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.
Algorithm Nonlinear Shooting
Algorithm Nonlinear Shooting (cont. )
Algorithm Nonlinear Shooting (cont. )
Algorithm Nonlinear Shooting (cont. )
Algorithm Nonlinear Shooting (example) Consider the boundary value problem The algorithm requires solving two initial value problems
Algorithm Nonlinear Shooting (example) If the iteration stops for this problem requires four iterations and Numerical exact and results:
Thomas Algorithm
Thomas Algorithm (cont. )
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