Complex Systems Of Differential Equations Research Articles

The analysis of fractional neutral stochastic differential equations in <inline-formula id="math-09-07-845-M1"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="math-09-07-845-M1.jpg"/></inline-formula> space

<abstract><p>After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of <inline-formula id="math-09-07-845-M2"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="math-09-07-845-M2.jpg"/></inline-formula> space. This exploration involved the utilization of Jensen's, Gröenwall-Bellman's, Hölder's, Burkholder-Davis-Gundy's inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes.</p></abstract>

A Computationally-Efficient, Zero-Dimensional Stack Model for Simulating Redox Flow Battery Performance

Redox flow batteries (RFBs) are an electrochemical energy storage platform with potential to support grid-scale decarbonization and resiliency efforts. The RFB architecture allows for decoupled energy and power, long lifetimes with simplified maintenance, and a range of potential chemistries. Despite their promising characteristics, present embodiments are too expensive for widespread adoption.1 , 2 Significant research efforts have focused on advancing new redox couples and constituent materials (e.g., electrodes, flow fields, membranes), but these contributions primarily focus on individual lab-scale cells, due, at least in part, to the time, material, and expertise required to investigate performance and durability in larger device formats (e.g., multi-cell stacks). This results in knowledge gaps within the field—specifically, electrochemical reactor scaling relationships are not yet well-understood and the relative importance of many tunable molecular and material properties at the stack level remains unclear.To aid with RFB stack design and understanding, computational models of varying dimensionality and complexity have been developed to simulate charge-discharge cycling.3 Zero- and one-dimensional models, which make simplifying assumptions about underlying electrochemistry and fluid dynamics, are particularly useful for capturing general trends in cell performance at a much lower computational cost than comprehensive multi-dimensional frameworks. However, previous models still employ numerical methods to solve complex systems of differential equations, which slows simulation time and impedes broader inquiry (e.g., durational cycling, parametric property sweeps).4 ,5 Further, existing models are typically designed to interrogate specific chemistries and, as such, cannot easily incorporate the array of properties and operating conditions possible for different redox couples and constituent materials.In this presentation, we will discuss the formulation and application of a computationally-lightweight zero-dimensional RFB stack model suitable for linking stack-level electrochemical and fluid dynamic performance characteristics to molecular- and cell-level property sets. By constructing analytical expressions for mass balances and cell voltages under galvanostatic cycling conditions,6 this framework facilitates connections between system design, component material properties, operating conditions, and cycling performance at the stack level with little computational expense. To validate the functionality of the model, we simulate durational performance (i.e., capacity fade, energy efficiency, power density) across a range of different representative chemistries, operating conditions, and reactor scales. Additionally, we evaluate the impact of typical parasitic losses—including shunt currents, crossover, and species decay —on cycling characteristics over hundreds of charge-discharge cycles. Finally, we will discuss pathways to integrate this framework into techno-economic assessments to enable physics-informed cost predictions that support more holistic technology comparisons. More broadly, the tools described here have the potential to advance understanding of cell-to-stack design principles by enabling assessments of system viability early in the innovation pipeline. Acknowledgments This work was supported by the Joint Center for Energy Storage Research, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences. B.J.N gratefully acknowledges the NSF Graduate Research Fellowship Program under Grant Number 1122374. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. References B. R. Chalamala et al., Proceedings of the IEEE, 102, 976–999 (2014).Z. Yang et al., Chem. Rev., 111, 3577–3613 (2011).B. Kumar Chakrabarti et al., Sustainable Energy & Fuels, 4, 5433–5468 (2020).M. Pugach, M. Kondratenko, S. Briola, and A. Bischi, Applied Energy, 226, 560–569 (2018).J. L. Barton and F. R. Brushett, Batteries, 5, 25 (2019).B. J. Neyhouse, J. Lee, and F. R. Brushett, J. Electrochem. Soc., 169, 090503 (2022).

ABOUT THE INFLUENCE OF VIBRATIONS OF THE CONVEYED LIQUID ON THE AREA OF STABILITY OF A CLOSED SYSTEM AUTOMATIC VEHICLE COURSE CONTROL

The existing publications that investigate vehicle course stability optimization were analyzed. A mathematical model, which describes the disturbed movement of a car with a tank, was compiled. This model allows to consider the liquid free surface oscillations and determine their effect on the car course stability during constant motion or emergency braking. There was described the main information regarding the car that was used to perform mathematical calculations. An algorithm was developed for deriving the characteristic equation for a complex system of differential equations describing dynamic changes in the movement parameters of a car, oscillations of partial layers of liquid in a tank and the operation of an electromagnetic drive of the control valve and an electronic PID controller for a two-circuit system for ensuring course stability. Based on the developed mathematical model, the influence of forced oscillations of the fluid on the stability area of the system built in the plane of variable parameters of the controller is investigated. It is shown that low-frequency oscillations of the free surface of a liquid lead to a significant reduction in the stability area, which indicates the need to consider such oscillations when solving problems of analysis and synthesis of this system. It was found that for a car with a tank, where low-frequency transverse oscillations of the liquid occur, which are accompanied by a redistribution of mass and disturb the movement, an increase of the speed unambiguously leads to a deterioration in directional stability. That enables exclusion of speed from the number of variable parameters and significantly simplify the problem being solved. The calculations for cases with different loading levels were performed. It was found out that the level of liquid in the tank, considering its relationship with the speed, has an ambiguous effect on the car course stability, and it is unacceptable to limit the research calculations to the case with 50 % load. Instead of this, it is necessary to find a line that bends from above the stability boundaries that correspond to many liquid levels.
 Keywords: fluid vibrations; exchange rate stability system; area of stability; tank; PID-controller; parameters.

Investigation of the process of pneumatic vibration separation of grain by density during one-dimensional movement of grain flow

Annotation Purpose. Formation of a mathematical description of the process of moving grains with different density in vibro-pneumatic liquefied grain medium under the action of pulsating air flow to determine the parameters of movement and the possibility of their separation into fractions. Methods. Theoretical determination of the dependences of the motion of grains with different densities in a vibro-pneumatic grain medium with pulsating air flow determines the mathematical method of research based on the addition and numerical solution of the equations of grain motion in a pneumatic vibrating separator. The mathematical model is formed as a result of drawing up the scheme of force interaction on grain in the vibro-pneumatic liquefied grain environment and the differential equations of movement of grains which are presented by a material point and indicators characterizing physical and mechanical properties of grain environment are received. Results. The motion of the center of mass of grains is considered as the motion of a certain particle, the coordinates of which coincide with the coordinates of the moving coordinate system (x, y), and the mass of a single grain m differs from the grains of the medium in density. The change of air flow velocity and the influence of pulsating air flow on the flow of grain material moving on a vibrating perforated surface are determined. The discrete-periodic change of the flow velocity, which is fed through the perforated support surface into the layer of grain material, is analytically described. The parameters of the grain motion process in an oscillating fluidized medium are determined, which are determined by the solution of a complex system of differential equations using computational methods using a PC. Conclusions. As a result of theoretical research, the possibility of increasing the efficiency of separation of grain materials by specific gravity of grains, by intensifying the process of separation of grain material provided the use of pulsating air flows to separate mixtures on vibrating surfaces and compared with existing separators using horizontal and inclined decks grain separation efficiency material increases by 35–42%. Keywords: pulsating air flow, grain, density, trajectory, fluidized medium, separation process, harmonic law of motion.

Дослідження впливу навантаженості на конструктивні елементи шарнірних машин для дорожнього будівництва

On the basis of machines with an articulated frame it is possible to create a complex of machines for road construction, repair and maintenance. Therefore, a number of comprehensive studies are needed to increase their reliability and durability. This is especially true of the main joints of the articulated joint and the hydraulic steering, which ensure the rotation of the frame parts around the vertical hinge. The research was performed in two stages. At the first stage the mathematical model of dynamic interaction of the working equipment of the car with the developed environment was created and investigated. In the second stage, a machine experiment was performed to determine the loads on the bearing system. The mathematical model is based on the physical model of the loader, which consists of three elements: two hinged parts of the frame, one installed energy module, the other ‒ the technological module and the connecting and control module. The Lagrange equation of the second kind served as a basis for modeling the dynamics of the equivalent calculated dynamic model. The solution of the complex system of differential equations allowed to obtain all the required values, which have a significant impact on the formation of the load of the coupling-control device. According to the results of mathematical modeling, the ranking of factors that affect the load of the hydraulic drive of the coupling and control device. The machine experiment allowed to determine the load on the load-bearing system and to study the effect of the impact loads from the working equipment on the hinged connection of the frame parts. As the analysis of the mathematical model showed, the place of application of force to the bucket, the angle of assembly of the frame parts and the initial speed of the loader have a significant effect on the load of the hydraulic drive. The load of the hydraulic cylinders is determined by the place of application of force to the bucket. The change in the circular torque of the coupling and control module, taking into account the characteristics of the material being developed and the stiffness of the tires, must be taken into account when calculating the frame of the articulated machine and the hydraulic drive.

Frictionally Decaying Frontal Warm-Core Eddies

Purpose. The dynamics of nonstationary, nonlinear, axisymmetric, warm-core geophysical surface frontal vortices affected by Rayleigh friction is investigated semi-analytically using the nonlinear, nonstationary reduced-gravity shallow-water equations. The scope is to enlarge the number of known (semi)analytical solutions of nonstationary, nonlinear problems referring to geophysical problems and even to pave the way to their extension to broader geometries and/or velocity fields. Methods and Results. The used method to obtain the solutions is based on the decomposition of the original equations in a part expressing their prescribed spatial structure, so that they can be transformed into ordinary differential equations depending on time only. Based on that analytical procedure, the solutions are then found numerically. In this frame, it is found that vortices characterized by linear distributions of their radial velocity and arbitrary structures of their section and azimuthal velocity can be described exactly by a set of nonstationary, nonlinear coupled ordinary differential equations. The first-order problem (i. e., that describing vortices characterized by a linear azimuthal velocity field and a quadratic section) consists of a system of 4 differential equations, and each further order introduces in the system three additional ordinary differential equations and two algebraic equations. In order to illustrate the behavior of the nonstationary decaying vortices and to put them in the context of observed dynamics in the World Ocean, the system’s solution for the first-order and for the second-order problem is then obtained numerically using a Runge-Kutta method. The solutions demonstrate that inertial oscillations and an exponential attenuation dominate the vortex dynamics: expansions and shallowings, contractions and deepenings alternate during an exact inertial period while the vortex decays. The dependence of the vortex dissipation rate on its initial radius is found to be non-monotonic: it is higher for small and large radii. The possibility of solving (semi)analytically complex systems of differential equations representing observed physical phenomena is rare and very valuable. Conclusions. Our analysis adds realism to previous theoretical investigations on mesoscale vortices, represents an ideal tool for testing the accuracy of numerical models in simulating nonlinear, nonstationary frictional frontal phenomena in a rotating ocean, and paves the way to further extensions of (semi-) analytical solutions of hydrodynamical geophysical problems to more arbitrary forms and more complex density stratifications.

Generating Function Method for the solution of point reactor kinetic equations

Point kinetic equations (PKEs) represent a simplified approach to study the reactor transients at first hand, and help to develop an insight for reactor control, operation and safety. Several methods have been around for over last five decades for accurate, stable, and reliable solution of PKEs however, their realization has become possible with the advent of modern computers. Additionally, the stiffness of PKEs inspires researchers to seek further for more robust algorithms. In this paper, an explicit algorithm based upon Generating Function Method (GFM) is proposed for the solution of such a complex system of differential equations. The method was implemented to solve conventional reactivity problems, such as step, ramp, and sinusoidal reactivity insertions and compared with the other numerical algorithms. In these problems, GFM has been found to provide numerical estimates of neutron population with comparable accuracy to other algorithms, and it is computationally efficient and relatively stable. In order to explore the feasibility of the GFM for problems in reactor dynamics, a delayed-supercritical process with temperature feedback was also simulated, and results found were numerically equivalent to available techniques, however with lesser computational effort and time.

Векторний спосіб визначення кінематичних параметрів ідеального інерційного модуля у стоповому режимі

The presence of a generalized scheme of the pulse mechanism created by A. I. Leonov does not have a universal mathematical description of parameters by continuous analytical functions, therefore each design is described, for the most part, by complex systems of differential equations, but the lack of clarity in presenting the results makes it difficult to intuitively understand dynamic processes. The objective of the research is the clarity in the study of the kinematic parameters of imbalance of the ideal inertia module during stop mode operation for predicting the dynamic indexes as continuous in time functions. The research method is based on a vector simulation of spherical motion parameters. To do this, a fixed Cartesian coordinate system is introduced so that the plane of the base circle of the initial cone of the jet conical wheel is aligned with the horizontal plane, and the axis of the satellite drive (geometric axis of the carrier) is aligned with the applicate axis. The central axis of the mechanism crossing the diameter of the jet wheel is aligned with the abscissa axis on the complementary branch of which at the initial moment there is the mass center of imbalance. In the stop mode the absolute motion of a point is determined by the result of its rotation around the satellite axis at the speed lying in the plane of the base of the initial cone of the satellite and around the drive axis at a speed which parallel to the plane of the initial cone of the jet wheel. Projections of the absolute velocity and the absolute acceleration of imbalance point on the coordinate axis, in turn, determine the projections of the components of the velocity vector and acceleration vector on the planes of the fixed Cartesian coordinate system, respectively. Determining the arms of these components in the planes of projections relative about the center of the axis makes it possible to further predict the dynamic parameters at certain points of the trajectory (at particular time or another). Analytical calculations of modular values of linear and angular velocities and accelerations, as well as drawing of the appropriate diagrams were performed using the operators of MathCAD software.

Total Poisson Boundedness of Solutions of $${\cal P}$$-Perturbed Complex Systems of Differential Equations

We introduce the concepts of $${\cal P}$$-perturbed system and, in particular, $${\cal P}$$-perturbed complex system. Based on the method of Lyapunov functions, we obtain the sufficient condition of total Poisson boundedness of solutions to the $${\cal P}$$-perturbed system with respect to any linear system with constant coefficients. Based on the method of vector Lyapunov functions and the above-stated condition, we obtain sufficient conditions of total Poisson boundedness of solutions to the $${\cal P}$$-perturbed complex system and solutions to the $${\cal P}$$-perturbed complex system with feedback loop.

Numerical solution of filtration in porous rock

The filtration problem is one of the most relevant in the design of retaining hydraulic structures, water supply channels, drainage systems, in the drainage of the soil foundation, etc. Construction of transport tunnels and underground structures requires careful study of the soil properties and special work to prevent dangerous geological processes. The model of particle transport in the porous rock, which is based on the mechanical-geometric interaction of particles with a porous medium, is considered in the paper. The suspension particles pass freely through large pores and get stuck in small pores. The deposit concentration increases, the porosity and the permissible flow of particles through large pores changes. The model of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and fractional flow through accessible pores is determined by the quasi-linear equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. This complex system of differential equations has no explicit analytical solution. An equivalent differential equation is used in the paper. The solution of this equation by the characteristics method yields a system of integral equations. Integration of the resulting equations leads to a cumbersome system of transcendental equations, which has no explicit solution. The system is solved numerically at the nodes of a rectangular grid. All calculations are performed for non-linear filtration coefficients obtained experimentally. It is shown that the solution of the transcendental system of equations and the numerical solution of the original hyperbolic system of partial differential equations by the finite difference method are very close. The obtained solution can be used to analyze the results of laboratory research and to optimize the grout composition pumped into the porous soil.

Тотальная ограниченность по Пуассону решений P-возмущенных сложных систем дифференциальных уравнений

Тотальная ограниченность по Пуассону решений P-возмущенных сложных систем дифференциальных уравнений

Mathematical model of the system “manual vibration shock shaker – fruit branch”

Purpose. Development of mathematical model of oscillating system “manual vibration shock shaker – fruit branch” for the purpose of theoretical substantiation of the parameters of the shaker. Methods. The basic positions of mathematics, theoretical mechanics, mathematical modeling, program development and numerical calculations on the PC using methods of constructing mathematical models of functioning of agricultural machines are used. Results. The paper proposes a mathematical system model “manual vibration shock shaker – fruit branch” of six differential equations describing the motion of five separate masses (the mass of branch and four masses of individual shaker strings) and differential equations of the transverse and rotational motion of the system as whole. The mathematical system model determines the regularity of the motion of all masses, as well as the reactions of the viscals of the oscillatory system to the impact and after the impact that is generated in the shock mechanism. The proposed nonlinear, complex system of differential equations solves the numerical Runge-Kutta method of the fourth order of accuracy. On the basis of the calculated data the theoretical regularities of change of movement, speed and acceleration of a branch in the place of capture are received, which confirm that in the case of interaction of the cups of the shock mechanism there is blow that is accompanied by an increase in the acceleration of the branch, which is 4–5 times greater than the acceleration of the vibration mode of operation. Conclusions 1. The mathematical model of oscillating system “manual vibration shock shaker – fruit branch” is proposed in the form of system of six differential equations that allows to theoretically substantiate the basic modes of work of the manual shaker in the vibration shock mode to provide the agrotechnical necessary extraction completeness. 2. The received theoretical regularities of change of displacement, speed and acceleration of branch at the place of capture confirm the effectiveness of the vibration shock mode of the shaker. Due to the vibration-shock mode, the acceleration of the branch at the point of transmission of disturbing forces is 4–5 times higher than the acceleration of the vibrational operation mode. Keywords: manual shakes, vibration shocking process, oscillation oscillators, mathematical model, fruit branch, harvesting.

Measuring Activation and Luminescence Time Scales of Upconverting NaYF4:Yb,Er Nanocrystals

Accurate determination of upconversion and luminescence lifetimes requires kinetic modeling of the complete time-resolved response for upconversion luminescence of NaYF4:Yb,Er nanocrystals. Prior investigations typically perform exponential fitting of the tail in the time profile or employ complex systems of differential equations to extract lifetimes. To simplify analysis while fitting the entire time-resolved response, this work introduces a set of simplified models that model the response as a convolution of upconversion and fluorescence processes. Models for two- and three-photon upconversion processes are developed and tested for NaYF4:Yb,Er nanocrystals excited by a 980 nm laser. The results are presented for the transitions 2H11/2, 4S3/2 → 4I15/2 (530 nm, green emission), 4F9/2 → 4I15/2 (650 nm, red emission), and 2H9/2 → 4I15/2 (400 nm, blue emission). Even with the same number of fitting parameters, the two- and three-photon models resulted in better fitting than the simpler models of single and ...

Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit.

Toward Multi-Input Control: A Dual-Feedback Loop Model of the Mammalian Circadian Clock

Many mathematical models have been used to describe the mammalian circadian clock, ranging from phase-only models to complex systems of differential equations. Here, we develop a 14 state model of the molecular clock, which provides sufficient detail to consider multi-input control. Assuming mass action and Michaelis-Menten kinetics, we derive a model of the transcription-translation feedback loops in the mammalian clock. Using a genetic algorithm, we fit the parameters of the model to capture the appropriate peak to trough ratios, relative abundances, and phase differences of the model species. We show that this in silico model captures much of the behavior that is observed in vitro under knockout conditions of different molecular species, validating our model. Thus, this model gives a simple but valid model of both feedback loops in the mammalian clock, providing the opportunity for enhanced control of both the phase and amplitude of the clock through multi-input control in a combination of small molecules which act on the species in both the positive and negative feedback loops.

Transversely diode-pumped alkali metal vapour laser

We have studied theoretically the operation of a transversely diode-pumped alkali metal vapour laser. For the case of high-intensity laser radiation, we have obtained an analytical solution to a complex system of differential equations describing the laser. This solution allows one to exhaustively determine all the energy characteristics of the laser and to find optimal parameters of the working medium and pump radiation (temperature, buffer gas pressure, and intensity and width of the pump spectrum).

Combining the ‘bottom up’ and ‘top down’ approaches in pharmacokinetic modelling: fittingPBPKmodels to observed clinical data

Pharmacokinetic models range from being entirely exploratory and empirical, to semi-mechanistic and ultimately complex physiologically based pharmacokinetic (PBPK) models. This choice is conditional on the modelling purpose as well as the amount and quality of the available data. The main advantage of PBPK models is that they can be used to extrapolate outside the studied population and experimental conditions. The trade-off for this advantage is a complex system of differential equations with a considerable number of model parameters. When these parameters cannot be informed from in vitro or in silico experiments they are usually optimized with respect to observed clinical data. Parameter estimation in complex models is a challenging task associated with many methodological issues which are discussed here with specific recommendations. Concepts such as structural and practical identifiability are described with regards to PBPK modelling and the value of experimental design and sensitivity analyses is sketched out. Parameter estimation approaches are discussed, while we also highlight the importance of not neglecting the covariance structure between model parameters and the uncertainty and population variability that is associated with them. Finally the possibility of using model order reduction techniques and minimal semi-mechanistic models that retain the physiological-mechanistic nature only in the parts of the model which are relevant to the desired modelling purpose is emphasized. Careful attention to all the above issues allows us to integrate successfully information from in vitro or in silico experiments together with information deriving from observed clinical data and develop mechanistically sound models with clinical relevance.

Comparison of Semi-empirical and Dislocation Density based Material Equations for Fast Modeling of Multistage Hot Working of Steel

Work hardening in metal forming can be modeled using different material model types. To capture the material response to plastic deformation semi-empirical models developed by e.g. Sellars and others compete with the physically-based – or internal variable – models developed by e.g. Bergstrøm or Kocks and Mecking. The more physical nature of the internal variable models means that they typically consist of complex systems of differential equations build upon a multitude of parameters (often > 20) to describe the various specific phenomena involved. The question thus open to debate is whether those internal variable models are more an advantage or a burden from an application point of view. In order to shed light on this question a direct comparison between semi-empirical and internal variable material models is drawn. Both model types are assessed using the following categories: model complexity, effort of model calibration, performance in compression tests, and applicability to hot rolling. The general quality of the model fitted to the same high manganese steel is demonstrated by a double hit compression tests. Additionally the material models are used to predict flow stress, recrystallized fractions and roll forces in a typical hot strip rolling schedule i.e. a complex multistage hot working operation. A difference in the best approach and necessary effort was exposed during model calibration. Validation trials reveal a good agreement between both model types. When modeling a complex rolling operation, however, profound differences in the microstructure predictions become apparent.

Modification of the Streeter-Phelps system with the aim to account for the feedback between dissolved oxygen concentration and organic matter oxidation rate

A fundamental feature of the classic Streeter-Phelps system of equations is the lack of a feedback between dissolved oxygen concentration and the rate of organic matter oxidation. This may result in physically incorrect solutions, in which dissolved oxygen concentration is negative. Such physical incorrectness is still potentially hazardous for more complex systems of differential equations, resulting from “evolutionary” complication of the classic version. An example of such solutions is given and a criterion (in the form of an inequality) is derived, providing a sufficient condition for the solution of a classic system to be a priori declared physically incorrect. It is proposed to supplement the classic system by an equation relating the concentration of dissolved oxygen and the oxidation rate of organic matter. The new system, called “closed Streeter-Phelps system” will not yield a physically incorrect solution. It can be recommended as an alternative to the classic system, which is used as the core in many complex models of water quality. An additional effect of such change will be a considerable simplification of the calibration of model parameters and the extension of their application range. Two versions of the closed system are suggested, one of which is proposed to be referred to as the system of Streeter-Phelps-Shishkin.

Numerical solutions of Burgers' equation in a B-polynomial basis

A new algorithm for approximating numerical solutions of Burgers' equation in a modified B-polynomial basis is presented. The desired solutions are expanded in terms of a set of continuous polynomials over the entire interval for the spatial part with only the time variable requiring discretization. The algorithm makes use of the Galerkin method to determine the expansion coefficients to construct initial values for expansion coefficients. Matrix formulation is used to construct system of equations in time variable which are solved using a fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. The current algorithm is implemented in solving two cases of homogeneous and non-homogeneous boundary conditions. Excellent agreement is found between exact and approximate solutions. This procedure has great potential to be implemented in more complex systems of differential equations where no exact solution is available.