logistic

logistic - Numerical Solution Sudarshan Balakrishnan August...

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Numerical Solution Sudarshan Balakrishnan August 24, 2011 Logistic Equation Given the logistic equation u 0 = u ( u - 1) we notice that the associated operators were A ( u ) = - u and B ( u ) = u 2 [1].The exact solutions associated with A , B and C = A + B are given by: Φ c ( t ) u 0 = u 0 u 0 + e t (1 - u 0 ) , Φ A ( t ) u 0 = u 0 e - t , Φ B ( t ) u 0 = u 0 1 - u 0 t . (1) Splitting Algorithm for Linear Ordinary Differential Equations In order to understand the splitting associated with the logistic equation, let us first consider the linear ODE given by: du dt = Au, du dt = Bu, du dt = Cu. (2) In (2), we have that C = A + B . Solving the each of them we notice that the solution for a given time l , using the separation of variables is: ϕ l ( u ) = e Al e Bl . (3) Using the solution, we can estimate the local error ρ ( u ; l ) by: ρ ( u ; l ) = ϕ l ( u ) - e Cl . (4) In order to estimate (4), we have that: ρ ( u ; l ) = ( I + Cl + 1 2 C 2 l 2 + ... ) - ( I + Al + 1 2 A 2 l 2 + ... )( I + Bl + 1 2 B 2 l
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logistic - Numerical Solution Sudarshan Balakrishnan August...

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