Chemical Engineering Hand Written_Notes_Part_53

Chemical - 108 3 LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 7 Solutions of ODE-BVP and PDEs by Orthogonal Collocation In this section

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108 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 7. Solutions of ODE-BVP and PDEs by Orthogonal Collocation In this section, we demonstrate how combination of Weierstrass theo- rem and Newton-Raphson method can be used to solve ODE boundary value problem and PDEs. Application of Weierstrass theorem facilitates conversion of ODE-BVP and certain class of PDEs to a set of nonlinear algebraic equations, which can be solved using Newton-Raphson method. E f ectively, the ODE-BVP / PDE is solved by forming a sequence of linear algebraic sub-problems. Consider ODE-BVP described by (7.1) Ψ [ d 2 y/dz 2 ,dy/dz,y,z ]=0 ; z (0 , 1) (7.2) f 1 [ dy/dz, y, z ]=0 at z =0 (7.3) f 2 [ dy/dz, y, z ]=0 at z =1 The true solution to problem is a function, say y ( z ) C ( 2 ) [ 0 , 1 ] , which be- longs to the set of twice di f erentiable continuous functions . According to the Weierstrass theorem, any continuous function over an interval can be approxi- mated with arbitrary accuracy using a polynomial function of appropriate de-
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical - 108 3 LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 7 Solutions of ODE-BVP and PDEs by Orthogonal Collocation In this section

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