Form Of Differential Equations Research Articles

Equivalence for Various Forms of Quadratic Functional Equations and Their Generalized Hyers–Ulam–Rassias Stability

In this manuscript, we study the properties and equivalence results for different forms of quadratic functional equations. Using the Brzdȩk and Ciepliński fixed‐point approach, we investigate the generalized Hyers–Ulam–Rassias stability for the 3‐variables quadratic functional equation in the setting of 2‐Banach space. Also, we obtain some hyperstability results for the 3‐variables quadratic functional equation. The results obtained in this paper extend several known results of the literature to the setting of 2‐Banach space.

Optimization of Chemotherapy Using Hybrid Optimal Control and Swarm Intelligence

This study aimed to minimize the tumor cell population by using minimal medicine for chemotherapy treatment while maintaining the effector-immune cell population at a healthy threshold. Therefore, a mathematical model was developed in the form of ordinary differential equations (ODE), and the solution to Multi-Objective Optimal Control Problem (MOOCP) was obtained using Multi-Objective Optimization algorithms. In this study, the interaction of the tumor-effector cell population with chemotherapy was investigated using Pure MOOCP and Hybrid MOOCP methods. The handling of constraints and the Pontryagin Maximum Principle (PMP) differ among these methods. Swarm Intelligence (SI) and Evolutionary Algorithms (EA) were used to process the results of these methods. The numerical outcomes of SI and EA are displayed via Pareto Optimal Front. In addition, the solutions from these algorithms were further analyzed using the Hypervolume Indicator. The findings of this study demonstrate that the hybrid method outperforms Pure MOOCP via Multi-Objective Differential Evolution (MODE). The MODE produces a point on the Pareto Optimal Front with minimal distance to the origin, where the distance represents the best solution.

New Perspective to Isogeometric Analysis: Solving Isogeometric Analysis Problem by Fitting Load Function

Isogeometric analysis (IGA) is introduced to establish the direct link between computer-aided design and analysis. It is commonly implemented by Galerkin formulations (isogeometric Galerkin, IGA-G) through the use of nonuniform rational B-splines (NURBS) basis functions for geometric design and analysis. Another promising approach, isogeometric collocation (IGA-C), working directly with the strong form of the partial differential equation (PDE) over the physical domain defined by NURBS geometry, calculates the derivatives of the numerical solution at the chosen collocation points. In a typical IGA, the knot vector of the NURBS numerical solution is only determined by the physical domain. A new perspective on the IGA method is proposed in this study to improve the accuracy and convergence of the solution. Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry (right-hand side) with derivatives of the NURBS numerical solution (left-hand side). Moreover, the design of the knot vector has a close relationship to the NURBS functions to be fitted in the area of data fitting in geometric design. Therefore, the detected feature points of the load function are integrated into the initial knot vector of the physical domain to construct the new knot vector of the numerical solution. Then, they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency, which we call isogeometric collocation by fitting load function (IGA-CL). In numerical experiments, we implement our method to solve 1D, 2D, and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method. We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method, which we call isogeometric Galerkin by fitting load function (IGA-GL).

Advances in state estimation: Finding a killer application [In My View

Electrical power systems’ operation and control functions rely on analytical tools for setting safe and secure operating limits and for maintaining reliable service. These tools utilize mathematical models of the power system components and equipment in the form of differential and algebraic equations. These mathematical models are formulated and coded in the form of computer programs used to determine the power system operating state or its response to various changes and disturbances, such as generation and load variations; switching operations; and faults. The results of these analyses are heavily relied upon for the operation and control of power systems and for managing and monitoring their performance.

Direct connection between Navier and spherical harmonic kernels in elasticity

<abstract> <p>Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.</p> </abstract>

An Effective Spectral Approach to Solving Fractal Differential Equations of Variable Order Based on the Non-singular Kernel Derivative

A new differential operators class has been discovered utilising fractional and variable-order fractal Atangana-Baleanu derivatives that have inspired the development of differential equations' new class. Physical phenomena with variable memory and fractal variable dimension can be described using these operators. In addition, the primary goal of this study is to use the operation matrix based on shifted Legendre polynomials to obtain numerical solutions with respect to this new differential equations' class, which will aid us in solving the issue and transforming it into an algebraic equation system. This method is employed in solving two forms of fractal fractional differential equations: non-linear and linear. The suggested strategy is contrasted with the mixture of two-step Lagrange polynomials, the predictor-corrector algorithm, as well as the fractional calculus methods' fundamental theorem, using numerical examples to demonstrate its accuracy and simplicity. The estimation error was proposed to contrast the results of the suggested methods and the exact solution to the problems. The proposed approach could apply to a wider class of biological systems, such as mathematical modelling of infectious disease dynamics and other important areas of study, such as economics, finance, and engineering. We are confident that this paper will open many new avenues of investigation for modelling real-world system problems.

MHD Nanofluid boundary layer flow over a stretching sheet with viscous, ohmic dissipation

The objective of this research is to examine the steady incompressible two-dimensional hydromagnetic boundary layer flow of nanofluid passing through a stretched sheet in the influence of viscous and ohmic dissipations. The present problem is obtained with the help of an analytical technique called DTM-Pade Approximation. The mathematical modeling of the flow is considered in the form of the partial differential equation and is transformed into a differential equation through suitable similarity transformation. The force of fixed parameters like thermophoresis number Nt, Brownian motion number Nb, Prandtl number Pr, Lewis number Le, Magnetic field M, suction/injection S and Eckart number Ec are displayed with the aid of Figures. Our outcomes showed a greater trend in the velocity profile for the parameters of magnetics M, suction S, and nonlinear stretching parameter n. While the reverse trend is found against the temperature profile when the Prandtl number increases. Lewis number and other parameters have shown increasing behavior in the concentration profile.

A Generalized Multistep Dynamic (GMD) TOPMODEL

AbstractThere is a lack of Ordinary Differential Equation (ODE) formulations in numerical hydrology, contributing to the lack of application of canned adaptive timestep solvers; hence the continued dominance of fixed (e.g., Euler) timestep techniques despite their fundamental problems. In this paper, we reformulate Dynamic‐TOPMODEL into a constraint‐handling ODE form and use MATLAB's advanced adaptive ODE‐solvers to solve the resulting system of equations. For wider applicability, but based on existing research and/or first principles, we developed Generalized Multistep Dynamic TOPMODEL which includes: iso‐basin spatial discretization, diffusion wave routing, depth‐dependent overland flow velocity, relaxing the assumption of water‐table parallelism to the ground surface, a power‐law hydraulic conductivity profile, new unsaturated zone flux, and a reference frame adjustment. To demonstrate the model we calibrate it to a peat catchment case study, for which we also test sensitivity to spatial discretization. Our results suggest that (a) a five‐fold improvement in model runtime can result from adaptive timestepping; (b) the additional iso‐basin discretization layer, as a way to further constrain spatial information where needed, also improves performance; and (c) the common‐practice arbitrary Topographic Index (TI) discretization substantially alters calibrated parameters. More objective and physically constrained (e.g., top‐down) approaches to TI classification may be needed.

MATHEMATICAL MODEL OF THE DYNAMIC MODES OF THE MICROCLIMATE SYSTEM IN THE LIVESTOCK BUILDING WITH A HEAT EXCHANGER

The article determines the relevance of research options for the use of heat exchange equipment for the utilization of the heat of the exhaust air of livestock building. It has been analyzed that for the efficient functioning of heat-utilizers, it is necessary not only to choose optimal design parameters, but also to implement rational energy-efficient modes. It was determined that for the synthesis of the system of automatic control of the thermo-humidity mode in the room, it is necessary to determine the dynamic characteristics of the control object based on the mathematical model of dynamic modes. The analysis of the existing heat utilization systems is presented and it is stated that they have only analytical dependencies for their description, which describe the stationary regimes of heat utilization systems that determine the statics of heat and mass exchange (moisture condensation on the surface) processes. In the considered works, the combined functioning of the heat recovery unit and the livestock building in non-stationary mode is not considered. On the basis of the analysis of the heat-moisture regime of the livestock building, the ventilation system of which is equipped with a recuperative type heat recovery unit, a mathematical description in the form of differential equations of the heat and material balance of the livestock building with the heat utilization ventilation system was compiled. Mathematical models of the heat and material balance for the livestock building have been developed in the form of a system of three differential equations, which contain two interrelated parameters: temperature and air humidity. But since the parameters of the air in the room with the waste ventilation air heat utilizer depend on the parameters and mode of operation of the heat exchanger, the equation of the thermal regime of the room was developed with the equations that determine the non-stationary thermal regime of the heat utilizer. The formulated mathematical model describes the non-stationary process of heat exchange in the microclimate creation system and can be used to justify the design and operating parameters of the heat exchanger regardless of design features. The mathematical model defines the transition process in the "room-heat-utilizer" system and can be used to create a system for automatic temperature-humidity control.

Theoretical investigation of injection-locked differential oscillator

A preliminary analysis of published works on this topic showed that at present there is no sufficiently substantiated theory of such devices, and the approximate approaches used are rough and do not always meet the requirements of practice. The proposed transition from a differential self-oscillator to an equivalent single-circuit oscillator has not received a convincing justification.
 This article presents a methodology for studying a synchronized differential oscillator using rigorous methods. A mathematical model of such oscillator is presented in the form of nonlinear differential equations obtained on the basis of Kirchhoff's laws. Their analysis made it possible to substantiate the transition to the model of a single circuit LC oscillator, equivalent to a differential one. A technique for such a transition is proposed, including the procedure for determining the nonlinear characteristics of the amplifying element of this self-oscillator, based on the nonlinear characteristics of two amplifying elements of the differential oscillator.
 The mathematical model of an equivalent oscillator is represented by a non-linear differential Van der Pol equation in a dimensionless form, it is simple and accurate. This form of representation made it possible to single out a small parameter and estimate its value. In the case of small values of the small parameter, as is known, traditional methods can be used for its analysis. Thus, the task of studying the synchronization process of a differential oscillator is reduced to the study of the synchronization process of a Van der Pol oscillator. The presented results can be useful in the development of various devices based on synchronized differential oscillators.

QUASI-ANALYTIC METHOD OF LINEAR DYNAMIC SYSTEMS INVERSION

The problem of inversion of dynamic systems has become widespread while solving the problems of control, identification, and measurement problems arising during the design and research of electrical and mechanical dynamic systems. Inverting is an effective way of implementing disturbance control processes, as well as in combined control systems with a predictive model. The analysis of information sources showed that in the practical solution of most inversion problems, a number of difficulties arise, which are associated with the high sensitivity of the results in relation to the accuracy of the parameters of the mathematical model of the control object, the instability of the inverse model of non-minimum-phase objects, and the violation of the conditions of physical feasibility. The work offers an effective method of inverting linear stationary dynamic systems, free from the mentioned shortcomings in many respects. The basis of the method is the presentation of input and output signals in the form of infinite linear combinations of their derivatives. A method of determining the sequence of matrix coefficients of linear representations of input and output signals is proposed. The main theoretical result is obtaining relationships between matrix coefficients of input and output signals. The work considers mathematical models of linear dynamic systems in the form of differential equations in the state space and in the equivalent "input-output" form. The considered systems must meet the conditions of asymptotic stability, as well as the condition of equal dimensions of the input and output vectors. Requirements for mathematical models of input and output signals are given, the fulfillment of which allows, instead of infinite sums representing signals, to be limited to a finite number of terms.

SIMULATION OF THE SYSTEM OF MINERAL NUTRITION OF PLANTS IN CLOSED SOIL

One of the important factors in the high productivity of growing vegetable crops by the method of small volume of this system of vegetable production is the accurate and timely supply of mineral nutrition to their root system. During hydroponic cultivation, all mineral substances of plants are obtained from their aqueous solutions. In closed soil, the water regime should be regulated taking into account the biological characteristics of the culture, the growth phase and the intensity of light. To successfully implement the plant nutrition process, it is necessary to formulate a mathematical model of the process and equivalent transfer functions. This will make it possible to choose a rational way of management and determine the parameters of management systems. The analysis of the latest researches in the field of automation of processes of nutrition of plants grown in closed soil shows that for each specific case, a scheme of automation of this process was proposed, which led to different results of the implementation of the process of nutrition. To generalize the principles of regulating the supply of nutrient solutions to plants in closed soil, it is necessary to develop real mathematical models of processes and equivalent transfer functions that will allow choosing a rational method of management. The specificity of the object of research determines the need to use the analytical method of obtaining dynamic models as the main one. The mathematical description of the dynamics of mass transfer during drip irrigation includes the value of the evaporative capacity of plants. This value significantly depends on the parameters of the microclimate in the building. To determine the influence of these parameters on the dynamics of liquid transpiration, a mathematical model was obtained in the form of differential equations of heat and material balances. The obtained mathematical models and the structural schemes built on their basis will serve as the basis for creating a system of mineral nutrition of plants in closed soil.

A simple model for viral decay dynamics and the distribution of infected cell life spans in SHIV-infected infant rhesus macaques

The dynamics of HIV viral load following the initiation of antiretroviral therapy is not well-described by simple, single-phase exponential decay. Several mathematical models have been proposed to describe its more complex behavior, the most popular of which is two-phase exponential decay. The underlying assumption in two-phase exponential decay is that there are two classes of infected cells with different lifespans. However, with the exception of CD4+ T cells, there is not a consensus on all of the cell types that can become productively infected, and the fit of the two-phase exponential decay to observed data from SHIV.C.CH505 infected infant rhesus macaques was relatively poor. Therefore, we propose a new model for viral decay, inspired by the Gompertz model where the decay rate itself is a dynamic variable. We modify the Gompertz model to include a linear term that modulates the decay rate. We show that this simple model performs as well as the two-phase exponential decay model on HIV and SIV data sets, and outperforms it for the infant rhesus macaque SHIV.C.CH505 infection data set. We also show that by using a stochastic differential equation formulation, the modified Gompertz model can be interpreted as being driven by a population of infected cells with a continuous distribution of cell lifespans, and estimate this distribution for the SHIV.C.CH505-infected infant rhesus macaques. Thus, we find that the dynamics of viral decay in this model of infant HIV infection and treatment may be explained by a distribution of cell lifespans, rather than two distinct cell types.

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

Black-Scholes (BS) equations, which are in the form of stochastic partial differential equations, are fundamental equations in mathematical finance, especially in option pricing. Even though there exists an analytical solution to the standard form, the equations are not straightforward to be solved numerically. The effective and efficient numerical method will be useful to solve advanced and non-standard forms of BS equations in the future. In this paper, we propose a method to solve BS equations using an approach of optimization problems, where a metaheuristic optimization algorithm is utilized to find the best-approximated solutions of the equations. Here we use the Adaptive Differential Evolution with Learning Parameter (ADELP) algorithm. The BS equations being solved are meant to find values of European option pricing that is equipped with Barrier option pricing. The result of our approximation method fits well to the analytical approximation solutions.

Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plotted.

Ternary hybrid nanofluids flow on a spinning disk with nonlinear thermal radiation

AbstractResearchers are interested in the thermophysical properties of nanofluids because of the successful use of nanofluids in state‐of‐the‐art technology and engineering. Hence, hybrid and tri‐hybrid nanofluid is used to ensure the required thermal features. An incompressible ternary hybrid nanofluid (NF) flow comprised of (Al2O3) aluminum oxide, (CuO) copper oxide, and (TiO2) titanium dioxide nanoparticles across an impermeable infinite spinning disc are addressed in the present study. The magnetic field and second‐degree thermal radiation were used in the proposed model. The main aim of this assessment is to raise the consciousness of energy exhaustion in the industrial and technical sectors. The numerous applicability of ternary nanoparticles (NPs) make the proposed model more plausible, such as, Al2O3 has society‐enhancing and life‐extending applications, CuO has unique electrical, optical, and magnetic properties, while TiO2 is excessively used in plastic and paper production. For this purpose, the sensations have been expressed in the form of differential equations and the consequences are obtained using the homotopy analysis method (HAM). The graphics and statistical outcomes of the analytical simulation of the proposed model are reported through Figures and Tables.

New solutions of the soliton type of shallow water waves and superconductivity models

For uni-directional wave transmission in the smooth bottom of shallow sea water and the superconductivity of nonlinear media with dispersion systems, the (1 + 1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations are of particular interest in research to the academics. Analytical wave solutions to the stated models have been successfully constructed in this study, which might have considerable implications in describing the nonlinear dynamical behavior associated with the phenomena. The models we aim to uncover have been put into the form of differential equations with one characteristic variable expending the wave transformation coordinate and, thereupon, the rational (G′/G)-expansion technique is executed. Using the considered technique, diverse soliton solutions in suitable forms arrayed to trigonometric, rational, and hyperbolic functions have been determined. The achieved solutions are figured out in 3D profiles, assigning the free parameters involved in solutions to particular values and discussed their physical significance to bring out the inner context of the tangible incidents in the natural domain. The rational (G′/G)-expansion approach is efficient, concise, and capable of finding analytical solutions to other nonlinear models that can be considered in subsequent studies.

STABILITY ANALYSIS OF FINITE DIFFERENCE METHOD FOR ONE WAY WAVE EQUATION

There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. In this paper we consider the convergence of finite difference method Lax -Wondroff one step, Lax- Wondroff two step methods and Backward time central space for solving one dimensional, time-dependent hyperbolic equation with Drichlet boundary condition. We present the derivation of the schemes and develop a computer program using python software to implement it. By the support of the numerical problems convergence of the schemes have been determined. The explicit scheme is convergent and conditionally stable and implicit scheme is convergent and unconditionally stable for any value of growth factor G .

Hydrothermal and mass aspects of MHD non-Darcian convective flows of radiating thixotropic nanofluids nearby a horizontal stretchable surface: Passive control strategy

Numerous rheological examinations attested that many products employed frequently in our daily life exhibit a thixotropic rheological behavior with a shear-thinning tendency (e.g., toothpaste, hair gels, ketchup, gypsum, and paints). Motivated by the importance of nanofluids in the thermal and mass transfer processes and their rheological trends, this article aimed to analyze convective boundary layer flows of radiating thixotropic nanofluids over a horizontal stretching surface by considering the retardational influences of Darcy-Forchheimer and Lorentz forces and invoking the effective contribution of Brownian and thermophoresis diffusions in the energy and concentration equations. By adopting the boundary layer approximations along with the zero mass flux and convective conditions, the conservation equations of the present non-homogeneous nanofluid flow model are derived mathematically in the form of partial differential equations (PDEs) based on Buongiorno's approach. After converting the leading PDEs into a set of nonlinear coupled ordinary differential equations (ODEs) via feasible local similarity alterations, the resulting highly nonlinear system of governing ODEs is then solved numerically using an efficient differential quadrature algorithm. As findings, it is found that the thixotropy feature reinforces the nanofluid motion. Besides, the significant impact of the radiative term is noticed in the heat flux rate. In contrast, the induced Darcy-Forchhemier and Lorentz forces impact unfavorably on the nanofluid flow and consequently alter the intensity of drag forces. Further results are provided also graphically and tabularly.

Precise integration solutions for the static and dynamic responses of axially graded solid beams

By virtue of the scaled boundary finite element method (SBFEM), the precise integration solutions (PIS) to deflections, stresses and transverse vibration parameters of the functionally graded solid beams are provided. It is worth to note that the mechanical parameters of the graded beams are altered across the x-axis according to power-law, exponential, trigonometric or other arbitrary theories. In SBFEM, the in-homogeneous beam is simplified into the one-dimensional model and just two high-order linear spectral elements are exploited to discretize the axially non-homogeneous beam. Nevertheless, the high accuracy can be ensured. It is essential to remind that the displacement-based beam spectral element is characterized by owning only two degrees of freedom. Additionally, the changing patterns of the displacement and stress field through the transverse direction can be expressed as an analytical matrix exponent. Aided by the two-dimensional fundamental equations of elastic materials and the employed scaled boundary coordinate, the governing equation of the axially graded solid beam is derived in the form of second order ordinary differential equation. Furthermore, the stiffness matrix and PIS of displacements and stresses are acquired from evaluating the governing equation by means of the dual vector technology and precise integration methodology (PIM). Applying the kinetic energy theory constructs the mass matrix of the axially heterogeneous beam. Afterwards the PIS to natural frequencies are obtained according to solving the eigenvalue equation. Finally, numerical exercises are presented to portray the excellent agreement and fast convergence ratio of the introduced PIS and discuss influences of supported conditions, gradient exponents, length-to-thickness ratios and formulae of gradation functions on the flexural behaviors and free vibration responses of the axially graded elastic beams.