In Science and engineering, so many nonlinear phenomena are presented as partial differential, ordinary differential and integral equation models.
Nonlinear partial differential equations of integer order play an important role in describing many nonlinear phenomena such as mathematical biology, electromagnetic theory, fluid mechanics, signal processing, engineering, solid state physics, and other fields of science.
The class of problems (1.1) appears in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mappings [1 3], in the generalized reaction-diffusion theory [4], in the turbulent flow of a gas in a porous medium, and in the non-Newtonian fluid theory [5].
Note that the upper-hybrid resonance altitude listed in the last line of Table 1 was included in the calculation interval for the O-mode waves, as many nonlinear phenomena, such as field-aligned irregularities, optical emissions, and stimulated electromagnetic emissions, are influenced by the upper-hybrid resonance.
The class of problems (1.1) appears in many nonlinear phenomena, for instance, in the theory of quasiregular and quasiconformal mappings [1 3], in the generalized reaction-diffusion theory [4], in the turbulent flow of a gas in porous medium and in the non-Newtonian fluid theory [5].
The generalized KP (GKP) equations with an arbitrary nonlinear term model and characterize many nonlinear physical phenomena.
Many nonlinear diffusion phenomena are described by the cooperation and interaction between the nonlinear source term and absorption term during the diffusion.
The fractional calculus has already become a powerful tool which describes many nonlinear complex phenomena arising in fluid mechanics, thermodynamics, plasma dynamics, continuum mechanics, quantum mechanics, electrodynamics and biological systems [1, 2].
The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations.
Recently, the fractional calculus has been used successfully to describe many complex nonlinear phenomena and dynamic processes in physics, engineering, electromagnetics, acoustics, viscoelasticity, electrochemistry, material science, etc. [24 40].
Nature is full of many nonlinear and random phenomena.
