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# nonlins - Introduction Solution of a nonlinear equation...

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Introduction Solution of a nonlinear equation I Goal: solve an equation of the form (find x ) f ( x ) = 0 (1) I Approximations I Graphical Representation I Bisection Method I Newton Raphson Method

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Example Used Throughout I Given σ, h , g , find k such that σ 2 = gk tanh kh I σ = 2 π T is the wave frequency I h is the water depth I g gravitational acceleration
Approximations Non-dimensionalization I Normalize the problem: σ 2 h g = kh tanh kh I Set ω = σ 2 h g and x = kh , and solve: x tanh x = ω I Equation becomes f ( x ) = x tanh x - ω = 0 I Intersection of f 1 ( x ) = ω x and f 2 ( x ) = tanh x 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 f ( x ) = 3 . 5 x f ( x ) = x f ( x ) = 0 . 1 x f ( x ) = 1 x f ( x ) = tanh x x

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Approximations Solution by approximations I If x 1, tanh x 1 and so x ω I If x 1, tanh x and so x ω I What to do if ω 1 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 f ( x ) = 3 . 5 x f ( x ) = x f ( x ) = 0 . 1 x f ( x ) = 1 x f ( x ) = tanh x x
Graphical Representation Graphical Representation I Draw graph of the function f ( x ) as a function of x . I Locate where the function crosses the y = 0 axis I Approximate the root(s) graphically

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Bisection Method Bisection Algorithm Bisection Method I Rewrite equation as f ( x ) = x tanh x - ω = 0 I f changes sign somewhere between a and b : f ( a ) f ( b ) < 0 I If f ( x ) is a continuous function of x then a α b I
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• Fall '08
• Staff
• Bisection Method, newton-raphson method, Bisection Properties, Bisection Method Bisection, Method Newton-Raphson Example

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