Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single.

Spectral theory of first-order systems: From crystals to Dirac operators

Let [Formula: see text] be a constant coefficient first-order partial differential system, where the matrices [Formula: see text] are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the non-homogeneous case it is assumed that the operator is isotropic. The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: • Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular, the Dirac and Maxwell systems are explicitly treated. • The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation.

Mass transfer and Cattaneo-Christov heat flux for a chemically reacting nanofluid in a porous medium between two rotary disks

This paper examines the transport of a chemically reacting nanofluid in a porous medium between two rotary disks with Cattaneo-Christov?s heat flux. The non-linear ordinary differential system formed under Vonn Karman transformation of a non-linear partial differential system is solved via a shooting method with MATLAB bvp4c. The nanofluid thermodynamics profiles with variation in physical properties of thermal relaxation time, thermal radiation, porosity, and chemical reaction are observed. Axial, radial, and tangential velocities are found to be increasing functions of porous medium. A decrease in the fluid temperature is perceived as thermal radiation and thermal relaxation increase since more heat can be transported to neighboring surroundings. The concentration is enhanced with intensified Cattaneo-Christov?s thermal relaxation but it oscillates with reacting chemicals. The rotary disks bound the oscillating nanofluid from downward to up-ward directions and vice versa. The axial velocity represents the change in force due to porosity and radial stretching of the disks.

Integral representation of solution manifolds for over determined systems with one singular line

In this paper, an over determined system of second-order partial differential equations with one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular line. Under the condition of compatibility, introducing a new function, we come to a over determined system of partial differential equations of the second order with one singular line of a simpler form. The integral representation of the manifold of solutions of the redefined second-order partial differential system with one singular line is found explicitly through three arbitrary constants, for which initial data problems (Cauchy type problems) can be posed.