Chemical Engineering Hand Written_Notes_Part_68

Chemical - 138 4 ODE-IVPS AND RELATED NUMERICAL SCHEMES Here the matrix et is limit of innite sum(2.21 1 et = I t 2 t2 2 t e1 0 0 0 2 t 0 e = 0 0 0

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138 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES Here, the matrix e Λ t is limit of in f nite sum (2.21) e Λ t = I + t Λ + 1 2! t 2 Λ 2 + ... = e λ 1 t 0 ... 0 0 e λ 2 t ... 0 ... ... ... ... 00 0 e λ m t Thus, equation (2.17) reduces to (2.22) x ( t )= Ψ e Λ t Ψ 1 x (0) With this de f nition, the solution to the ODE-IVP can be written as (2.23) x ( t )= Ψ e Λ t Ψ 1 x (0) = e At x (0) 2.3. Asymptotic behavior of solutions. Inthecaseofl inearmu lt ivar i- able ODE-IVP problems, it is possible to analyze asymptotic behavior of the solution by observing eigenvalues of matrix A . (2.24) x ( t )= c 1 e λ 1 t v (1) + c 2 e λ 2 t v (2) + ....... + c m e λ m t v ( m ) C = Ψ 1 x (0) Let λ j = α j + j represent j’th eigenvalue of matrix A. Then, we can write (2.25) e λ j t = e α j t .e j t = e α j t [cos β j t + i sin β j t ] As (2.26) ¯ ¯ [cos β j t + i sin β j t ] ¯ ¯ 1 for all t
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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 - 138 4 ODE-IVPS AND RELATED NUMERICAL SCHEMES Here the matrix et is limit of innite sum(2.21 1 et = I t 2 t2 2 t e1 0 0 0 2 t 0 e = 0 0 0

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