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Chemical Engineering Hand Written_Notes_Part_56

Chemical Engineering Hand Written_Notes_Part_56 - 114 3...

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114 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES At the boundaries, we have u 0 ,j = u ; ( j = 0 , 1 , ...n x +1 ) (7.51) u 1 ,j = u ; ( j = 0 , 1 , ...n x +1 ) (7.52) u i, 0 = u ; ( i = 0 , 1 , ...n y +1 ) (7.53) k £ s ( n y +1) y ¤ T U ( i ) = h ( u u ( x i , 1)) (7.54) ( i = 1 , ...n y ) The above procedure yields ( n x +1) × ( n y + 1) nonlinear equations in ( n x +1) × ( n y + 1) unknowns, which can be solved simultaneously using Newton-Raphson method. Remark 3 . Are the two methods presented above, i.e. fi nite di ff erence and collocation methods, doing something fundamentally di ff erent? Suppose we choose n 0 th order polynomial (4.7), we are essentially approximating the true solution vector y ( z ) C ( 2 ) [ 0 , 1 ] by another vector (i.e. the polynomial function) in ( n + 2 ) dimensional subspace of C ( 2 ) [ 0 , 1 ] . If we choose n inter- nal grid points by fi nite di ff erence approach, then we are essentially fi nding a vector y in R n +2 that approximates y ( z ) . In fact, if we compare the Approach 2 presented above and the fi nite di ff erence method, the similarities are more apparent as the underlying ( n + 2
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