X2 =
14. 12
It can be checked that substituting (3.8) into (3.4) one recovers (3.3)From the
previous example it might not be clear what the advantage of regular perturbation
theory is, since one can obtain (3.3) more directly by Taylor expansion of the roots
in (3.2). To see the strength of regular perturbation theory, consider the following
equation
X2
-1 = (3.9)
15. 13
Figure 2: The solid line is the graph of two of the three solutions of (3.9) obtained
numerically and plotted as a function of (solid line). Also plotted are the
approximations by truncation of the Taylor series at O(
2
), x1=1+
∈
(dotted line),
and O(
3
), x1= 1+
+
(dashed line).
The solutions of this equation are not available; therefore the direct method
is inapplicable here. However, the Taylor series expansion of these solutions can
be obtained by perturbation theory. We introduce the expansion (3.4). We use
(recall that ez
= 1+z+ + O (z3
))
16. 14
= ( )
= ( )
= + eXo
+O(
3
) (3.10)
Substituting this expression in (3.9) written as x2
-1-
x
= 0 and using (3.5), we
obtain
X2
0 – 1+ (2X0X1-eXo
)+
2
(X2
1+2X0X1- X1eXo
)+O(
3
) = 0. (3.11)
Thus, the sequence of equations obtained is
O (
0
) : X2
0-1 = 0,
O (
1
) :2X0X1 – eX0
= 0
O (
2
) : X2
1 + 2X0X1-X1 eX0
= 0 (3.12)
O (
3
) : …
from which we obtain
Xo = 1, X1 = , X2 = (3.13)
X0 = -1, X1 = , X2=
or, equivalently,
17. 15
x1 =1+
+ +O( 3
)
x2 =1+
− +O( 3
)
The expression for x1 is compared to the numerical solution of (3.9) on figure 2.
Remark: In fact (3.9) has three solutions for 0 < < , with ≈ 0.43, and only
one for > . The solution which exists for all > 0 is the one with expansion
given in x2 in (3.14) ; the solution with the expansion given in x1 in (3.14)
disappears for > ; and the third solution (see figure 2: the solid line is the graph
of a two-valued function) cannot be obtained by regular perturbation.
Example 3.1.2
Consider the cubic equation
x3
–x+ = 0 (3.15)
We look for a solution of the form
x= x0+ x1+
2
x2 + O( 3
) (3.16)
Using this expansion in the equation, expanding, and equating coefficients of
to zero, we get
x3
0 – x0 = 0
3x2
0 x1 - xl + 1 = 0
3x0 x2 – x2 + 3x0 x2
1 = 0
18. 16
Note that we obtain a nonlinear equation for the leading order solution x0, and
nonhomogeneous linearized equations for the higher order corrections x1, x2,..This
structure is typical of many perturbation problems.
Solving the leading-order perturbation equation, we obtain the three roots
X0 = 0, 1.
Solving the first-order perturbation equation, we find that
x1 =
The corresponding solutions are
x = +O( 2
), x = 1- + O ( 2
)
Continuing in this way, we can obtain a convergent power series expansion about
= 0 for each of the three distinct roots of (3.15). This result is typical of regular
perturbation problems. .
An alternative but equivalent method to obtain the perturbation series is to
use the Taylor expansion
x( ) = x(0) + x(0) +
!
x (O) ( 2
)+…
where the dot denotes a derivative with respect to . To compute the coefficients,
we repeatedly differentiate the equation with respect to and set = 0 in the
result. For example, setting = 0 in (3.15), and solving the resulting equation for
x(0), we get x(0) = 0, 1. Differentiating (3.15) with respect to , we get
3x2
x - x +1 = 0.
Setting = 0 and solving for x(0), we get the same answer as before.
19. 17
Example 3.1.3
Consider the quadratic equation
(1- )x2
– 2x+1 = 0 (3.18)
Suppose we look for a straight forward power series expansion of the form
x= x0 + x1 + O( 2
)
We find that
x2
0 – 2x0 + 1 =0,
2(x0 – 1) x1 =
Solving the first equation, we get x0 = 1. The second equation then becomes 0= 1.
It follows that there is no solution of the assumed form.
This difficulty arises because x= 1 is a repeated root of the unperturbed
problem. As a result, the solution
x=
±
does not have a power series expansion in , but depends on
An expansion
x= x0 + x1 + x2 + O( )
leads to the equations x0= 1, x2
1 = 1, or
x= 1 + O( )
20. 18
in agreement with the exact solution.
3.2 Solution Of First Order Differential Equitions
Example 3.2.1
Consider the differential equation
y′ + y2
= 0 (3.19)
which has been disturbed by a small effect, so that (3.19) has to be modified to
read
y′ + y2
= x, y(1) =1 (3.20)
where is small. If then become necessary to determine by how much the solution
of (3.19) has been altered because of the presence of the disturbing function x.
We refer to this change in the solutions as a perturbation.
A precise perturbation theory is extremely difficult. In this example, we shall aim
to give a rough outline of a method by which this problem can be handled. Call
y0(x) a solution of satisfying y(1) = 1, and denote the solution of (3.20) by
y(x) = y0(x) + p(x) (3.21)
where p(x) is the perturbation. We next expand y(x) in a series in powers of , so
that
y(x) = y0(x) + y1(x) + 2
y2(x)+… (3.22)
Comparing (3.21) with 3.22), we see that
p(x) = y1(x) + 2
y2(x)+… (3.23)
21. 19
The term y1(x) is called the first order perturbation; the second term 2
y2(x) is
called the second order perturbation, and so on.
Substituting (3.22) in (3.20), we obtain
(yo
′
+ y1′+ 2
y2′+ …+(yo+ y1+ 2
y2+…)2
= x (3.24)
Carrying out the indicated multiplication, then collecting coefficient of like powers
of , we have
(yo
′
+ y0
2
) + (y1′+ 2y0y1) + (y2′+ 2y0 + y2 + y1
2
) 2
+ …= x (3.25)
Next we take like powers of . There results
yo
′
+ y0
′
= 0,
(y1
′
+ 2y0y1) = x,
(y2
′
+ 2y0y2 + y12
) = 0 (3.26)
…………………………
By solving each equation (3.26) is succession, we can thus determine the functions
y1(x), y2(x), y3(x)… in (3.22). Each of these functions, however, must satisfies an
initial condition. Since the initial condition associated with the original equation
(3.19) is y(1) = 1 and since y0 is a solution of (3.19) so that yo(1) = 1, this is initial
condition will be satisfied if in (3.22), we assume
yo(1) = 1, y1(1) = 0, y2(1) = 0,…
We illustrate the details of the above method in the next example
In practice the first and second order perturbations terms of (3.23) are usually
sufficient
22. 20
Example 3.2.2
Find the first and second order perturbation terms in the solution of
y′ + y2
= 0, for which y(1) = 1 (3.28)
due to the precense of a disturbing function x, where is small Solution. Because
of the disturbing function x, (3.28) must be modified to read
y′ + y2
= x (3.29)
For which y(1) = 1. Following the procedure outlined above we let, see (3.22) and
(3.27)
y(x) = y0(x) + y1(x) + 2
y2(x) +… (3.30)
with initial conditions
y0(1) = 1, y1(1) = 0, y2(1) = 0… (3.31)
Substituting (3.30) in (3.31), we obtain,
(y0′+ y0
2
) + (y1′+ 2y0y1) + (y2′+ 2y0y2 + y1
2
) 2
+…= x (3.32)
Equating coefficients of like powers of 0
, , 2
, we obtain from (3.32) the system
of equations
y0′+ y0
2
= 0
(y1′+ 2y0y1) = x (3.33)
(y2′+ 2y0y2 + y1
2
) = 0
A solution of the first equation of (3.33), satisfying the initial condition y0(1)=1 of
(3.31), is
23. 21
y0 = (3.34)
Substituting (3.34) in the second equation of (3.33)
y1′ + y1 = x (3.35)
A solution of (3.35) satisfying y1(1) = 0 of (3.31) is
y1 = ( − ) (3.36)
Substituting (3.34) and (3.36) in the third equation of (3.33), we obtain
y2′+ = −
( − 2 + )
A solution of (3.37) satisfying y2(1) = 0 of (3.31), is
y2 = − ( − − ) − (3.38)
Substituting (3.34), (3.36), (3.38) and (3.30), we obtain
y= + − − 3 − 14 − + (3.39)
The solution of (3.28) satisfies y(1) = 1, that is, its solution if there were no
disturbing function x present, is . Because of the disturbing function x, the first
and second order perturbation terms are, respectively, the second and third term in
(3.39).
24. 22
3.3 Eigenvalue problems
Spectral perturbation theory studies how the spectrum of an operator is perturbed
when the operator is perturbed. In general this question is a difficult one, and
subtle phenomena may occur, especially in connection with the behavior of the
continuous spectrum of the operators. Here, we consider the simplest case of the
perturbation in an eugenvalue.
Let ℋ be a Hilbert space with inner product <.,.>, and
: ( ) ⊂ ℋ → ℋ a linear operator in ℋ, with domain D(A), depending
smoothly on a real parameter . We assume that :
(a) is self adjoint, so that
(x,
y) = (
x,y) for all x, y ∈ D( )
(b)
has a smooth branch of simple eigenvalues ⋋ ∈ ℝ with eigenvectors
∈ ℋ, meaning that
=⋋ . (3.40)
We will compute the perturbation in the eigenvalue from its value at = 0
when is small but nonzero
A concrete example is the perturbation in the eigenvalues of a symmetric
matrix. In that case, we have ℋ= ℝn
with the Euclidean inner product
<x,y> = xT
y,
and
: ℝn
→ ℝn
is a linear transformation with and n × n symmetric
matrix (aij). The perturbation in the eigenvalues of a Hermitian matrix
corresponds to ℋ = ℂn
with inner product <x,y> = x-T
y. A we illustrate
below with the schrodinger equation of quantum mechanics, spectral
problems for differential equations can be formulated in terms of unbounded
operators acting in infinite – dimensional Hilbert spaces.
25. 23
We use the expansions.
= A0 + A1+…+ An+…
= x0 + x1+…+ xn+…
⋋ = ⋋0 + ⋋1+…+ ⋋n+…
in the eigenvalue problem (3.40), equate coefficients of , and rearrange
the result. We find that
(A0 - ⋋0I)x0 = 0, (3.41)
(A0 - ⋋0I)x1 = -A1x0 +⋋1x0, (3.42)
(A0 - ⋋0I)xn = ∑ {-Aixn-i+ ⋋ixn-i}. (3.43)
Assuming that x0 ≠ 0, equation (3.41) that ⋋0 is an eigenvalue of A0 and x0
is an eigenvector. Equation (3.42) is then a singular equation for x1. The
following proposition gives a simple, but fundamental, solvability condition
for this equation
Proposition 3.3.1
Suppose that A is a self – adjoint operator acting in a Hilbert space ℋ and
⋋ ∈ ℝ. If z ∈ ℋ, a necessary condition for the existence of a solution by
the y ∈ ℋ of the equation
(A-⋋I)y = z (3.44)
is that
(x, z) = 0,
For every eigenvector x of A with eigenvalue ⋋
Proof:-Suppose z ∈ ℋ and y ia a solution of (3.44). If x ia an eigenvector
of A, then using (3.44) and the self – adjointness of A-⋋ , we find that
<x,z> = (x,(A-⋋ )y)
= <(A-⋋ ) , >
= 0
26. 24
In many cases, the necessary solvability condition in this proposition is also
sufficient, and then we say trhat A - ⋋ satisfies the fredholm alternative; for
example, this is true in the finite dimensional case, or when A is an elliptic partial
differential operator.
Since A0 is self – adjoint and ⋋0 is a simple eigenvalue with eigenvector x0,
equation (3.44) it is solvable for x1 only if the right hand side is orthogonal to x0,
which implies that
⋋=
〈 , 〉
〈 , 〉
This equation gives the leading order perturbation in the eigenvalue, and is the
most important result of the expansion.
Assuming that the necessary solvability condition in the proposition is
sufficient, we can then solve (3.42) for x1. A solution for x1 is not unique, since
we can add to it an arbitrary scalar multiple of x0. This nonuniqueness is a
consequence of the fact that if is an eigenvector of , then is also a
solution for any scalar . If
= 1+ c1+O( 2
)
then
= x0 + (x1+c1x0) + O( 2
).
Thus, the addition of c1x0 to x1 corresponds to arescaling of the eigenvector by a
factor that is close to one.
This expansion can be continued to any order. The solvability condition for
(3.43) determines ⋋n, and the erquation may then be solved for xn, up to an
27. 25
arbitrary vector cnx0. The appearance of singular problems, and the need to impose
solvability conditions at each order which determine parameters in the expansion
and allow for the solution of higher order corrections, is a typical structure of many
perturbation problems.
28. 26
Bibliography
[1] John.K.Hunter., Asymptotic Analysis And Singular Perturbation Theory,
University of California,2004. .
[2] James.G.Sinunonds.James.E.Nlann Jr, A First Look at Perturbation Theroy,
second edition.
[3] J.Kevorkian, J.D.Cole, Perturbation Methods In Applied Math-ematics,1981.
[4] Ravi.P.agarwal, Donal.O.Regan, Ordinary And Partial Differential Equations.