A two-parameter continuation algorithm is implemented here to trace the solution curve of the CNLSE.
Similarly, the solution of a differential equation is obtained as a sequence of discrete points determined by approximating the true solution curve by a sequence of tangential line segments.
When ε varies from a fixed ε=ε0, bifurcation occurs to the solution curve (λ(s), u(s)).
The solution curve with respect to some parameter of the NAEP is then followed by the proposed method.
Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve.
Similarly, the solution of a differential equation is obtained as a sequence of discrete points determined, in simplistic terms, by approximating the true solution curve by a sequence of tangential line segments.
The equation y′ = f (x,y) gives a direction, y′, associated with each point (x,y) in the plane that must be satisfied by any solution curve passing through that point.
The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve.
The weight function in the integral of adaptation is constructed, so that the grid lines are accumulated near strong gradients and curvatures of the solution curve.
For spherical flame propagation, the trajectory is shown to change significantly with the fuel Lewis number and a C-shaped solution curve of flame propagation speed as a function of flame radius is observed for large fuel Lewis numbers.
First, this paper investigates, for a tokamak design with large poloidal field-shaping coil (PFC) to plasma distance, the bifurcated numerical solution curve as a function of flux loop position and relates this curve to the practical existence of ideal magnetohydrodynamic equilibria and practical tokamak design.
