The approximation on the right hand side of Eq. (20) seems save for small values DS << 1.
where L ϕ i. j k is the approximation of the right-hand side of Equation (24) by the above spatial difference scheme.
where ϕ i. j k = ϕ i∇x, j∇y, k∇t with k ≥ 0 and L ϕ i. j k is the approximation of the right-hand side in Equation (22) by the above spatial difference scheme.
$$\end{document} where F m −1, i ≈ f(t m −1+ c i τ) denotes a given approximation of the right-hand side for 1≤ i≤ s and 1≤ m≤ M. Thus, at each time step, solving for the internal stages U m −1=(U m −1,1,…, U m −1, s )T yields the new approximation value u m = U m −1, s.
This discretization method is an approximation for the right hand side of the differential equation has the formula (D^{alpha}x ( t ) =f ( x ( t ) ) ), (t>0), (alphain ( 0,1 ) ).
We give the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) with the stable state (H_{1}^=(2.2682, -0.0559)) in the double and resonant eigenvalue case.
Figure 9 gives the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) in the double and non-resonant eigenvalue case.
We give the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) with the only stable state (R^= 0, 1.9391)) under double and resonant eigenvalue case.
Figure 10 gives the comparison between the numerical solution of system (1.3) (left) and the weakly nonlinear first order approximation of the solution (right) under double and non-resonant eigenvalue case.
There are no more dire indictments of ordinary human life than in the early work, "The American Scholar," where Emerson states that "Men in history, men in the world of to-day, are bugs, are spawn, and are called 'the mass' and 'the herd.' In a century, in a millennium, one or two men; that is to say, one or two approximations to the right state of every man" (CW1: 65).
