Purchase Solution

Lipschitz Continuity and Initial Value Problems for ODE's

Not what you're looking for?

Ask Custom Question

PROBLEM 1. Find a Lipschitz constant, K, for the function f (u, t) = u^3 + t u^2 which shows that f is Lipschitz in u on the set 0 ? u ? 2, 0 ? t ? 1.

PROBLEM 2. Show that the function f (u, t) = t u^(1/2), is not Lipschitz in u on [0, 1] × [0, 2].

PROBLEM 3. Find two solutions to the initial value problem y = |y|^(1/2) , y(0) = 0. What hypothesis of the Picard-Lindelöf Theorem is violated?

Attachments
Purchase this Solution
Solution Summary

Lipschitz continuity and its role in the existence and uniqueness of solutions to ordinary differential equations is investigated. Three problems are solved. The first is to show that a given function is Lipschitz continuous. The second problem is to show that another given function is not Lipschitz continuous. The third problem shows that an initial value problem based on the function of the second problem fails to have unique solutions, in general. This shows that, without the Lipshitz condition, solutions to an initial value problem may fail to be unique.

Solution Preview

The Picard Lindelöf Theorem
__________________________

We need two basic facts for the three problems discussed below. The first is the definition of Lipschitz continuity that is required for the theorem that follows.

DEFINITION . Suppose the function f (y, t) is defined on [a, b] × [c, d] . Then it is Lipschitz continuous in y on this domain if

(1) |f (x, t) ? f (y, t)| ? ...

Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Probability Quiz

Some questions on probability

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

View More Free Quizzes
Related BrainMass Solutions

  • Existence and Uniqueness of Solution to an ODE

    (R)), f is continuous with respect to 1st variable and Lipschitz with respect to 2nd variable. Lipschitz Continuity: A function g:Iâ?'R is Lipschitz continuous if â??Î?>0 such that â?-g(â-x)-g(â-y)â?-â?¤Î?â?-â-x-â-yâ?-â??â-x,â-yâ??I.

  • Using the C++ Code to Obtain the Numerical Approx to ODE

    Using Euler's method is the simplest numerical method for approximation solving initial value ODE'S.

  • Existence and uniqueness of solutions for IVPs

    44056 Existence and uniqueness of solution What is the best statement that you can make about the existence and uniqueness of the solutions of the following initial value problems? (Please see attached).

  • Local or Uniform Lipschitz Constants

    50875 Local or Uniform Lipschitz Constants Determine if the following functions satisfy local or uniform Lipschitz condition. 1). te^y My work: I found d/dy (te^y) = te^y, and this is not bounded above for any value of y, so this made me conclude

  • Wave equation

    Please refer to these notes to maintain continuity for the solution needed. Thank you for your time and consideration in these matters. 1.)

  • Cauchy Problem: One-Parameter Family of Solution Curves

    (Remember, when solving for the constants in each of the ODE's, we set s=0 and let the initial curve be the initial value.) Then I will have to solve for s and t in terms of x and y. Then I plug that back into u(s,t) to get my answer: u(x,y).

  • Lipschitz functions

    29427 Lipschitz functions A function f:A->R is called Lipschitz if there exists a bound M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x, y belong to A.

  • Real analysis : Lipschitz Criterion

    30037 Real analysis : Lipschitz Criterion A function f:A->R is Lipschitz on A if there exists an M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x,y belong to A. show that if f is differentiable on a closed interval [a,b] and if f' is continous

  • Lipschitz functions

    To prove that f(t)∈ C_[a,b] we use triangle inequalities to construct inequality |f(t)-f(t^' )|≤|f(t)-f_n (t)|+|f_n (t)-f_n (t^' )|+|f_n (t^' )-f(t^' )| (a.1) to see that the condition of continuity can be satisfied for any t ∈[a,b] by taking

View More Related Solutions