Ordinary Differential Equations

Spring 2009, Wednesdays 5:00-7:00 pm, WWH Room 102

Instructor: Olof B. Widlund

  • Coordinates
    Office: WWH 612
    Telephone: 998-3110
    Office Hours: Drop by any time, or send email or call for an appointment
    Email: widlund at cims.nyu.edu
    Course URL: http://math.cims.nyu.edu/courses/spring09/G63.2470.001/index.html

  • Announcements
  • Class will meet every Wednesday except on March 18. The last class will be on April 29 and the in-class final on May 6 starting at 5:00 pm. Here is a Practice Final.

  • Blackboard
    A Blackboard account has been activated and you can use it to keep track of homework scores. I will also use it to send e-mail occasionally. The assignments will only be available, in pdf format, via this course home page.

  • Requirements
    There will be regular homework assignments. It is important that you do the homework yourself, but when you get stuck, I encourage you to consult with other students, or me, to get help when necessary. However, when you get help, it's important to acknowledge it in writing. Passing off other people's work as your own is not acceptable.

  • Homework
  • Lectures These are short descriptions of the content of each lecture, after the fact.
    1. January 21. Initial value problems for first order ordinary differential equations with one dependent variable x'=f(t,x). If f(t,x) is continuous, there exists at least one solution locally. x' = x^2 provides an example of an ODE with a solution which cannot be continued for all values of t. x'=x^{1/2} and x'=x^{1/3} illustrate that there can be more than one solution. Uniqueness is guaranteed if f(t,x) is Lipschitz continuous with respect to the second variable. Picard iteration and an existence proof for f(t,x) which satisfies the Lipschitz condition. An example, with f(t,x) not Lipschitz and for which the Picard iteration does not converge. Several integration techniques: (i) f(t,x) a function of t only; (ii) separation of variables; (iii) homogeneous equations.
    2. January 28. An existence proof for f(t,x), which are not Lipschitz; approximate this function by a family of polynomials in t and x and show, by using Arzela's theorem (given in the C & L book) that when the polynomial approximation converges to f(t,x), then a family of solutions converges to a solution of x'=f(t,x). Another proof of the same result in given in the C & L book; is uses a finite difference approach. Euler' method, a method using Taylor series, and two other finite difference methods to solve the ODEs numerically. We can obtain better than linear convergence in the time step if we use the more advanced methods. Turning higher order ODEs into first order systems of ODE. There are no surprises as far as existence and uniqueness are concerned. Continuing solutions of an ODE if two solutions are defined on different intervals and the two solutions are the same on a common interval of the t-axis. A nonlinear spring problem for which we can show existence for all times by establishing a bound on x and x', which is valid for all t. Solutions of ODE can cease to exist only if |x(t)| goes to infinity. Additional solution techniques: (iv) solving x'(t)=f((a_1 t + b_1 x + c_1)/a_2 t + b_2 x + c_2)); (v) using the transformation x=y^m; (vi) linear equations by "variation of the constant".
    3. February 4. First order systems of ODE with initial values and right hand sides which depend smoothly on a number of parameters. A method of perturbation which allow us to compute a linear approximation of solution of such a system valid for small values of the parameters. The cost is the solution of the unperturbed problem and a set of linear systems of ODE. A proof of the continuity of the solutions as a function of the parameters. Under suitable assumptions, we can also show that we have continuous derivatives with respect to the parameters. This latter result can be used to give a full justification of the perturbation method. Linear systems of ODE. Their solution sets define a fundamental solution matrix and these sets are also vector spaces of the same dimension as the systems. The solution matrices can be used to extend the variation of constant method previously developed for scalar problems. The Wronskian of a linear system; it satisfies a scalar linear ODE with a coefficient obtained from the trace of the matrix of the linear system. The form of the Wronskian if the system originates from a scalar ODE of higher order. How to use information of special solutions and the Wronskian to obtain other columns of the fundamental solution matrix.
    4. February 11. How to turn Bernoulli equations into linear ODEs. Ricatti equations and how they can be turned into Bernoulli equations if we know on particular solution. Euler's equations and how an exponential transformation of the independent variable turns them into linear equations with constant coefficients. Finding information on two linearly independent solutions of a particular linear ODE of second order by first developing one solution as a power series and then, by using the Wronskian, showing that the second solution has a logarithmic singularity. Reducing the size of a linear system of equations if one solution is known; to be revisited. Linear systems of ODE's with constant coefficients. The use of eigenvalues and eigenvectors of the coefficient matrix. The deficient case when we can use Jordan's canonical form. Solving the equations when there are "forcing terms"; to be revisited. The exponential of matrices defined in terms of convergent power series. The exponential of a Jordan box. Solving complete differentials; results in an algebraic equation satisfied by the solution. How to turn other problems into complete differentials by multiplying by a common factor.
    5. February 18. Reducing the construction of the fundamental solution matrix for an n-by-n linear systems to an (n-1)-by-(n-1) problem if one solution is known. Details about how a related transformation can be computed. An attempt to solve a 3-by-3 problem with two known solutions; more rapid progress seems to be possible using the Wronskian. Linear system with constant coefficients. The real Jordan form. Finding the solutions for special "forcing functions"; different cases depending on the size of the Jordan blocks, etc. Liapunov's construction of a norm, equivalent to the Euclidean norm, and a proof that all solutions of a linear system with constant coefficients approach the zero solution even in the presence of a small nonlinear perturbation provided that all eigenvalues of the matrix have strictly negative real parts and the initial values are small.
    6. February 25. Why is it interesting to study the effect of linear or nonlinear perturbations of linear systems of ODE? We can linearize many nonlinear problems around a solution of the linear problem and ask interesting questions about the stability of the linear systems. This can be done by using Taylor expansions. Simple examples which illustrates the importance of the linearized problem to determine the behavior for large values of the independent variable t. The Schur normal form as an alternative to the Jordan normal form. Linear problems with periodic coefficients; the fundamental matrix of solutions has a special from. It is a product of a matrix with periodic coefficients and an exponential factor given by a constant matrix. This matrix determines the behavior for large t. Linear perturbations of the coefficient matrix by a matrix B(t) and the effect on the such perturbations if the unperturbed problem has constant or periodic coefficients.
    7. March 4. How to diagonalize the matrix of linear system of ODE; it can be done successfully, if the matrix is smooth enough and has distinct eigenvalues. An extra term appears in the system written in the transformed variables and this affects the formulation of a result which establishes asymptotic stability of a perturbed problem given that the unperturbed problem is asymptotically stable. A counter example to a proposed general stability result on asymptotic stability under the assumption that the norm of the perturbation matrix is integrable. Adding a lower finite bound on the integral of the trace of the coefficient matrix assures that the inverse of the fundamental matrix of solutions will remain bounded; a computation of the Wronskian is involved. Autonomous systems; they can be converted it a system, which normally is not autonomous, and which has one equation less. Any system can, at the expense of increasing its order by 1 be converted into an autonomous system. Singular points and singular solutions: system at rest. Periodic solution; they have well defined period and can be mapped one-to-one and continuously onto the unit circle. Other solutions; a third option.
    8. March 11. Linear autonomous systems in the 2-by-2 case; there are six cases. Adding nonlinear perturbations; the type can change unless we restrict the non-linearity to vanish faster than o(r). An autonomous system with solutions on a torus and which is very sensitive to perturbations of the initial data; problems with periodic solutions can change to orbits on the surface of the torus which come arbitrarily close to any point.
    9. March 25. Further comments on the 3-by-3 autonomous system. The Poincare- Bendixson theorem.
    10. April 1. Boundary value problems for systems of ordinary differential equations; in the nonlinear case there might be no solutions or several solutions even if the system is defined by very regular functions. Scalar second order problems on a finite interval. The adjoint of the operator and conditions for self-adjointness. The Green's function, its properties and an explicit construction; it provides a representation of the solution of inhomogeneous problems in integral form. Eigenvalue problems and a determinant condition; the determinant is an entire function of the eigenvalue parameter which gives important insight. For selfadjoint problems, the eigenfunctions of different eigenvectors are orthogonal and the eigenvalues are real. Using the integral operator to establish the existence of eigenfunctions; the compactness of that operator plays a crucial role in establishing the smoothness of elements in its range. An outline of how the first eigenfunction is found and how we then can proceed to find the next, etc.
    11. April 8. Further comments on eigensystems of Sturm-Liouville problems. Completeness of expansions using eigenfunctions. Separation and comparison theorems due to Sturm. Orthogonal polynomials. Orthogonality with respect to a non-negative weight function. Examples of well-known sets of orthogonal polynomials; Legendre, Chebyshev, Jacobi, Laguerre, and Hermite. The Legendre ODE. Three term recurrences to compute orthogonal polynomials. Oscillations and zeros of orthogonal polynomials.
    12. April 15. Gaussian quadrature. Power series solutions of second order scalar ODE in the regular case. A proof that the power series solutions have the same radius of convergence as the series that define the coefficients. Linear systems of equations defined by a single-valued matrix function with complex-valued elements defined in a simply connected domain. The case of an isolated singularity at the origin. The form of the solution; it can be analytic everywhere, it can have a branch point or a pole, or even have an essential singularity. Using the Jordan form of a matrix to determine a fundamental matrix of solutions.
    13. April 22. Linear systems with a singularity of the first kind at the origin of the complex plane; solutions have at most a pole at the origin. Construction of solutions in terms of power series and powers of log(z). When we have a singularity of the first kind, more can be established of the form of the solution, in particular, we can at times prove that z^{A_0} is a factor in the fundamental matrix of solutions. Equations with a singularity of the first kind only at infinity; the proper definition of such a singularity by using a simple change of variables. This assumption restricts the coefficients considerably. Remarks on cases when we have one of two additional singularities of the first kind.
    14. April 29. The coefficients of a second order ordinary differential equation with two independent power series solutions can be calculated and it can be shown that the singularity is of the first kind. Bessel functions; finding two independent solutions to Bessel's equation. The presence of a logarithmic factor in the case of integer values of the parameter. Second order linear equations with coefficients analytic everywhere including infinity: there are none. Equations with only one singularity of the first kind at infinity or at finite point in the complex plane: this is a very special problem. Problems with two singularities of the first kind; the coefficients have to be of a special form and the problem can be reduced to a second order differential equation with constant coefficients. The case with three singular points, Riemann's differential equation, and hypergeometric series. Comments on special cases which gives us Jacobi polynomials.
  • Required Text
    Coddington and Levinson, Theory of Ordinary Differential Equations.

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