Simple Differential Equation Research Articles

Sharp Cone-Broad Cone-Disk: Analytical Solutions in the Tunnel Mathematics Space to the Steady Navier-Stokes Equations in the Area of Boundary Layer for Incompressible Symmetric Flows Entrained by these Rotating Bodies

More than 150 years of history of efforts to solve the Navier-Stokes equation have clearly shown that, applying standard mathematical tools, it is possible to do this in only a small number of simple cases. Therefore, to solve such equations, we can try non-standard methods of mathematical modeling. In this case, the emphasis should be placed not on the mathematical accuracy of the proposed solutions, but on their correspondence to experimental data or solutions to the Navier-Stokes equations obtained by numerical methods. We believe that tunnel mathematics is such a method of mathematical modeling. Main theorem of tunnel mathematics allows us to reduce a system of the steady Navier-Stokes equations to simple ordinary differential equations that give solutions in planes parallel to the basic xy plane. Collecting such solutions, we finally obtain full 3D solution of a system of the steady Navier-Stokes equations. Approximate solutions for a system Sharp cone—Broad cone—Disk in the area of boundary layer can be obtained without use of specific software (including case of turbulent motion of fluid). We get solution for a rotating disk as a limit transition for a broad cone. If such solution will be similar with famous Karman`s solution for a rotating disk (we mean laminar flows), then we could conclude that our theory is successive.

Construction and analysis of symmetric Sprott B multi-attractors with electric implementation

Abstract In chaos theory, a number of systems and models which apparently contain simple ordinary differential equations (ODEs) turn out to show a dynamic characterized by complicated behaviors and complex trajectories. One of such systems is the Sprott B model. We construct some set of multi-attractors based on the Sprott B model where additional parameters and operators are considered. After summarizing important preliminaries relevant to simple chaotic differential systems, the model is firstly solved analytically and numerically, then graphical simulations are provided. The later show coexistence and evolution of two chaotic attractors in a symmetrical representation. Lastly, similar results and expected outcomes are recovered via an electrical circuit implementation, realized using the Field Programmable Gate Array (FPGA) board, the Digital-to-Analog Converter (DAC) and the Rigol Oscilloscope. They also show progressing sets of coexisting multi-attractors.

Nonlinear Perturbations and Weak Shock Waves in Isentropic Atmospheres

Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low-amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane-parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham’s approximation can be used to solve for the shock’s dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate (∼few–15) Mach numbers, where the accuracy increases with the adiabatic index.

Numerical calculation of groundwater geofiltration processes in multilayer porous media

In this The article presents the modeling of multilayer hydrogeological processes in porous media, the results of fundamental and applied research, numerical modeling of hydrogeological processes, computational mathematics and methods of finite-difference schemes for solving simple differential equations, the development and improvement of mathematical models, and considers computational algorithms and software tools for solving problems analysis and forecasting of processes.

Evaluating single multiplicative neuron models in physics-informed neural networks for differential equations

Machine learning is a prominent and highly effective field of study, renowned for its ability to yield favorable outcomes in estimation and classification tasks. Within this domain, artificial neural networks (ANNs) have emerged as one of the most powerful methodologies. Physics-informed neural networks (PINNs) have proven particularly adept at solving physics problems formulated as differential equations, incorporating boundary and initial conditions into the ANN’s loss function. However, a critical challenge in ANNs lies in determining the optimal architecture, encompassing the selection of the appropriate number of neurons and layers. Traditionally, the Single Multiplicative Neuron Model (SMNM) has been explored as a solution to this issue, utilizing a single neuron with a multiplication function in the hidden layer to enhance computational efficiency. This study initially aimed to apply the SMNM within the PINNs framework, targeting the differential equation y′-y=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y'-y=0$$\\end{document} with boundary conditions y(0)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(0) = 1$$\\end{document} and y(1)=e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(1) = e$$\\end{document}. Upon implementation, however, it was discovered that while the conventional SMNM approach was theorized to offer significant advantages, multiplicative aggregate function led to a failure in convergence. Consequently, we introduced a “mimic single multiplicative neuron model (mimic-SMNM)” employing an architecture with a single neuron, designed to simulate the SMNM’s conceptual advantages while ensuring convergence and computational efficiency. Comparative analysis revealed that the real-PINNs accurately solved the equation, the true SMNM failed to converge, and the mimic model was highlighted for its architectural simplicity and computational feasibility, directly implying it is faster and more efficient than real PINNs for the solution of simple differential equations. Furthermore, our findings demonstrated that our proposed mimic-SMNM model achieves a five-times increase in computational speed compared to real PINNs after 30,000 epochs.

A stochastic approximation for the finite-size Kuramoto–Sakaguchi model

We perform a stochastic model reduction of the Kuramoto–Sakaguchi model for finitely many coupled phase oscillators with phase frustration. Whereas in the thermodynamic limit coupled oscillators exhibit stationary states and a constant order parameter, finite-size networks exhibit persistent temporal fluctuations of the order parameter. These fluctuations are caused by the interaction of the synchronised oscillators with the non-entrained oscillators. We present numerical results suggesting that the collective effect of the non-entrained oscillators on the synchronised cluster can be approximated by a Gaussian process. This allows for an effective closed evolution equation for the synchronised oscillators driven by a Gaussian process which we approximate by a two-dimensional Ornstein–Uhlenbeck process. Our reduction reproduces the stochastic fluctuations of the order parameter and leads to a simple stochastic differential equation for the order parameter.

Truncated Taylor Solvers for Simple First-Order Ordinary Differential Equations

Certain exact solutions of first-order ordinary differential equations naturally become corresponding solvers for simple firstorder ordinary differential equations whose form is the equality of a twice continuously differentiable function of the dependent variable to the derivative with respect the independent variable of the dependent variable. When the twice continuously differentiable function of the dependent variable is replaced by its truncated Taylor expansions through second order about its initial value, the resulting first-order ordinary differential equations have exact solutions that naturally become corresponding solvers for those simple first-order ordinary differential equations

A Model of Linear Response with Progressive Decay Equally Suited for RiboGreen® Quantitation of RNA and Dissolution of Solid Oral Dosage forms

A new regression model is presented which offers flexibility, freedom from subjective determinations of linear range, and very wide applicability to measurement systems of industrial importance. This “progressive decay” model starts as a deceptively simple ordinary differential equation. We show here that its solution faithfully describes real but seemingly unconnected data from a plate-based assay for quantitation of RNA with RiboGreen® and dissolution data for a triple fixed-dose combination solid oral dosage form.

Pemodelan Konveksi Panas Pada Fluida Statis dan Dinamis Dalam Bentuk Persamaan Diferensial Orde 1

This article discusses the importance of understanding heat convection in fluids in the context of physics and engineering. Using first order Simple Differential Equations (PDS), we can analyze the temperature distribution in a fluid over time and space in high detail. PDS allows modeling heat convection by considering parameters such as temperature differences and fluid flow velocity. Numerical methods are used to complete the PDS computationally, while data collection techniques through literature studies provide an in-depth understanding of relevant theories and previous findings. With the application of PDS and numerical methods, we can better understand and predict heat transfer in fluids, which has wide applications in engineering, biology, and physics. In conclusion, this article provides a comprehensive insight into the use of PDS in the analysis of heat convection in static and dynamic fluids, with a focus on mathematical and computational approaches to better understand this phenomenon.

Short-time expansion of one-dimensional Fokker-Planck equationswith heterogeneous diffusion.

We formulate a short-time expansion for one-dimensional Fokker-Planck equationswith spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation(the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equationswhich can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

Periodic Solutions in a Simple Delay Differential Equation

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations.

Solving a Predictive Model of Infectious Disease in Epidemiology and Epizootiology by Assigning Scores to Individuals

Every mathematical solution to a problem corresponds to a model that more or less corresponds to reality. One of the many possibilities is the solution to epidemiological problems (processes) using the susceptible-infected-removed (SIR) model. Like any model, this model also has its advantages and benefits that allow us to close such a complicated reality as the origin and course of the epidemic into simple differential equations (DE) and through the model’s parameters modify it and compare it with reality. This is possible only in the case of a comprehensive understanding of reality in all its breadth and depth. This study focuses on another solution to the epidemiological or epizootiological processes based on the newly designed simulation, which assigns individuals a certain score in time intervals. In this case, the score from the simulation has a normal probability distribution with certain parameters N (µ, σ) in given time intervals. This score level also divides the given individuals into certain groups based on the given (selected) critical level. This determines whether individuals are with manifestations of the disease or without signs of the disease. In the study, the epidemic curve (EC) simulation results could be used to solve the current pandemic caused by COVID-19. It can be stated that the newly proposed model could be more suitable because it permanently confirms the agreement of experimental and expected frequencies.

Approximate Solution of Fractional Order of Partial Differential Equations Using Laplace-Adomian Decomposition Method in MATLAB

This article presents the application of the Laplace-Adomian Decomposition Method (LADM) for solving partial differential equations (PDEs) in the context of heat conduction and wave propagation. The LADM combines Laplace transform and Adomian decomposition to approximate solutions to PDEs efficiently in MATLAB. The procedure involves transforming the PDE into simpler differential equations, which are then solved iteratively using the Adomian decomposition method. The advantages of LADM include simplicity, flexibility, and applicability to a wide range of PDEs. We demonstrate the effectiveness of LADM through numerical experiments solving the heat equation and wave equation using MATLAB. The results show good agreement with analytical solutions and highlight the efficiency and accuracy of LADM for solving PDEs.

In situ study and assessment of the phosphorus-induced solute drag effect on the grain boundary mobility of austenite

The solute drag effect of phosphorus (P) in the single-phase austenite (γ-Fe) region was studied under isothermal annealing conditions at temperatures of 1050 °C, 1150 °C, 1250 °C and 1350 °C. High-temperature laser scanning confocal microscopy (HT-LSCM) was used to observe and quantify in situ grain growth on the surface of three different samples containing 0.026 wt.-% P, 0.044 wt.-% P and 0.102 wt.-% P. The dependence of the arithmetic mean grain sizes on time and temperature were modeled mathematically using a simple ordinary differential equation (ODE) according to normal grain growth (NGG) theory. As no other major effects, i.e., Zener-pinning by precipitates, occurred under the selected experimental conditions, grain growth interference was only considered by grain boundary (GB) segregation of P. Thus, the total grain boundary mobility (M) was directly determined depending on the P content and isothermal annealing temperature. The fitted GB mobility values enabled the determination of an average binding energy value between impurity P atoms and grain boundaries (E0 = -1.1 - -0.6 eV) in the system. Finally, the GB segregation of P in γ-Fe was derived from the observed grain growth kinetics. The results showed reasonable agreement with calculations using segregation enthalpies from the literature.

Multilevel optimization for policy design with agent-based epidemic models

Epidemiological modeling has a long history and is often used to forecast the course of infectious diseases or pandemics. These models come in different complexities, ranging from systems of simple ordinary differential equations (ODEs) to complex agent-based models (ABMs). The former allow a fast and straightforward optimization, but are limited in accuracy, detail, and parameterization, while the latter can resolve spreading processes in detail, but are extremely expensive to optimize. Epidemiological modeling can also be used to propose and design non-pharmaceutical interventions such as lockdowns. In general, their optimal design often leads to nonlinear optimization problems. We consider policy optimization in a prototypical situation modeled as both ODE and ABM, review numerical optimization approaches, and propose a heterogeneous multilevel approach based on combining a fine-resolution ABM and a coarse ODE model. Numerical experiments, in particular with respect to convergence speed, are given for illustrative examples.

Continuum Modeling and Boundary Control of a Satellite with a Large Space Truss Structure

Due to its advantages of easy deployment and high stiffness-to-mass ratio, the utilization of truss structures for constructing large satellites presents an appealing solution for modern space missions, including Earth observation and astronomy. However, the dimensions of the traditional finite element model for a satellite with a large space truss structure become exceedingly large as the structure’s size increases. The control system design process based on the finite element model is complex and time-consuming. This paper employs the continuum modeling method to represent the truss structure as a continuous entity. The bending vibrations of the truss structure are encapsulated by a simplified partial differential equation (PDE), as opposed to the more intricate traditional finite element model. Simultaneously, the satellite’s attitude motion is characterized by an ordinary differential equation (ODE). Building upon this coupled PDE-ODE model, a boundary control law that only requires sensors/actuators at the boundary is formulated to effectively mitigate structural vibrations and regulate the satellite’s attitude. The exponential stability of this closed-loop system is scrutinized using Lyapunov’s direct method. The simulation results affirm that the continuum modeling method is indeed well-suited for satellites endowed with substantial truss structures, and the proposed boundary law proves to be highly effective in both attitude tracking and vibration suppression.

In biology, solving a problem coming to a differential equation in the maple program

We know that not only the problems of mathematics, but also the mathematical model of a number of processes that occur in nature can been reduced to a differential equation. Most of the quantities found in nature have their own laws. Finding these, laws directly have more complicated matter. Finding the relationship between the quantity in question, its rate of change and acceleration is quite easy by nature. Simple differential equations have formed as a mathematical expression of this connection. It is important and significant to use modern computer programs to find a quick and accurate solution to such equations. Found in Maple.

INSIGHT INTO THE IMPACT OF MELTING HEAT TRANSFER AND MHD ON STAGNATION POINT FLOW OF TANGENT HYPERBOLIC FLUID OVER A POROUS ROTATING DISK

Melting heat transfer plays a crucial role in many industrial devices, including heat exchangers, air conditioning, and metal casting. Considering these uses the heat transmission in three-dimensional tangent hyperbolic fluid flow is evaluated. The effects of magnetohydrodynamics (MHD), Ohmic heating, porous medium and melting heat transfer at the boundary are applied to the stretching rotating disk. The governing equations are transformed into a nondimensional form after applying a similarity transformation. The simplified ordinary differential equations contain various dimensionless terms, and the results of these variables are obtained by the bvp4c method. The graphical and tabular results for existing parameters are displayed. For the validation of our results, a comparison is done. From the outcomes, it is noticed that velocity and temperature profiles are enhanced with melting heat transfer at the boundary. The porosity parameter reduces the velocity of the tangent hyperbolic fluid. Moreover, the Eckert number demonstrates the dual nature of temperature profiles.

A mathematical model of visceral leishmaniasis transmission and control: Impact of ITNs on VL prevention and elimination in the Indian subcontinent.

Visceral Leishmaniasis (VL) is a deadly, vector-borne, parasitic, neglected tropical disease, particularly prevalent on the Indian subcontinent. Sleeping under the long-lasting insecticide-treated nets (ITNs) was considered an effective VL prevention and control measures, until KalaNet, a large trial in Nepal and India, did not show enough supporting evidence. In this paper, we adapt a biologically accurate, yet relatively simple compartmental ordinary differential equations (ODE) model of VL transmission and explicitly model the use of ITNs and their role in VL prevention and elimination. We also include a game-theoretic analysis in order to determine an optimal use of ITNs from the individuals' perspective. In agreement with the previous more detailed and complex model, we show that the ITNs coverage amongst the susceptible population has to be unrealistically high (over 96%) in order for VL to be eliminated. However, we also show that if the whole population, including symptomatic and asymptomatic VL cases adopt about 90% ITN usage, then VL can be eliminated. Our model also suggests that ITN usage should be accompanied with other interventions such as vector control.

Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations

Abstract Recently, integral transforms are a powerful tool used in many areas of mathematics, physics, engineering, and other fields and disciplines. This article is devoted to the study of one important integral transform, which is called the modified degenerate Laplace transform (MDLT). The fundamental formulas and properties of the MDLT are obtained. Furthermore, as an application of the acquired MDLT, we solved a simple differential equation and fractional-order kinetic equations. The outcomes covered here are general in nature and easily reducible to new and known outcomes.