Hidden Properties of Mathematical Physics Equations. Double Solutions. The Realization of Integrable Structures. Emergence of Physical Structures and Observable Formations

With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.

Modeling and Simulations in Symmetrical Supercapacitors Using Time Domain Mathematical Expressions

This study presents the deduction of time domain mathematical equations to simulate the curve of the charging process of a symmetrical electrochemical supercapacitor with activated carbon electrodes fed by a source of constant electric potential in time ε and the curve of the discharge process through two fixed resistors. The first resistor RCo is a control that aims to prevent sudden variations in the intensity of the electric current i1(t) present at the terminals of the electrochemical supercapacitor at the beginning of the charging process. The second resistor is the internal resistance RA of the ammeter used in the calculation of the intensity of the electric current i1(t) over time in the charging and discharging processes. The mathematical equations generated were based on a 2R(C + kUC(t)) electrical circuit model and allowed to simulate the effects of the potential-dependent capacitance (kUC(t)) on the charge and discharge curves and hence on the calculated values of the fixed capacitance C, the equivalent series resistance (ESR), the equivalent parallel resistance (EPR) and the electrical potential dependent capacitance index k.

EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION

Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.

Progress on Heat Transfer in Fractures of Hot Dry Rock Enhanced Geothermal System

Hot Dry Rock(HDR)is the most potential renewable geothermal energy in the future.Enhanced Geothermal System(EGS)is the most effective method for the development and utilization of HDR resources,and fractures are the main flow channels and one of the most important conditions for studying heat transfer process of EGS.Therefore,the heat transfer process and the heat transfer mechanism in fractures of EGS have been the hot spots of research.Due to the particularity of the mathematical models of heat transfer,research in this field has been at an exploratory stage,and its methods are mainly experimental tests and numerical simulations.This paper introduces the progress on heat transfer in fractures of Hot Dry Rock EGS in detail,provides a comparative analysis of the research results and prospects for future research directions:It is suggested that relevant scholars should further study the mathematical equations which are applicable to engineering construction of seepage heat transfer in irregular fractures of the rock mass,the unsteady heat transfer process between multiple fractures of the rock mass and the heat transfer mechanism of the complex three-dimensional models of EGS.

Fruit Mass of <i>Carica papaya</i>L. from Cultivars Aliança and THB from the Width and Length of the Fruit

Papaya (Carica papaya L.) is a typical plant with a tropical climate, but also grown in subtropical regions. Using mathematical models well-adjusted allows with good precision to estimate characteristics of interest. The objective was to adjust an equation that estimates the fruit mass for each cultivar of papaya, Aliança and THB, using only one measure, length or width. The experiment was conducted in the municipality of Linhares in the state of Espírito Santo, Brazil. Seedlings were planted on the same day, spaced 3.6 × 1.5 m and in rows side by side. Initially, the equations were modeled, they were linearized and then the covariance analysis was performed in order to verify the possibility of an equation that would serve both cultivars. As the covariance was significant, it was necessary to develop equations for each cultivar. To obtain the growth equations, 350 fruits of cultivar Aliança and 550 of THB were used. The validation was performed with 50 fruits of each. The characteristics evaluated were the largest width (W in mm), the longest fruit length (L, in mm) and the observed mass (OM in g). The equations that best fit were those of the power model that use width (W) as an independent variable.

Numerical Simulation of Bone Remodeling Coupling the Damage Repair Process in Human Proximal Femur

Microdamage is produced in bone tissue under the long-termeffects of physiological loading,as well as age,disease and other factors.Bone remodeling can repair microdamage,otherwise this damage will undermine bone quality and even lead to fractures.In this paper,the damage variable was introduced into the remodeling algorithm.The new remodeling algorithm contains a quadratic term that can simulate reduction in bone density after large numbers of loading cycles.The model was applied in conjunction with the 3Dfinite elementmethod(FEM)to the remodeling of the proximal femur.The results showed that the initial accumulation of fatigue damage led to an increase in density but when the damage reached a certain level,the bone density decreased rapidly until the femur failed.With the accumulation of damage,bone remodeling was coupled with fatigue damage to maintain the function of bone.When the accumulation of damage reached a certain level,bone remodeling failed to repair the accumulated fatigue damage in time,and continued cyclic loading significantly weakened the loadbearing capacity of the bone.The new mathematical model not only predicts fatigue life,but also helps to further understand the compromise between damage repair and damage accumulation,which is of great significance for the prevention and treatment of clinical bone diseases.

Solving the Burgers-Huxley Equation by G'/G Expansion Method

By introducing and extending the G'/G expansion method with the aid of computer algebraic system “Mathematics”, the exact general solutions were obtained for the Burgers-Huxley equation and special form. Final results were represented in hyperbolic function, trigonometric function and rational function with arbitrary parameters.