Solution Of Equation Research Articles

The extended Adomian decomposition method and its application to the rotating shallow water system for the numerical pulsrodon solutions

The rotating shallow water system is an important physical model, which has been widely used in many scientific areas, such as fluids, hydrodynamics, geophysics, oceanic and atmospheric dynamics. In this paper, we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin. By introducing some special initial values, we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings, which take the form of realistic series solutions. Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An, which fully shows the efficiency and accuracy of the proposed method. Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations. The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography, oceanic and atmospheric science.

Study of MHD Casson Nanofluid Flow Over an Inclined Stretching Sheet in Porous Media with Thermal Slip, Chemical Reaction and Radiation Effects

In the current study, the behaviour of Casson nanofluid subjected to magnetohydrodynamic (MHD) flow across an inclined stretched sheet within a porous medium has been investigated numerically. The governing equations are transformed into ordinary differential equations with the corresponding boundary conditions by employing similarity transformations. The solutions of essential equations are achieved by using the 4th-order Runge-Kutta method combined with the shooting technique. The novelty and innovative contribution are showcased through illustrative graphs that scrutinize the effect of factors that impact the velocity, temperature, and concentration profiles within assorted flow scenarios. The primary focal points of the study encompass examining variations in magnetic field strength, angle of inclination, and suction intensity that affect the fluid's velocity moderation, while improved porosity and radiation parameters lead to a rise in fluid temperature. Higher Biot numbers correlate with an increase in fluid temperature. The implications of positive coefficients of heat transfer are crucial across various fields to ensure efficient thermal management ensuring optimal performance and longevity. The numerical data presented is aligned with earlier published results for comparison. Furthermore, the variations in skin friction, Nusselt, and Sherwood numbers driven by different parameters are displayed in tables to highlight significant modifications.

EFFECT OF NONUNIFORM POROSITY IN CURVED BREAKWATER ON WATER WAVES

Abstract Assuming linear theory, the phenomenon of scattering of waves by a circular arc shaped barrier with nonuniform porosity is studied. The water region is considered to be of infinite or finite depth. Based on a judicious application of Green’s integral theorem, the corresponding boundary value problem is reduced to a hypersingular integral equation of second kind. The boundary element method and the collocation method are adopted to solve the hypersingular integral equation, and we ensure a good matching of the solutions obtained by the two methods. The reflection coefficient and energy dissipation are evaluated by using the solution of the integral equation which is then studied graphically. Different choices of distributions of pores on the barrier are considered, and we observe that the nonuniform porosity of the barrier has significant effect on the reflected wave and the energy dissipation.

Discrete-time hybrid control with risk-sensitive discounted costs

AbstractThis paper studies discrete-time hybrid control problems where the controller’s payoff function follows a risk-sensitive discounted model. The associated discount factor can depend on state and action variables, which can even take values of zero or one. We prove the existence of optimal control policies under two different hypotheses. The following technique is the dynamic programming method, where we characterize the minimum cost as the solution of the so-called dynamic programming equations and find the above optimal policies through these equations. We illustrate our theory with practical examples involving inventory-manufacturing systems, and pollution management.

Symbolic equation solving via reinforcement learning

Machine-learning methods are rapidly being adopted in a wide variety of social, economic, and scientific contexts, yet they are notorious for struggling with exact mathematics. A typical example is computer algebra, which includes tasks like simplifying mathematical terms, calculating formal derivatives, or finding exact solutions of algebraic equations. Traditional software packages for these purposes are commonly based on a huge database of rules for how a specific operation (e.g., differentiation) transforms a certain term (e.g., sine function) into another one (e.g., cosine function). These rules have usually needed to be discovered and subsequently programmed by humans. Efforts to automate this process by machine-learning approaches are faced with challenges like the singular nature of solutions to mathematical problems, when approximations are unacceptable, as well as hallucination effects leading to flawed reasoning. We propose a novel deep-learning interface involving a reinforcement-learning agent that operates a symbolic stack calculator to explore mathematical relations. By construction, this system is capable of exact transformations and immune to hallucination. Using the paradigmatic example of solving linear equations in symbolic form, we demonstrate how our reinforcement-learning agent autonomously discovers elementary transformation rules and step-by-step solutions.

Modelling pressure dynamics in a gas pipeline with an in-line sampling

Gas pipelines, whether main, field, or urban, often operate in non-stationary modes. Changes in the operating modes of pumping stations, equipment start-up and shutdown, an in-line sampling or pumping and various factors are causes of instability of pressure, velocity, gas flow rate. Main pipelines are complex engineering systems with the pipeline itself being the main element. Fail-safety is the major factor in the operation of this system. Ensuring operational safety requires studying the movement regimes of the transported medium, particularly study of the dynamics of pressure duringstart-up or shutdown and at specific extraction points. This article aims to build a mathematical model and study the pressure dynamics in a gas pipeline with a sampling point. The theory of non-stationary motion liquid in round pipes has been strongly developed in the works of I. A. Charnyj. These works consider a large complex of engineering tasks taking into account viscous properties of the transported medium and pipe resistance in the hydraulic approximation. In the article on the basis of I. A. Charnyj's researches on the motion of real liquid in circular pipes the equation in partial derivatives of hyperbolic type is compiled. The equation describes the unsteady pressure of a horizontal section of gas pipeline with an extraction point. Using the Dirac delta function allows the formulation of the problem in the form of a single equation. Pressures are set at the ends of a given section, and the initial velocity is related to the Dirac delta function. By applying the finite Fourier sine transform, the partial differential equation is transformed into an ordinary differential equation and solved. Solution of equation is vision of a solution to the initial task. Inverse transform formulas based on Fourier theory allowed us to proceed to the solution of this task. Explicit dependences for the dynamics of unsteady pressure are obtained. The qualitative analysis of the formulas indicates the wave motion of a medium during the initial phase of operation, transitioning into a stationary mode after a brief period. The duration of the transition period depends on factors such as the hydraulic resistance coefficient and the velocity of the transported medium. Coefficient of hydraulic resistance and the velocity of the transported medium are the main factors. An example is considered for a horizontal section under isothermal flow conditions. With assumed numerical parameters, the transition to a stationary state occurs approximately 17 minutes after the process begins, as illustrated in the graphs provided in the article. Engineers can use mathematical models of the liquid and gas motion through pipes in the design of pipelines, as well as in solving tasks that arise during their operation. These tasks include monitoring the condition of the pipeline system, optimizing the operation, accumulation capacity estimates and others.

Semi-Analytical Solution of Brent Equations

Semi-Analytical Solution of Brent Equations

The photons and gluons sedimentation in the Svedberg ultracentrifuge

We consider the case of rotating black body (including the rotation of the black bath with gluons) where photons and gluons can perform the settling and sedimentation. The Planck formula for photons must be, in this situation, replaced by the Exner solution of the Lamm equation for photons.

Stability of stochastic difference equations with multiplicative and additive fading noise

In this paper, the effects of multiplicative and additive fading stochastic perturbations on the solutions of stochastic difference equations are studied, in which the strength of fading stochastic perturbations is square summable. Different from the existing research, this paper mainly studies the influence of two kinds of stochastic perturbations on the stability of the system, and applies the Lyapunov functional method, the asymptotic mean square trivial sufficiency conditions are obtained for the solutions of stochastic difference equations which based on the influence of multiplicative and additive fading noises. At the same time, the conditions of stochastic perturbation fading rate are obtained. Finally, the above conditions are verified by numerical simulation examples, and the results of numerical simulation show that: (1) the additive perturbations have less influence on the system than the multiplicative perturbations; (2) the fading stochastic perturbations can make the system reach a stable state in a faster time.

Numerical search for the effective thermal conductivity of cracked media

Spacecraft parts accumulate damage during operation and defects that are invariably present even in new designs may grow. This leads to changes in the behavior of individual parts of the space vehicle and, consequently, to the risk of fracture. A more accurate assessment of spacecraft safety requires internal defects to be included in the material models under consideration. One of the main hazardous effects on space objects is multiple temperature heating and cooling due to periodic action of solar rays. This paper presents a study of thermal conduction of media containing cracks. It is carried out with the help of a technique developed by the authors to determine the effective thermal conductivity of materials and based on approximate numerical solution of the steady-state thermal conduction problem for a three-dimensional medium with cracks by the boundary element method. This technique allows to obtain the distribution of the temperature field and heat flux density at any point of the body under consideration, as well as to calculate the effective parameters of materials with high accuracy at relatively low calculation time using ordinary personal computers of average power. The basis of the numerical method presented in this paper is the decomposition of the desired solution into a series of some pre-calculated analytical solutions of the heat conduction equations. The dependence of the effective thermal conductivity on the density of thermally insulated cracks was considered. The formula of this dependence is proposed. Verification of the proposed methodology was carried out by comparing the numerical results of a number of problems with the results of other authors.

SDEs with two reflecting barriers driven by optional processes with regulated trajectories

We study the existence, uniqueness, and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by optional semimartingales. We do not assume that the probability space has to satisfy the usual conditions. We define and solve an appropriate version of the deterministic Skorokhod problem for regulated functions. Applications to currency option pricing in financial models are given.

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution.Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O(nlogn) instead of the typical O(n3) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations

AbstractIn this article, we study a stochastic optimal control problem in the pathwise sense, as initially proposed by Lions and Souganidis in [C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 735-741]. The corresponding Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a non-adapted stochastic partial differential equation, is analyzed. Making use of the viscosity solution framework, we show that the value function of the optimal control problem is the unique solution of the HJB equation. When the optimal drift is defined, we provide its characterization. Finally, we describe the associated conserved quantities, namely the space-time transformations leaving our pathwise action invariant.

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

Acceleration of Numerical Modeling of Uranium In Situ Leaching: Application of IDW Interpolation and Neural Networks for Solving the Hydraulic Head Equation

The application of In Situ Leaching (ISL) has significantly boosted uranium production in countries like Kazakhstan. Given that hydrodynamic and chemical processes occur underground, mining enterprises worldwide have developed models of reactive transport. However, modeling these complex processes demands considerable computational resources. This issue is particularly significant in the context of numerical analyses of mining processes or when modeling production scenarios in uranium mining by the ISL technique, given that a substantial portion of computational resources is allocated to solving the hydraulic head equation. This work aims to explore the applicability of PINNs to accelerate hydrodynamic simulations of the ISL process. The solution of the Poisson equation is accelerated by generating an initial approximation for the iterative method through the application of the Inverse Distance Weighting (IDW) interpolation and PINNs. The impact of various factors, including the computational grid and the spacing between wells, on both the accuracy and efficiency of initial approximation and the overall solution of the elliptic equation are explored. Employing the hydraulic head distribution obtained through PINNs as the initial approximation led to a significant reduction in computation time and a decrease in the number of iterations by a factor of 2.8 to 7.10.

Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

Rainbow Solutions of a Linear Equation with Coefficients in $${\mathbb {Z}/p\mathbb {Z}}$$

Rainbow Solutions of a Linear Equation with Coefficients in $${\mathbb {Z}/p\mathbb {Z}}$$

Efficient Solutions for Stochastic Fractional Differential Equations with a Neutral Delay Using Jacobi Poly-Fractonomials

This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties of the proposed method are rigorously analyzed. Afterward, the practical utility of the proposed numerical technique is validated through a series of test examples. These examples illustrate the method’s capability to produce accurate and efficient solutions.

TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

Analytical Solutions of the D-dimensional Klein-Gordon equation with Hua Potential via N-U Method

Analytical Solutions of the D-dimensional Klein-Gordon equation with Hua Potential via N-U Method