Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems

This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.

Interaction of a singular surface with a strong shock in the interstellar gas clouds

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions. doi:10.1017/S1446181121000328

Generalized F(R, T) cosmological models with fermionic fields

We investigated the gravity model F (R, T), which interacts with a fermion field in a uniform and isotropic at spacetime FLRW. The main idea and purpose of the work donewas to create a mathematical model and find a particular solution for the scale factor a, since it describes the dynamics of the evolution of the Universe. The solutions for this universe are obtained using the Noether symmetry method. With its help, a specific form of the Lagrangian is obtained. And the possible types of the scale factor were found. The evolution of the resulting cosmological model has been investigated.

Analytical modeling of the binary dynamic circuit motion

The binary dynamic circuit, which can be a design scheme for a number of technical systems is considered in the paper. The dynamic circuit is characterized by the kinetic energy of the translational motion of two masses, the potential energy of these masses’ elastic interaction and the dissipative function of energy dissipation during their motion. The free motion of a binary dynamic circuit is found according to a given initial phase state. A mathematical model of the binary dynamic circuit motion in the canonical form and the corresponding characteristic equation of the fourth degree are compiled. Analytical modeling of the binary dynamic circuit is carried out on the basis of the proposed particular solution of the complete algebraic equation of the fourth degree. A homogeneous dynamic circuit is considered and the reduced coefficients of elasticity and damping are introduced. The dependence of the reduced coefficients of elasticity and damping is established, which provides the required class of solutions to the characteristic equation. An ordered form of the analytical representation of a dynamic process is proposed in symmetric determinants, which is distinguished by the conservatism of notation with respect to the roots of the characteristic equation and phase coordinates.

LOCALLY ORTHOTROPIC LAY-UP AS AN OPTIMAL SOLUTION FOR VAT POST-BUCKLED COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE VON KARMAN LIMITS

The present paper deals with the optimization of post-buckled VAT (variable angle tow) composite plates with large deflections. The Kirchhoff assumptions are used. The plates have a symmetric lay-up. The large deflection geometrically nonlinear theory above the von Karman limits is employed. The structural potential energy is treated as a measure of structural stiffness. For the plate stiffness maximization problem, the first-order necessary conditions of the local optimality are derived. The mathematical analysis of the conditions is performed. The conditions contain two terms. One of them corresponds to the mid-plane strains; another one corresponds to the generalized plate curvatures. A locally orthotropic lay-up is identified as an optimal solution. The local ply material direction is clearly coupled with the principal directions of 2D-strains and generalized curvatures. A particular solution of the linear combination of the ply optimality conditions is indicated. For the solution two pairs of the structural tensors are co-axial: the force and the strain tensors, as well as the moment and the generalized curvature tensors.

INTERACTION OF A SINGULAR SURFACE WITH A STRONG SHOCK IN THE INTERSTELLAR GAS CLOUDS

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions.

Exclusion regions for parameter-dependent systems of equations

This paper presents a new algorithm based on interval methods for rigorously constructing inner estimates of feasible parameter regions together with enclosures of the solution set for parameter-dependent systems of nonlinear equations in low (parameter) dimensions. The proposed method allows to explicitly construct feasible parameter sets around a regular parameter value, and to rigorously enclose a particular solution curve (resp. manifold) by a union of inclusion regions, simultaneously. The method is based on the calculation of inclusion and exclusion regions for zeros of square nonlinear systems of equations. Starting from an approximate solution at a fixed set p of parameters, the new method provides an algorithmic concept on how to construct a box $${\mathbf {s}}$$ s around p such that for each element $$s\in {\mathbf {s}}$$ s ∈ s in the box the existence of a solution can be proved within certain error bounds.

Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

On generalized Lemaitre–Tolman–Bondi metric: Fractal matter at the end of matter–antimatter recombination

Many recent researches have investigated the deviations from the Friedmannian cosmological model, as well as their consequences on unexplained cosmological phenomena, such as dark matter and the acceleration of the Universe. On one hand, a first-order perturbative study of matter inhomogeneity returned a partial explanation of dark matter and dark energy, as relativistic effects due to the retarded potentials of far objects. On the other hand, the fractal cosmology, now approximated by a Lemaitre–Tolman–Bondi (LTB) metric, results in distortions of the luminosity distances of SNe Ia, explaining the acceleration as apparent. In this work, we extend the LTB metric to ancient times. The origin of the fractal distribution of matter is explained as the matter remnant after the matter–antimatter recombination epoch. We show that the evolution of such a inhomogeneity necessarily requires a dynamical generalization of LTB, and we propose a particular solution.

Temporal evolution of a radiating star via Lie symmetries

In this work we present for the first time the general solution of the temporal evolution equation arising from the matching of a conformally flat interior to the Vaidya solution. This problem was first articulated by Banerjee et al. (Phys Rev D 40:670, 1989) in which they provided a particular solution of the temporal equation. This simple exact solution has been widely utilised in modeling dissipative collapse with the most notable result being prediction of the avoidance of the horizon as the collapse proceeds. We study the dynamics of dissipative collapse arising from the general solution obtained via the method of symmetries and of the singularity analysis. We show that the end-state of collapse for our model is significantly different from the widely used linear solution.