Chemical Engineering Hand Written_Notes_Part_54

Chemical Engineering Hand Written_Notes_Part_54 - 110 3....

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110 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES where (7.12) θ =[ θ 0 ........... θ n +1 ] T Substitute for y , y 0 and y 00 in equation (3.1) and enforcing residual to be equal to zero at the grid points, we get (7.13) Ψ [ y 00 i ( θ ) ,y 0 i ( θ ) ,y i ( θ ) ,z i ]=0 i =1 , ...... n Similarly, enforcing residuals at boundary points equal to zero yields (7.14) f 1 [ y 0 0 ( θ ) ,y 0 ( θ ) , 0] = 0 (7.15) f 2 [[ y 0 1 ( θ ) ,y 1 ( θ ) , 1] = 0 Thus, we have n +2 nonlinear equations in n +2 unknowns, which can be solved simultaneously using Newton-Raphson method for estimating θ . Approach 2 The above approach is not convenient from computational viewpoint. Note that we have to select an initial guess for vector θ to start the Newton-Raphson method. Now, unlike linear algebraic equations, a set of nonlinear equations can have multiple solutions and the solution we reach by applying Newton Raphson method is governed by the choice of the initial guess vector θ (0) . Instead of
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Chemical Engineering Hand Written_Notes_Part_54 - 110 3....

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