Sentence examples for many nonlinear equations from inspiring English sources

Many nonlinear equations are naturally formulated as fixed point problems, x = T x, (1.1).

The proposed methods are straightforward, concise and effective, and can be applied to many nonlinear equations arise in applied sciences.

The solutions of many nonlinear equations can be expressed in the form [34] u ( x, t ) = { λ cos β , | ξ | ≤ π 2 μ, 0, otherwise, (2.4).

Many nonlinear equations are naturally formulated as fixed point problems, begin{aligned} x =Tx, end{aligned} (1.1 where T, the fixed point mapping, may be nonlinear.

2. The solutions of many nonlinear equations can be expressed in the form [34] u ( x, t ) = { λ cos β , | ξ | ≤ π 2 μ, 0, otherwise, (2.4)  .

An interesting fact is that many nonlinear equations and systems are related to linear ones although they are of relatively complex forms, which usually do not suggest it on the first site.

(1.9) is a class of most important auxiliary equations because many nonlinear evolution equations can be converted to this equation making use of the traveling wave reduction.

The complex method is a very important tool in finding the exact solutions of nonlinear evolution equation, and Eq. (3) is one of the most important auxiliary equations, because many nonlinear evolution equations can be converted to it.

The simply periodic solutions w 1 s ( z ) = − 6 A C α 2 coth 2 α 2 ( z − z 0 ) − A 2 C α 2 − B 2 C, where 4 D C = − A 2 α 4 + B 2, z 0 ∈ C. The rational function solutions w 1 r ( z ) = − 6 A C ( z − z 0 ) 2 − B 2 C, where 4 C D = B 2, z 0 ∈ C. Equation (1) is an important auxiliary equation, because many nonlinear evolution equations can be converted to Eq. (1) using the traveling wave reduction.

There exist many nonlinear wave equations in fluid mechanics, such as the KdV equation, the Boussinesq equation, the (2+1 -dimensional dispersive long wave equation in shallow water, etc.

In this manuscript, we consider an initial-boundary-value problem governed by a (1+1 -dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics.