t 6 3 t 2 n n By math induction assume true for n k φ k 1 t Z t 2 s 1 φ k s ds

T 6 3 t 2 n n by math induction assume true for n k

This preview shows page 3 - 5 out of 6 pages.

+ t 6 3! + ... + t 2 n n ! , By math induction, assume true for n = k φ k +1 ( t ) = Z t 0 2 s (1 + φ k ( s )) ds = Z t 0 2 s (1 + s 2 + ... + s 2 k k ! ) ds = t 2 + t 4 2! + t 6 3! + ... + t 2 k +2 ( k + 1)! which is what we needed to show The limit exists if the series converges or lim n →∞ φ n ( t ) exists Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (11/23) Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Picard Iteration - Example 3 Apply the Ratio test lim k →∞ t 2 k +2 ( k + 1)! k ! t 2 k = t 2 k + 1 0 which shows this series converges for all t Since this is a Taylor’s series, it can be integrated and differentiated in its interval of convergence. Thus, it is a solution of the integral equation Note that this is the Taylor’s series for φ ( t ) = e t 2 - 1, which can be shown to satisfy the IVP Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (12/23)
Image of page 3

Subscribe to view the full document.

Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Picard Iteration - Example 4 First 4 Picard Iterates -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 t φ ( t ) Picard Iterates φ 1 φ 2 φ 3 φ 4 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (13/23) Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Example - Uniqueness 1 Example - Uniqueness - Suppose there are two solutions, φ ( t ) and ψ ( t ) satisfying the integral equation φ ( t ) - ψ ( t ) = Z t 0 2 s ( φ ( s ) - ψ ( s )) ds Take absolute values and restrict 0 t A/ 2 ( A arbitrary). then | φ ( t ) - ψ ( t ) | = Z t 0 2 s ( φ ( s ) - ψ ( s )) ds Z t 0 2 s | φ ( s ) - ψ ( s ) | ds A Z t 0 | φ ( s ) - ψ ( s ) | ds for 0 t A/ 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (14/23) Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Example - Uniqueness 2 Let U ( t ) = R t 0 | φ ( s ) - ψ ( s ) | ds , then U (0) = 0 and U ( t ) 0 for t 0 U ( t ) is differentiable with U 0 ( t ) = | φ ( t ) - ψ ( t ) | We have the differential inequality U 0 ( t ) - AU ( t ) 0 , 0 t A/ 2 Multiplying by positive function e - At , then integrating gives d dt ( e - At U ( t ) ) 0 , 0 t A/ 2 , e - At U ( t ) 0 , 0 t A/ 2 Hence, U ( t ) 0 with A arbitrary. It follows that U ( t ) 0 or φ ( t ) = ψ ( t ) for each t , so the functions are the same, giving uniqueness Joseph M. Mahaffy, h [email protected] i Lecture Notes – Existence and Uniqueness — (15/23) Introduction Linear Differential Equation Nonlinear Differential Equation Existence and Uniqueness Picard Iteration Uniqueness Examples Existence and Uniqueness Theorem 1 We leave the details of the proof of the Existence and Uniqueness Theorem to the interested reader, but give a sketch of the key steps 1 Restrict the time interval | t | ≤ h a Since f is continuous in the the rectangle R : | t | ≤ a, | y | ≤ b , the function f is bounded on R , so there exists M such that | f ( t, y ) | ≤ M ( t, y ) R Let h = min ( a, b M )
Image of page 4
Image of page 5
  • Fall '08
  • staff

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes