With no general solution we need an indirect approach Technique uses

With no general solution we need an indirect approach

• 23

This preview shows page 6 - 12 out of 23 pages.

With no general solution we need an indirect approach Technique uses convergence of a sequence of functions with methods from advanced calculus Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (6/23)

Subscribe to view the full document.

Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Existence and Uniqueness A change of coordinates allows us to consider y 0 = f ( t, y ) , with y (0) = 0 (1) Theorem If f and ∂f/∂y are continuous in a rectangle R : | t | ≤ a, | y | ≤ b , then there is some interval | t | ≤ h ≤ | a | in which there exists a unique solution y = φ ( t ) of the initial value problem (1). Motivation: Suppose that there is a function y = φ ( t ) that satisfies (1). Integrating, φ ( t ) must satisfy φ ( t ) = Z t t 0 f ( s, φ ( s )) ds, (2) which is an integral equation . A solution to (1) is equivalent (2). Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (7/23)
Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Picard Iteration 1 Show a solution to the integral equation using the Method of Successive Approximations or Picard’s Iteration Method Start with an initial function, φ 0 = 0 (satisfying initial condition) φ 1 ( t ) = Z t 0 f ( s, φ 0 ( s )) ds Successively obtain φ 2 ( t ) = Z t 0 f ( s, φ 1 ( s )) ds . . . φ n +1 ( t ) = Z t 0 f ( s, φ n ( s )) ds Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (8/23)

Subscribe to view the full document.

Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Picard Iteration 2 The Picard’s Iteration generates a sequence, so to prove the theorem we must demonstrate 1 Do all members of the sequence exist? 2 Does the sequence converge? 3 What are the properties of the limit function? Does it satisfy the integral equation 4 Is this the only solution? ( Uniqueness ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (9/23)
Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Picard Iteration - Example 1 Consider the initial value problem (IVP) y 0 = 2 t (1 + y ) , with y (0) = 0 , and apply the Method of Successive Approximations Let φ 0 = 0, then φ 1 ( t ) = Z t 0 2 s (1 + φ 0 ( s )) ds = t 2 Next φ 2 ( t ) = Z t 0 2 s (1 + φ 1 ( s )) ds = Z t 0 2 s (1 + s 2 ) ds = t 2 + t 4 2 Next φ 3 ( t ) = Z t 0 2 s (1 + φ 2 ( s )) ds = t 2 + t 4 2 + t 6 2 · 3 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (10/23)

Subscribe to view the full document.

Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Picard Iteration - Example 2 The integrations above suggest φ n ( t ) = t 2 + t 4 2!
• Fall '08
• staff

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern