BoundaryValue Problems for ODE BoundaryValue Problems for ODE

Boundary-Value Problems for ODE ( )בעיות הגבול

Boundary-Value Problems for ODE (The Uniqueness Theorem) Theorem: Suppose the function f in the boundary-value problem satisfies the following conditions Then the boundary-value problem has a unique solution.

The Linear Shooting Method Theorem: If the linear boundary-value problem satisfies the following conditions Then the problem has a unique solution.

The Linear Shooting Method (cont. ) To solve the linear boundary-value problem we solve two initial-value problems: Then the solution to the boundary-value problem is:

The Linear Shooting Method (cont. ) We illustrate this method graphically:

The Shooting Method for Nonlinear Problems The solution to a nonlinear problem cannot be expressed as a linear combination of the solution to initial-value problem. Instead we need to use the solutions to a sequence of initial -value problems of the form: We choose , so that

The Shooting Method for Nonlinear Problems denotes the solution to the initial-value problem. Shooting begins with:

The Shooting Method for Nonlinear Problems and continues with: denotes the solution to the initial-value problem, then the problem is to determine t so that

The Shooting Method for Nonlinear Problems is a nonlinear equation! The Secant method ( )מיתר : The Newton-Raphson method: ?

The Shooting Method for Nonlinear Problems (Newton-Raphson) We rewrite the initial-value problem, emphasizing that the solution depends on both x and t as Since we need with respect to t we take the partial derivative =0 and…

The Shooting Method for Nonlinear Problems (Newton-Raphson) …and: By defining and reversing the order of differentiation of x and t, we have This is the initial-value problem for z(x, t).

The Shooting Method for Nonlinear Problems (Newton-Raphson) Finally, NR shooting method implies solving two initialvalue problems: If the problem satisfies the uniqueness theorem, any choice of will give convergence.