Post by matthorr on Oct 7, 2015 19:44:08 GMT
The representative equation is \( u'_n = \lambda u_n+ae^{\mu hn}\). Applied to the predictor
\[\tilde{u}_{n+\frac{1}{2}} = u_n+\frac{1}{2}h\left(\lambda u_n+ae^{\mu hn}\right) \]
The representative equation for the derivative of the predictor is
\[\tilde{u}'_{n+\frac{1}{2}} = \lambda \tilde{u}_{n+\frac{1}{2}}+ae^{\mu h(n+\frac{1}{2}})\]
applied to the corrector
\[u_{n+1}=u_n+h\left[\lambda\bigg(u_n+\frac{1}{2}h\left(\lambda u_n+ae^{\mu hn}\right) \bigg)+ae^{\mu h(n+\frac{1}{2})}\right]\]
Rearranging terms and applying the displacement operator
\[\left[E-\left(1+\lambda h+\frac{1}{2}\lambda^2h^2\right)\right]u_n = \left[hE^{\frac{1}{2}}+\frac{1}{2}\lambda h^2\right]ae^{\mu hn}\]
The characteristic polynomial
\[P(E) = E-\left(1+\lambda h+\frac{1}{2}\lambda^2h^2\right)\]
whose root gives
\[\sigma_1 = 1+\lambda h+\frac{1}{2}\lambda^2h^2 \]
The particular polynomial
\[Q(E) = hE^{\frac{1}{2}}+\frac{1}{2}\lambda h^2\]
The transient solution is given by \( u_{n_t} = c_1(\sigma_1)^n \)
\[u_{n_t} = c_1(1+\lambda h+\frac{1}{2}\lambda^2h^2)^n\]
The particular solution is given by \( u_{n_p} = ae^{\mu hn}\frac{Q(e^{\mu h})}{P(e^{\mu h})} \)
\[u_{n_p}=ae^{\mu hn}\frac{he^{\frac{\mu h}{2}}+\frac{1}{2}\lambda h^2}{e^{\mu h}-1-\lambda h-\frac{1}{2}\lambda^2h^2}\]
\[\tilde{u}_{n+\frac{1}{2}} = u_n+\frac{1}{2}h\left(\lambda u_n+ae^{\mu hn}\right) \]
The representative equation for the derivative of the predictor is
\[\tilde{u}'_{n+\frac{1}{2}} = \lambda \tilde{u}_{n+\frac{1}{2}}+ae^{\mu h(n+\frac{1}{2}})\]
applied to the corrector
\[u_{n+1}=u_n+h\left[\lambda\bigg(u_n+\frac{1}{2}h\left(\lambda u_n+ae^{\mu hn}\right) \bigg)+ae^{\mu h(n+\frac{1}{2})}\right]\]
Rearranging terms and applying the displacement operator
\[\left[E-\left(1+\lambda h+\frac{1}{2}\lambda^2h^2\right)\right]u_n = \left[hE^{\frac{1}{2}}+\frac{1}{2}\lambda h^2\right]ae^{\mu hn}\]
The characteristic polynomial
\[P(E) = E-\left(1+\lambda h+\frac{1}{2}\lambda^2h^2\right)\]
whose root gives
\[\sigma_1 = 1+\lambda h+\frac{1}{2}\lambda^2h^2 \]
The particular polynomial
\[Q(E) = hE^{\frac{1}{2}}+\frac{1}{2}\lambda h^2\]
The transient solution is given by \( u_{n_t} = c_1(\sigma_1)^n \)
\[u_{n_t} = c_1(1+\lambda h+\frac{1}{2}\lambda^2h^2)^n\]
The particular solution is given by \( u_{n_p} = ae^{\mu hn}\frac{Q(e^{\mu h})}{P(e^{\mu h})} \)
\[u_{n_p}=ae^{\mu hn}\frac{he^{\frac{\mu h}{2}}+\frac{1}{2}\lambda h^2}{e^{\mu h}-1-\lambda h-\frac{1}{2}\lambda^2h^2}\]