Mathematical modeling of elastic inverted pendulum control system

Inverted pendulums are important objects of theoretical investigation and experiment in the area of control theory and engineering. The researches concentrate on the rigid finite dimensional models which are described by ordinary differential equations (ODEs) .Complete rigidity is the approximation of practical models ; Elasticity should be introduced into mathematical models in the analysis of system dynamics and integration of highly precise controller. A new kind of inverted pendulum, elastic inverted pendulum was proposed, and elasticity was considered. Mathematical model was derived from Hamiltonian principle and variational methods, which were formulated by the coupling of partial differential equations (PDE) and ODE. Because of infinite dimensional, system analysis and control of elastic inverted pendulum is more sophisticated than the rigid one.

Consistency and Validity of the Mathematical Models and the Solution Methods for BVPs and IVPs Based on Energy Methods and Principle of Virtual Work for Homogeneous Isotropic and Non-Homogeneous Non-Isotropic Solid Continua

Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (I) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (δI) set to zero is a necessary condition for an extremum of I. In this approach one could use δI = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from δI = 0, which is also satisfied by a solution obtained from δI = 0. The Euler’s equations obtained from δI = 0 indeed are the mathematical model associated with the energy functional I. In case of BVPs we follow the same approach except in this case, the energy functional I consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles i.e. conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.

SCALE AND TIME EFFECTS ON MATHEMATICAL MODELS FOR TRANSPORT IN THE ENVIRONMENT

The purpose of this paper is to analyse mathematical models used in environmental modelling. Following a brief survey of the development in modelling scale- and time-dependent dispersion processes in the environment, this paper compares three similarity solutions, one of which is a solution of the generalized Feller equation （GF） with fractal parameters, and the other two for the newly-developed generalized Fokker-Planck equation （GFP）. The three solutions are derived with parameters having physical significance. Data from field experiments are used to verify the solutions. The analyses indicate that the solutions of both GF and GFP represent the physically meaningful natural processes, and simulate the realistic shapes of tracer breakthrough curves.

Reducing the Range of Cancer Risk on BI-RADS 4 Subcategories via Mathematical Modelling

Breast Imaging Reporting and Data System,also known as BI-RADS is a universal system used by radiologists and doctors.It constructs a comprehensive language for the diagnosis of breast cancer.BI-RADS 4 category has a wide range of cancer risk since it is divided into 3 categories.Mathematicalmodels play an important role in the diagnosis and treatment of cancer.In this study,data of 42 BI-RADS 4 patients taken fromthe Center for Breast Health,Near East University Hospital is utilized.Regarding the analysis,a mathematical model is constructed by dividing the population into 4 compartments.Sensitivity analysis is applied to the parameters with the desired outcome of a reduced range of cancer risk.Numerical simulations of the parameters are demonstrated.The results of the model have revealed that an increase in the lactation rate and earlymenopause have a negative correlation with the chance of being diagnosed with BI-RADS 4 whereas a positive correlation increase in age,the palpable mass,and family history is distinctive.Furthermore,the negative effects of smoking and late menopause on BI-RADS 4C diagnosis are vehemently outlined.Consequently,the model showed that the percentages of parameters play an important role in the diagnosis of BI-RADS 4 subcategories.All things considered,with the assistance of the most effective parameters,the range of cancer risks in BI-RADS 4 subcategories will decrease.

Mathematical Modeling of Heat Flux Distribution in Raw Cotton Stored in Bunt

The scientific article examines the physical and mechanical properties of raw cotton stored in buntings in cotton palaces. Because during the storage of raw cotton in bunts, some of its properties deteriorate, some improvements. Therefore, the mathematical modeling of storage conditions of raw cotton in bunts and the physical and mechanical conditions that occur in it is of great importance. In the developed mathematical model, the main factor influencing the physical and mechanical properties of raw cotton is the change in temperature. Due to the temperature, kinetic and biological processes accumulated in the raw cotton in Bunt, it can spread over a large surface, first in a small-local state, over time with a nonlinear law. As a result, small changes in temperature lead to a qualitative change in physical properties. In determining the law of temperature distribution in the raw cotton in Bunt, Laplace’s differential equation of heat transfer was used. The differential equation of heat transfer in Laplace’s law was replaced by a system of ordinary differential equations by approximation. Conditions are solved in MAPLE-17 program by numerical method. As a result, graphs of temperature changes over time in raw cotton were obtained. In addition, the table shows the changes in density, pressure and mass of cotton, the height of the bun. As the density of the cotton raw material increases from the top layer of the bunt to the bottom layer, an increase in the temperature in it has been observed. This leads to overheating of the bottom layer of cotton and is the main reason for the deterioration of the quality of raw materials.

A Mathematical Model for Redshift

We have used model scaling so that the propagation of light through space could be studied using the well-known nonlinear Schrödinger equation. We have developed a set of numerical procedures to obtain a stable propagating wave so that it could be used to find out how wavelength could increase with distance travelled. We have found that broadening of wavelength, expressed as redshift, is proportional to distance, a fact that is in agreement with many physical observations by astronomers. There are other reasons for redshifts that could be additional to the transmission redshift, resulting in the deviation from the linear relationship as often observed. Our model shows that redshift needs not be the result of an expanding space that is a long standing view held by many astrophysicists. Any theory about the universe, if bases on an expanding space as physical fact, is open to question.

Mathematical Model of Seam-Cavity Growth of Underground Coal Gasification at High Temperature

A mathematical model to describe the seam-cavity growth during underground coal gasification(UCG) is developed. The effect of coal gasification or coal combustion on lateral growth of coal wall is mainly considered in this model. It is based on some basic assumptions and balances of heat and mass transfer in gasification zone of cavity,and considers the effect of dry distillation of seams. The gasifing zone of coal wall is assumpted to be a thin,highly permeable char layer and an ash layer seperating inert ash rubble of the main cavity from coal wall. The model can explain lateral growth mechanism of coal wall in entire gasification process,and determine the growth velocity v (x,t) of coal wall.

Non-Linear Mathematical Model of the Interaction between Tumor and Oncolytic Viruses

A mathematical modeling of tumor therapy with oncolytic viruses is discussed. The model consists of two coupled, deterministic differential equations allowing for cell reproduction and death, and cell infection. The model is one of the conceptual mathematical models of tumor growth that treat a tumor as a dynamic society of interacting cells. In this paper, we obtain an approximate analytical expression of uninfected and infected cell population by solving the non-linear equations using Homotopy analysis method (HAM). Furthermore, the results are compared with the numerical simulation of the problem using Matlab program. The obtained results are valid for the whole solution domain.

Mathematical Model of Cell Growth for Biofuel Production under Synthetic Feedback

In this paper, mathematical model for cell growth and biofuel production under synthetic feedback loop is discussed. The nonlinear differential equations are solved analytically for the maximum production of biofuel under synthetic feedback. The closed-form of analytical expressions pertaining to the concentrations of cell density, repressor proteins, pump expressions, intracellular biofuel and extracellular biofuel are presented. The constant pump model is compared with feedback loop model analytically to know the biofuel production. The numerical solution of this problem is also reported using Scilab/Matlab program. Also, the analytical results are compared with previous published numerical results and found to be in good agreement.

A two-parameter mathematical model for immobilizedenzymes and Homotopy analysis method

A two parameter mathematical model was developed to find the concentration for immobilized enzyme systems in porous spherical particles. This model contains a non-linear term related to reversible Michaelies-Menten kinetics. Analytical expression pertaining to the substrate concentration was reported for all possible values of Thiele module φ and α . In this work, we report the theoretically evaluated steady-state effectiveness factor for immobilized enzyme systems in porous spherical particles. These analytical results were found to be in good agreement with numerical results. Moreover, herein we employ new “Homotopy analysis method” (HAM) to solve non-linear reaction/diffusion equation.

Physico-Mathematical Models in Rotational Motions

This paper was the prologue to the first book published by Gabriel Barceló on the Theory of Dynamic Interactions, entitled: The Flight of the Boomerang. However, this text, is fully current, as well as the book itself. The book described mainly the initial historical analysis made by Dr. Barceló on the rotational dynamics. For Professor Garcia Moliner, the book contains an extremely positive message. It shows us that scientific research consists of constantly questioning hitherto accepted theories and seeing if they can always be applied to new situations.

Mathematical model for predicting fungal growth and decomposition rates based on improved Logistic equations

As the function of the decomposition of fungi has been clearly researched in the global carbon cycle,it is obviously of value to explore the decomposition rate of fungal populations.This study analyzed the relationship between environmental factors and biodiversity step by step.In order to explore the interaction between the fungi and the relationship between the decomposition rate of fungi with time,the model based on the Logistic model was built and the Lotka-Volterra model was employed in the condition of two kinds of fungi existing in an environment with limited resources.The changing trend of population number and decomposition rate of several fungi under different environmental conditions can be predicted through the model.To illustrate the applicability of the model,Laetiporus conifericola and Hyphoderma setigerum were applied as examples.The results showed that the higher the degree of population diversity,the greater the decomposition rate,and the higher the decomposition efficiency of the ecosystem.Its rich species diversity is conducive to accelerating the decomposition of litter,lignocellulose,and the circulation of the entire ecosystem.Based on the above model and using the data from measuring the mycelial elongation rate of each isolate at 10℃,16℃,and 22℃ under standardized laboratory conditions,the growth patterns of the five fungi combinations were simulated.The results revealed a general increase in growth rate with increasing temperature,which verifies the accuracy of the model.Moreover,it also revealed that the total decomposition rate after fungal incorporation was negatively correlated with the decomposition rate of a fungal single action.Based on the above model,predictions can be made for fungal growth in different environments,and suitable environments for fungal growth can be determined.In the future,the model can be further optimized,and lignin and cellulose decomposition factors can be added to fit the decomposition of logs.The application scenarios of the model can be further broadened,which can contribute to the restoration and management of the ecological environment,as well as produce good effects in the fields of fungi assisting the global carbon cycle and soil problem restoration.

Application of differential equations in mathematical modeling

This paper introduces the main methods and steps of modeling principle by ordinary differential equations, and is used to explore the differential equation model to solve some practical problems, some features of the related problems. With the development of science and technology and production practice, differential equation is more closely connected with other subjects, and a mathematical model for some practical problems of good.

A Comparative Study of Adomian Decomposition Method with Variational Iteration Method for Solving Linear and Nonlinear Differential Equations

This study compares the Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM) for solving nonlinear differential equations in engineering. Differential equations are essential for modeling dynamic systems in various disciplines, including biological processes, heat transfer, and control systems. This study addresses first, second, and third-order nonlinear differential equations using Mathematica for data generation and graphing. The ADM, developed by George Adomian, uses Adomian polynomials to handle nonlinear terms, which can be computationally intensive. In contrast, VIM, developed by He, directly iterates the correction functional, providing a more straightforward and efficient approach. This study highlights VIM’s rapid convergence and effectiveness of VIM, particularly for nonlinear problems, where it simplifies calculations and offers direct solutions without polynomial derivation. The results demonstrate VIM’s superior efficiency and rapid convergence of VIM compared with ADM. The VIM’s minimal computational requirements make it practical for real-time applications and complex system modeling. Our findings align with those of previous research, confirming VIM’s efficiency of VIM in various engineering applications. This study emphasizes the importance of selecting appropriate methods based on specific problem requirements. While ADM is valuable for certain nonlinearities, VIM’s approach is ideal for many engineering scenarios. Future research should explore broader applications and hybrid methods to enhance the solution’s accuracy and efficiency. This comprehensive comparison provides valuable guidance for selecting effective numerical methods for differential equations in engineering.

Some studies on mathematical models for general elastic multi-structures

The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.

NUMERICAL SIMULATION OF A MATHEMATICAL MODEL FOR WAVE PROPAGATION ON NON-UNIFORM CURRENT AND DEPTH

A water wave evolution equation is developed from the combinedrefraction-diffraction equation on non-uniform current in water of slowly varying topography byusing the perturbation method. A numerical model is presented with the governing equationdiscretized with an improved Alternating Direction Impicit (ADI) method involving a relaxationfactor which can improve convergent rate. The calculation results show that the model caneffectively reflect the effects of current on wave propagation.

Mathematical modeling of COVID-19 infection dynamics in Ghana: Impact evaluation of integrated government and individual level interventions

The raging COVID-19 pandemic is arguably the most important threat to global health presently.Although there is currently a vaccine,preventive measures have been proposed to reduce the spread of infection but the efficacy of these interventions,and their likely impact on the number of COVID-19 infections is unknown.In this study,we proposed the SEIQHRS model(susceptible-exposed-infectious-quarantine-hospitalized-recovered-susceptible)model that predicts the trajectory of the epidemic to help plan an effective control strategy for COVID-19 in Ghana.We provided a short-term forecast of the early phase of the epidemic trajectory in Ghana using the generalized growth model.We estimated the effective basic Reproductive number Re in real-time using three different estimation procedures and simulated worse case epidemic scenarios and the impact of integrated individual and government interventions on the epidemic in the long term using compartmental models.The maximum likelihood estimates of Re and the corresponding 95%confidence interval was 2.04[95%CI:1.82e2.27;12th March-7th April 2020].The Re estimate using the exponential growth method was 2.11[95%CI:2.00e2.24]within the same period.The Re estimate using time-dependent(TD)method showed a gradual decline of the Effective Reproductive Number since March 12,2020 when the first 2 index cases were recorded but the rate of transmission remains high(TD:Re=2.52;95%CI:[1.87e3.49]).The current estimate of Re based on the TD method is 1.74[95%CI:1.41 e2.10;(13th May 2020)]but with comprehensive integrated government and individual level interventions,the Re could reduce to 0.5 which is an indication of the epidemic dying out in the general population.Our results showed that enhanced government and individual-level interventions and the intensity of media coverage could have a substantial effect on suppressing transmission of new COVID-19 cases and reduced death rates in Ghana until such a time that a potent vaccine or drug is discovered.

Line-Art and its Mathematical Models

In this paper, the authors describe the principles of Straight Line Strokes illustration, present the mathematical model of the principles, and show how a great number of lines can be implemented as main part of an automated drawing system named Line-Art. Different from traditional drawing art, Line-Art generates pictures without curves, colors, ink marks, brushes, and oil paint, but only with Straight Line Strokes. Generated pictures are composed, clipped, and plotted. The paper also introduces how to use the initial value problem of the ordinary differential equation to describe a drawing art, e.g. Line-Art.

GC-MCSPT-A MODIFIED HARD-SPHERE THREE-PARAMETER EQUATION OF STATE WITH GROUP MODEL FOR VLE PREDICTION

The mixing rule for a new group-contribution equation of state was proposed by combining the excess Gibbs energy model with the modified Hard-Sphere Three-Parameter Equation of State designated as the MCSPT equation. Low-and high-pressure vapor-liquid equilibria of 28 binary and 9 ternary systems containing strongly polar substances were predicted by using the interaction parameters of the original and modified UNIFAC model. Predicted results have shown that the proposed GO-MCSPT equation has an extensive applicability with satisfactory accuracy.

An insight into the role of the association equations of states in gas hydrate modeling:a review

Encouraged by the wide spectrum of novel applications of gas hydrates,e.g.,energy recovery,gas separation,gas storage,gas transportation,water desalination,and hydrogen hydrate as a green energy resource,as well as CO2 capturing,many scientists have focused their attention on investigating this important phenomenon.Of course,from an engineering viewpoint,the mathematical modeling of gas hydrates is of paramount importance,as anticipation of gas hydrate stability conditions is effective in the design and control of industrial processes.Overall,the thermodynamic modeling of gas hydrate can be tackled as an equilibration of three phases,i.e.,liquid,gas,and solid hydrate.The inseparable component in all hydrate systems,water,is highly polar and non-ideal,necessitating the use of more advanced equation of states(EoSs) that take into account more intermolecular forces for thermodynamic modeling of these systems.Motivated by the ever-increasing number of publications on this topic,this study aims to review the application of associating EoSs for the thermodynamic modeling of gas hydrates.Three most important hydrate-based models available in the literature including the van der Waals-Platteeuw(vdW-P) model,Chen-Guo model,and Klauda-Sandler model coupled with and SAFT EoSs were investigated and compared with cubic EoSs.It was concluded that the CPA and SAFT EoSs gave very accurate results for hydrate systems as they take into account the association interactions,which are very crucial in gas hydrate systems in which water,methanol,glycols,and other types of associating compounds are available.Moreover,it was concluded that the CPA EoS is easier to use than the SAFT-type EoSs and our suggestion for the gas hydrate systems is the CPA EoS.