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The second edition of the symposium on Mathematical Problems in Engineering focuses on new results of engineering researches carried out by using mathematical techniques. The aim of the Session is to collect relevant papers dealing with important approaches of applied mathematics, which have relevance to the engineering in the field of Computer Engineering, Engineering Applications of Aerospace and Intelligent Transportation Systems in order to create new collaborations between mathematicians and engineers. Papers are characterized by innovative models, methods and approaches that can found practical application to the engineering field or by new useful application of existing models to solve engineering problems. [ABSTRACT FROM AUTHOR]

There are innumerable methods for solving continuous beams, including the slope and deflection method, the moment distribution method, the flexibility matrix method, and the stiffness matrix method. Each method has its own identity. The final results of all four methods are shear force, bending moment, slope, and deflection. Even if all approaches yield the same result. The issue now is the time duration of those methods. One assessment should take approximately over thirty minutes to complete. Another issue is that the entire analysis will be repeated even if only one parameter is changed. Computer-based analysis is the best solution to all of these problems. Python is a computer programming language that has recently gained popularity due to its syntax simplicity. The code can be executed as soon as it is written in this case. A four-span continuous beam will be analyzed using Python in this paper, and the results will be compared to the conventional stiffness method. Initially, inputs are sent to Python using finite element procedures. The analysis is performed with Numpy, and the results are printed. Numpy is a Python library that is capable of performing all mathematical analysis, particularly on arrays and matrices. The analysis in this paper is carried out using Google colab as a plot form and Numpy as a tool. [ABSTRACT FROM AUTHOR]

Advancements in the field of physics gave birth to remote sensing for analyzing climate problems effectively and smoothly in form of providing vulnerable and significant data. Mathematics and statistics play a pivot role in the succeeding domains to cater to data analysis in a structured and lucid manner. This paper incubates a climate-related study with the help of enrichments in mathematics and physics. The paper states clearly all about the temperature profile of an Indian city, Ajmer. The incorporation of data from 2011 to 2020 depicts a good picture of land skin temperature as a result. Duo-Satellite imageries of ten years had been examined carefully and software-derived data was acted upon by mathematical analysis to find a pattern meticulously. Significant findings were that the summer season was found out the hottest among all the seasons during both day and night time. The winter season had the least anomalies and the monsoon had the most. The paper and its findings were showed alarming information about the gradual rise in temperature which will create a difficult situation in near future for human beings. [ABSTRACT FROM AUTHOR]

It takes millions of years for a star to get formed. The evolutionary process of star formation undergoes various stages. Theoretical and astronomical calculations confirm the series of events from nebula to a white dwarf, neutron star, or black due to conversion of mass into energy. This paper deals with the stability analysis of a mathematical model which governs star formation during the ever-changing process of evolution. Variation matrix or Jacobian Matrix is developed out of the differential equation which arises from three components of the galactic region that is atomic, molecular, and stellar components. The mathematical model is the best way to explore and explain the system of three components in the galactic region. In fact, Mathematical Modelling-is the main, if not the only way one, way for theoretical research of astrophysical processes and objects. Various examples have been taken for comparative studies and computational purposes. [ABSTRACT FROM AUTHOR]

The task of developing effective granular aggregates, as well as the creation of a mathematical model and management of this process, is the most promising and relevant. The paper deals with the construction of mathematical models based on the results of experimental studies of granular aggregates (GA). We present the processing of the results of experimental studies on the production of granular aggregates using the classical method with the use of the high-level general-purpose Python programming language. The processing of research results is fully object-oriented. This approach to the generation of the obtained results allowed us to automate the processing of the array of data for the planning of the experiment. The results of mathematical modeling and nomograms reflecting the variability of the developed compositions, which allowed to obtain samples with the required technological, physical-mechanical and operational characteristics are presented. The paper calculates the values of reproducibility dispersions: for sand fraction ≤0.16 mm - 0.889; for sand fraction ≤0.315 mm - 0.967; for sand fraction ≤0.63 mm - 0.981. The analysis of nomograms has shown that the greatest strength values have the compositions with the sand fraction of 0.16 mm and the minimum content of sand (as an aggregate). For mixtures with sand fractions ≤0.315 mm it is established that the strongest will be the mixture with the minimum content of aggregate. It is revealed that when using sand fraction ≤0.63 mm the strength is reduced. [ABSTRACT FROM AUTHOR]

Through means of this specific paper, we have likely embraced an investigation towards leading an examination in regards to "flow and heat transfer qualities in a second grade liquid through a permeable medium over a straight extending surface with two general cases, in particular PST and PHF, including the impact of viscous dissipation". Subsequently, fractional differential conditions administering the idea of "flow and heat movement are changed over into non-linear customary differential conditions and limit conditions by utilizing suit-capable similitude change", was regularly being thought of. Accordingly, the proposed issue identifying with our paper, was fundamentally being addressed systematically, by methods for power arrangement strategy, wherein subsequently graphical outcomes for "speed, skin-grinding coefficient, and temperature for PST and PHF cases are also introduced and talked about, close by thought of mathematical qualities identifying with divider grating coefficient, divider temperature slope θη (0), and divider temperature θ (0)" were in like manner being determined, organized, & examined subsequently. [ABSTRACT FROM AUTHOR]

This paper deals with an analytical technique used to estimate and resolve the steady-state Frank-Kamenetskii problem. A homotopy perturbation method is applied to solve the nonlinear steady-state thermal explosion in a vessel. Additionally, using the Matlab software, the numerical simulation of the problem is described in this study. The interior temperature of the vessel is estimated and analysed to variations in the Frank-Kamenetskii and geometry parameters. A graphic representation of a vessel's temperature result is provided to demonstrate the solution, and theoretical discussion is had. The results of the analytical and numerical methods show good agreement. [ABSTRACT FROM AUTHOR]

Mathematical models of the COVID-19 pandemic have been utilized in a variety of settings as a core component of national public health responses. Often based on systems of ordinary differential equations, compartmental models are commonly used to understand and forecast outbreak trajectories. In view of the primarily applied nature of COVID-19 models, theoretical analysis can provide a global and long-term perspective of key model properties, and relevant insights about the infection dynamics they represent. This work formulates and undertakes such an investigation for a compartmental model of COVID-19, which includes the effect of interventions. More specifically, this paper analyzes the characteristics of the solutions of a compartmental model by establishing the existence and stability of the equilibrium points based on the value of the basic reproductive number R0. Our results provide insights on the possible policies that can be implemented to address the health crisis. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical investigation of ellipticity of the boundary value problem for isothermal finite strain elasto-plasticity. Ellipticity can be lost when softening occurs. A discontinuity surface then appears in the considered material body and this is associated with the ill-posedness of the boundary value problem. In the paper the condition for ellipticity loss is derived using the deformation gradient and the first Piola-Kirchhoff stress tensor. Next, the obtained condition is implemented and numerically tested within symbolic-numerical tools AceGen and AceFEM using the benchmark of an elongated rectangular plate with imperfection in plane stress and plane strain conditions. [ABSTRACT FROM AUTHOR]

In this paper, an identification problem on determining the unknown coefficients of nonlinear Schrödinger equation is studied. The first variation of the observation functional is obtained and a necessary condition in the form of variational inequality for the solution of the identification problem is given. [ABSTRACT FROM AUTHOR]

The hypothesis method, based on the adoption of certain assumptions that allow reducing the three-dimensional problem of the theory of elasticity to two-dimensional problem was first obtained at the end of the nineteenth century. Everyone knows the theory based on Kirchhoff's hypotheses, which until now has not lost its significance. In a number of works, an attempt has been made to refine the theory of plates by adopting less rigid hypotheses than those of Kirchhoff. In this way, the still popular theories of plate bending of the Timoshenko S.P. and E. Reissner e were obtained taking into account the transverse shear. These theories of plate bending are now widely used as refined theories for plate bending and for calculating thick plates. In the paper by the mathematical method, without using any assumptions, the three-dimensional equations of the theory of elasticity have been reduced to refined equations for the bending of plates. The obtained mathematical theory of plates is compared with popular theories of the Timoshenko type. It is shown that in order to refine the theory of plate bending, in addition to transverse shear, a number of additional terms should be taken into account, which are absent in theories of the Reissner E. and Timoshenko S.P. type obtained by the method of hypotheses. Mathematical analysis of three-dimensional elasticity equations confirms that the error of theories such as the theory of E. Reissner and S.P. Timoshenko is of the same order as the error of the classical theory of plate bending. This means that the popular theories discussed are not refined and they cannot be used to calculate thick plates. [ABSTRACT FROM AUTHOR]

The paper considers second-order nonlinear differential equation describing deformations of a cantilever structure under various loads. The corresponding mathematical model is studied using the analytical approximate method, as well as the calculation method implemented in the Mathematica system. Theoretical material is accompanied by numerical experiments. Cases of steel and wood structures are considered. Comparative analysis of the results of calculations carried out in software systems SCAD ++ and LIRA 10 along with the results of analytical approximate method and calculations by using Mathematica system is conducted, and the outcomes are presented. [ABSTRACT FROM AUTHOR]

Steel trihedral trusses are used as flat roofs of public and industrial buildings and structures. A modular system can be used to transport the structures of trihedral trusses. The paper presents the results of numerical studies of modular trihedral trusses for flat roofs of buildings with transversal partitioning into dispatch modules. Application of modular system for trihedral trusses is aimed at mass production with widespread delivery by various cargo transport. Numerical studies based on the computational mathematical model of a space-core truss allows to estimate deformability of a trihedral truss with pliable mounting junctions of adjacent modules for various forms of loading. We describe peculiarities of constructing variative computational models which take into account the peculiarities of structures with noncentered connection junctions of struts from single corners to the chords of a composite profile made of channel's and corner's rolled elements. The most stressed elements of the spatial and strut system of trihedral trusses with butt joint adjacency of struts to chords are determined. Results are presented in a table for comparative analysis of deformability for various calculation models. According to the comparative analysis of deformability of mathematical calculation models for trihedral roof trusses, the use of pliable mounting junctions of module coupling has been substantiated. [ABSTRACT FROM AUTHOR]

The main goal is to study nonlinear operators and find majorization for operators. Especially, Volterra operators are very important for us. This article has given a three-dimensional matrix and some properties nonlinear operators. Nonlinear operators from matrices are used, and trajectories are founded. The theory of boundary properties made considerable advances in the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions, and objects of study. This class also studied Lyubich. He found some results, but bistochastic operators have not opened yet. This all works related to quadratic stochastic operators. Three-dimensional matrices Pijk for us to study theories and research of some problems are very important. We use them to study quadratic stochastic operators and relate them to majorization. The main connection between them is as follows: V (x) ≺ x where V(x) operator called Volterra operator. The operator is written as follows: (V x) = ∑ i = 1 m P i j , k x i x j This article gave some properties for majorization and showed some connection with nonlinear operators. The article also gave some examples for nonlinear operators. The main goal is to find an operator and a corresponding matrix that satisfies the definition of majorization. This paper used the theory of majorization and some classes. This article's main result is, constructs nonlinear operators. It is also shown that the operators are bistochastic. The initial points are given from the simplex, and the result shows whether the operator belongs to the simplex or not. The theory of boundary properties made considerable advances in the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions, and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of real functional analysis, the theory of majorization, and Volterra operators. On the other hand, one of the main purposes of this article is to connect with the Volterra operators. The theory of boundary properties of majorization, which grew out of the works of the Moscow Mathematical School (G.G.Hardy, D.E.Littlewood), was developed in the further works of E. Beccenbach and R.Belman as well as in the works of and other Russian scientists. We will extend this class by constructing a nonlinear operators class for the Volterra operators. Not everything goes exactly without an analog. In such cases, calculations are carried out in other ways. [ABSTRACT FROM AUTHOR]

This paper shows the exhibiting and examination of Perturb and Observe (P and O) algorithm for isolating the power structure photovoltaic group (PVA). In perspective on non-linear characteristics of PV cell, the best power can be removed under one explicit voltage conditions. In this way, maximum power point (MPPT) computations are used in PVA to extend the yield control. In this paper the MPPT computations is executed using DC-DC booster converter for high voltage gain application. The component of PVA is reproduced at different sun situated irradiance's and cell temperatures. [ABSTRACT FROM AUTHOR]

The main purpose of the suspension system is to absorb the vibrations from wheels. To compromise between the two conflicting criteria i.e. road holding and ride comfort, passive suspension system is most suitable. The passenger discomfort is caused due to unwanted vibrations which are transmitted through unsprung mass. The numerical analysis using Simulink MATLAB software is used to determine Ride comfort of two suspension systems based on quarter car model. The experimental setup is developed and vibration analysis is performed using multichannel FFT analyser. The mathematical analysis result of the two different models of the suspensions is compared with experimental analysis. The aim of paper is to investigate effect of sprung mass on ride comfort at constant speed with input 0.1m peak stroke which generated a waveform resembling to half sine wave. The mathematical model is developed and analysed in Simulink MATLAB software. [ABSTRACT FROM AUTHOR]

The pandemic Covid-19 accelerated the evolution of education 4.0. The online teaching and learning activities are promoted by movement control which had been enforced globally. Teaching and learning in engineering, which very much relied on practical and face-to-face by nature, are challenged by rapid process of adoption and adaptation by both learners and teachers. In this paper, such process is reported on one of the most challenging subjects in engineering education – advanced calculus. The design and strategy of teaching, the teaching process, the tools, the assessment, and students' feedback are discussed. Majority of the students were able to accept the new norm of teaching and learning, with a few still opt to resume to face-to-face learning. As for teacher, multiple platforms and delivery methods, including interactive methods are suggested to retain the attention and maintain the performance of the students. [ABSTRACT FROM AUTHOR]

This paper recommend a new topology of 27 level Multi Level Inverter (MLI) and two different switching pattern scheme such as Multi Carrier Phase Disposition Pulse Width Modulation Technique (PD-PWM) and Half Height method (HH). The control signal is processed through fuzzy logic for performing closed loop speed control in the proposed inverter. The proposed inverter has been analyzed by developing a mathematical model through Fourier series and also the performance of the MLI such as RMS voltage and THD equations which have been derived for the theoretical validation with existing inverter. The features of the multilevel inverters (MLIs) are to produce a good quality of AC supply with low Total Harmonic Distortion (THD). The main feature of the proposed inverter is to reduce number of switching devices and reduction in gate driver circuits. In this study, the MATLAB software is utilized for the verification and validation of practical feasibility in the proposed inverter. The proposed MLI has been implemented in laboratory prototype model for practical feasibility and performance parameters have been analyzed and verified through simulation as well as prototype model. [ABSTRACT FROM AUTHOR]

This paper deals with mathematical analysis of nonlinear model prey predator system by means of presenting its approximate analytical expression by using HPM which is abbreviated as Homotopy Perturbation Method. The numerical simulation of the proposed system is also reported using Scilab program. It is amazing that our analytical solution is exactly matches with the numerical simulation. [ABSTRACT FROM AUTHOR]

Solubility phase diagram which consists of information on solute concentration, metastable and saturation limits against temperature, provides helpful insights in designing crystallization process to achieve desired crystal size distribution (CSD). Usually, the design of cooling crystallization process involves high supersaturation level at the beginning of the process in the metastable zone that is bounded by metastable and saturation limits. However, this high level of supersaturation causes an increment in both nucleation and crystal growth rates which induce the growth of the seed crystals as well as unwanted secondary nucleation that produce excessive fine crystals. Mitigation by employing proper temperature trajectory or policy along metastable zone to avoid unnecessary long operational time and fine crystals is needed. Thus, the purpose of this paper is to develop and simulate mathematical model of seeded batch crystallization process for the case of potash alum which is mainly used for purification in water treatment. Dynamic response of such process under open-loop operation for three cooling policies which are natural, linear and cubic is performed for evaluating the effect of different cooling policies on CSD. Simulation results based on the dynamic behaviour for all three cooling policies show cubic cooling policy obtained the best performance by achieving mean crystal size of 420 µm from the targeted CSD at 430 µm, and the mean crystal size for fine crystals is the lowest which is 35 µm compared to linear and natural cooling policies, at 40 µm and 55 µm, respectively. This information is prominent in deciding proper temperature trajectory of optimal cooling policy for potash alum crystallization process. [ABSTRACT FROM AUTHOR]

Listeriosis is a food-borne disease caused by a bacterium known as Listeria monocytogens. Humans can be infected by consuming contaminated food products. A transmission can also occur through contact with infected animals or people, however to a less extent. In this paper, a mathematical model for Listeriosis dynamics was developed. The steady states and their stability of the model system were determined and analyzed. The result shows that the disease-free equilibrium is asymptotically stable if the bacteria's growth rate is less than its removal rate, and also the growth rate of food contamination is less than its removal rate. It was further observed that we can still have Listeriosis driven by the contaminated food products even if the Listeria bacteria population in the environment is very small.The results indicate that Listeriosis can be effectively controlled by removing contaminated food products, which was the policy adopted by the South African government during the recent Listeriosis outbreak. [ABSTRACT FROM AUTHOR]

The first corona virus disease case was reported in Wuhan City, China in December 2019 and spreads worldwide very quickly. On 9th March 2020, World Health Organization declared COVID-19 as a global pandemic. Mathematical models can be very effective in understanding the transmission patterns of COVID-19. This paper proposes a fractional-order susceptible, exposed, infected, asymptomatic, hospitalized, removed (SEIAHR) model. The fractional models help understand the disease epidemic. To formulate its fractional derivative, we consider the Caputo operator and discuss the uniqueness and existence of the solution for the model by applying the Banach contraction mapping principle. We performed numerical simulation by the Adams- Bashforth-Moulton method, which shows that the fractional models provide a better understanding and more biological insights about the dynamics of disease than traditional integer-order models. [ABSTRACT FROM AUTHOR]

The paper is a continuation of research undertaken in paper [22]. The considered problem consists in presenting some condition sufficient for approximation of the selective points in normed spaces by subseries of the given series. [ABSTRACT FROM AUTHOR]

One of the aims of new circular collider projects is further increase in their luminosity. A high electromagnetic field of space charge at the meeting points limits the achievable current densities and consequently the luminosity. Non- linear focusing compensation in a storage ring done by the opposite-charge beam circulating in another storage ring was proposed and tested many years ago. Ya. S. Derbenev has shown that such a scheme suffers from tune shifts of coherent betatron oscillations, which move betatron frequencies toward the nearest integer or half-integer resonance. In this paper, the collider based on electron energy recovery linac (ERL) and "figure-8" positron storage ring with beams of equal currents is considered. Positrons are circulating in a two-loop storage ring (positron-positron collider), and electron- electron collider uses ERL, as in original Tigner's proposal. Thus, a collision of four bunches and space-charge compensation in a multi-bunch mode can be ensured. The mathematical and numerical analysis of this configuration is presented. [ABSTRACT FROM AUTHOR]

The paper is focused on reliability analysis of time-consuming mathematical models utilizing approximation in form of Taylor series expansion. Statistical analysis is crucial part of reliability analysis of structures but it is still challenging to analyze time-consuming mathematical models, e.g. represented by finite element method in implicit form. Efficient alternative is an approximation of original model by explicit function in specific form. The paper is focused on approximation by Taylor series expansion for statistical analysis of functions of random variables. Although it is common to use Taylor series expansion for functions of uncorrelated random variables, it is challenging to utilize Taylor series for correlated variables and highly non-linear functions. Therefore, possibilities and pitfalls of such approach are herein discussed from engineers point of view. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical analysis of the hybrid algorithm DESA, a combination of Differential Evolution (DE) and Simulated Annealing (SA) algorithm. DESA is a DE-based algorithm with SA-like selection operator. The discussion includes the detailed algorithmic frameworks and characteristics of the DESA-population and its comparison with the classical DE algorithm. [ABSTRACT FROM AUTHOR]

Computational modelling technique represents the only method, how to assess a functionality of a designed nuclear wastes repository in a long term period that accounts for hundreds of year. Such a technique is based on exploitation of results of short-term experiments that work as input data. Since the modelling outputs are strongly dependent on input data, the material parameters in particular, the precision of these measurement together with a consistency and stability of the mathematical model used play an important role in the reliability of results produced. This paper therefore aims at sensitivity analysis of a selected diffusion mathematical model on the second Fick's law basis, that is used for a description of radionuclides transport (90Sr) in an existing radioactive waste repository concept when a hypothetical accident is assumed. Within the sensitivity analysis, the parameters of particular materials involved in the repository concept are varied by 10% which corresponds to common measurement uncertainties. The results provided by the model showed very low deviations as the envelope curves differed by up to 10.5%. Such a difference can be considered as a sign of very good model consistency, especially in the light of the logarithmic axis used for the radionuclide concentration plotting and very rough time axis that covers a period of 2000 years. Such factors, in general, do not require an increased precision and mathematical model analyzed was proved to be precise enough to be applied in further research in this field. [ABSTRACT FROM AUTHOR]

The paper presents an approach for developing a web-based system for modelling and studying the leasing operations' parameters to assist managers in decision making and training. The report proposes a program model of the methods and means for the economic evaluation of the leasing project. But the main aim of the proposed web-based system is to facilitate engineering students who study software systems for business modeling and financial technologies. On the one hand, the application illustrates theory and combines it with practice, and on the other hand, represents accurate mathematical calculations and analysis. [ABSTRACT FROM AUTHOR]

An application of small aerial vehicles is described in this paper which presents an optimization of the camber position ever-changing using models and numerical analysis. During this study, the mechanics performance of a wing must be investigated over a profile of camber of 20 percent length of the chord from edge. The model was initially modelled using a software package and by adjusting the morphing wing system's modularity and reconfigurability. The angle of attack is also explored following a comparative analysis of the mechanics parameters. A considerable effect of morphing flap camber profiles on mechanics performance is observed in the results. As a supplementary validation, we provide an analysis of wind characteristics through ANSYS; we obtain the pressure coefficients, lift (CL), drag (CD), and lift-to-drag ratio (CL/ CD). The correlation between the CL, (CL/CD) and drag cut back simulations for several preferred shapes is analyzed when getting the results. For that reason forward in optimizing the cambered profile when morphing edges because the mathematical analysis associated with it has been shown improvements in CL from 15% to 30% when morphing edges. [ABSTRACT FROM AUTHOR]

In this paper, we consider some aspects of the numerical analysis of the mathematical model of fractional Duffing with a derivative of variable fractional order of the Riemann-Liouville type. Using numerical methods: an explicit finite-difference scheme based on the Grunwald-Letnikov and Adams-Bashford-Moulton approximations (predictor-corrector), the proposed numerical model is found. These methods have been verified with a test case. It is shown that the predictor-corrector method has a faster convergence than the method according to the explicit finite-difference scheme. For these schemes, using Runge's rule, estimates of the computational accuracy were made, which tended to unity with an increase in the number of calculated grid nodes. [ABSTRACT FROM AUTHOR]

The main aim of this paper is an experimental and numerical analysis of the dynamic behavior of an active rotor with three composite blades. The study focuses on developing an effective FE modeling technique of a macro fiber composite element (denoted as MFC or active element) for the dynamic tests of active structures. The active rotor under consideration consists of a hub with a drive shaft, three grips and three glass-epoxy laminate blades with embedded active elements. A simplified FE model of the macro fiber composite element exhibiting the d33 piezoelectric effect is developed using the Abaqus software package. The discussed transducer is modeled as quasi-homogeneous piezoelectric material, and voltage is applied to the opposite faces of the element. In this case, the effective (equivalent) piezoelectric constant d33 * is specified. Both static and dynamic tests are performed to verify the proposed model. First, static deflections of the active blade caused by the voltage signal are determined by numerical and experimental analyses. Next, a numerical modal analysis of the active rotor is performed. The eigenmodes and corresponding eigenfrequencies are determined by the Lanczos method. The influence of the model parameters (i.e., the effective piezoelectric constant d33 *, voltage signal, angular velocity) on the dynamics of the active rotor is examined. Finally, selected numerical results are validated in experimental tests. The experimental findings demonstrate that the structural stiffening effect caused by the active element strongly depends on the value of the effective piezoelectric constant. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the counting problems of I-tromino tilings of regions called holey squares. An Itromino is a 3×1 rectangle formed by connecting three unit squares edge-to-edge. A holey square is an n×n square grid with a hole of size m×m removed from the center, where 1⩽m

Aiming at the innovation of the contours of automobile side, this paper presents an innovative design method of vehicle side profile based on Fourier descriptor. The design flow of this design method is: pre-processing, coordinate extraction, standardization, discrete Fourier transform, simplified Fourier descriptor, exchange descriptor innovation, inverse Fourier transform to get the outline of innovative design. Innovative concepts of the innovative methods of gene exchange among species and the innovative methods of gene exchange among different species are presented, and the contours of the innovative design are obtained separately. A three-dimensional model of a car is obtained by referring to the profile curve which is obtained by exchanging xenogeneic genes. The feasibility of the method proposed in this paper is verified by various aspects. [ABSTRACT FROM AUTHOR]

Carl David Tolmé Runge was a German mathematician. In 1895 he brought about a revolution in the science of what has become numerical analysis. Numerical approximation to the solution of ordinary differential equations had been based on the method of Euler. but it suffered from a low convergence rate. Runge's discovery in his 1895 paper brought the so-called order from 1 up to 2. This development was followed within a few years by extensions of his techniques which brought the possible order up to 4. The analysis of order conditions for early work on Runge-Kutta methods was based on the use of a scalar test problem y' = f (x, y), but for high order method derivations and other applications, it is appropriate to use instead an autonomous high dimensional test problem y' = f (y). Because of the special requirements of stiff problems, implicit Runge-Kutta methods have an importance of their own and these developments are briefly surveyed. The intricate manipulations required to analyse the order of complicated methods has lead to an algebraic theory and its formulation as B-series. Finally, the special applications to structure-preserving algorithms are introduced. [ABSTRACT FROM AUTHOR]