Techniques for Diffusive Problems

Diffusive problems are characterized by second-order spatial derivatives with equations of the form 

To get the first-order time partial, we develop Taylor series expansions for the functions u(tj, r ) and u(tj+i, r ) about the Crank-Nicolson finite-difference point u(fj+i/2, a ). 

Following a similar derivation as presented in Section 8.1, we have 

The second-order correct second-spatial derivative analog is obtained by developing Taylor series expansions for u(ti+i/2,x 4.i) and u(ti+i/2,a i) about the point (tj+i/2)3 ), thus  

Dynamic Response of Tube Side to Step Disturbance 

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