With no general solution we need an indirect approach Technique uses

With no general solution we need an indirect approach

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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) 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)
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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) 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!
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