Taking α = ( P + P 2 + 4 Q ) / 2 and β = ( P − P 2 + 4 Q ) / 2 to be the roots of the characteristic equation x 2 − P x − Q = 0, we have the well-known expressions named Binet's formulas U n ( P, Q ) = ( α n − β n ) / and V n ( P, Q ) = α n + β n (1.2). for all n ∈ Z. Instead of U n ( P, Q ) and V n ( P, Q ), we use U n and V n, respectively.
Therefore, consider u : t ↦ u ( μ t ) instead of u.
We will write α̃ instead of (( s,t ) ) and β̃ instead of (( u,v ) ).
Here, we write u n iT instead of ( u n i ) T to lighten the notation.
Instead of u i,j k we can apply the gradient of the GREEN-LAGRANGE strain, E i j,k.
Due to the lack of regularity of u in time, we also use the Steklov average of u instead of u itself.
Meanwhile, since u ( x, t ) is not required to be nonnegative, we use | u | q − 1 u instead of u q in equation (1.1).
Hence, taking ku instead of u, for any k>0, we have ∫ Ω ∪ Γ u 1 p − k p u p dx ≤ 0, Open image in new window.
Using (tau=U-D) instead of U improves accuracy in regions of low energy and enables one to consistently recover the Newtonian limit.
Recently, Zhao and Zhou [44] discussed the symbolic analysis and exact traveling wave solutions of a modified Novikov equation, which is new in that it has a nonlinear term u 4 u x instead of u 2 u x.
For the simplicity of notation, we write u instead of u i and u k for |u|k- 1u where k > 0. Also, let C, C j be generic constants independent of k, i, n changeable from line to line.
