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# lecture34 - Lecture 34 Finite Dierence Method Nonlinear ODE...

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Lecture 34 Finite Difference Method – Nonlinear ODE Heat conduction with radiation If we again consider the heat in a metal bar of length L , but this time consider the effect of radiation as well as conduction, then the steady state equation has the form u xx - d ( u 4 - u 4 b ) = - g ( x ) , (34.1) where u b is the temperature of the background, d incorporates a coefficient of radiation and g ( x ) is the heat source. If we again replace the continuous problem by its discrete approximation then we get u i +1 - 2 u i + u i - 1 h 2 - d ( u 4 i - u 4 b ) = - g i = - g ( x i ) . (34.2) This equation is nonlinear in the unknowns, thus we no longer have a system of linear equations to solve, but a system of nonlinear equations. One way to solve these equations would be by the multivariable Newton method. Instead, we introduce another interative method. Relaxation Method for Nonlinear Finite Differences We can rewrite equation (34.2) as u i +1 - 2 u i + u i - 1 = h 2 d ( u 4 i - u 4 b ) - h 2 g i .

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lecture34 - Lecture 34 Finite Dierence Method Nonlinear ODE...

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