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

This preview shows pages 1–2. Sign up to view the full content.

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-

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online