Nonlinear Equations Research Articles

SOLUSI NUMERIK SISTEM PERSAMAAN NON LINEAR DALAM REAKSI STEAM REFORMING MENGGUNAKAN METODE NEWTON-RHAPSON

This study discusses the solution of a system of non linear equation from the steam reforming reaction which consist of two non-linear equation with two variables, namely the and variables which will be solved using the Newton-Raphson method. The results of the final iteration are obtained with a value of x is 0,754090792 and y is 0,700893797. These two values will be substituted into the mole fraction formula to find the equilibrium composition of the steam reforming reaction components. The research results were obtained by formulating the mole fraction of each component of a chemical compound that had been formulated, then the value of the mole fraction for each component of the chemical compound was obtained, the value of the mole fraction for CO = 0,00462, H2 = 0,25745, CO2 = 0,06089, H2O = 0,56875 and CH4 = 0,10827. Based on the results obtained, it can be concluded that the Newton-Raphson method can be applied to find the equilibrium composition of the components of the steam reforming reaction.

PETSc/TAO developments for GPU-based early exascale systems

The Portable Extensible Toolkit for Scientific Computation (PETSc) library provides scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization via the Toolkit for Advanced Optimization (TAO). PETSc is used in dozens of scientific fields and is an important building block for many simulation codes. During the U.S. Department of Energy’s Exascale Computing Project, the PETSc team has made substantial efforts to enable efficient utilization of the massive fine-grain parallelism present within exascale compute nodes and to enable performance portability across exascale architectures. We recap some of the challenges that designers of numerical libraries face in such an endeavor, and then discuss the many developments we have made, which include the addition of new GPU backends, features supporting efficient on-device matrix assembly, better support for asynchronicity and GPU kernel concurrency, and new communication infrastructure. We evaluate the performance of these developments on some pre-exascale systems as well as the early exascale systems Frontier and Aurora, using compute kernel, communication layer, solver, and mini-application benchmark studies, and then close with a few observations drawn from our experiences on the tension between portable performance and other goals of numerical libraries.

A framework for upscaling aboveground biomass from an individual tree to landscape level and qualifying the multiscale spatial uncertainties for secondary forests

ABSTRACT Secondary forests, a typical forest type in the sub-frigid zone of Northeast China, have significant potential for carbon sequestration. Accurate estimation of the Aboveground Biomass (AGB) of secondary forests and assessment of multiscale uncertainties are crucial for promoting Reduced Emissions from Deforestation and Degradation. This study developed a novel framework to upscale the AGB estimation from the tree to the landscape level and assessed multiscale uncertainties based on multi-platform laser scanning data and Unmanned Aerial Vehicle (UAV) hyperspectral images. The framework included two stages: (1) quantifying multiple uncertainties (uncertainties of individual tree crown delineation, individual tree parameters estimation, and tree species classification) in individual tree-based AGB estimation using Monte Carlo simulations; (2) upscaling the plot to the landscape level estimated AGB using the Nonlinear Simultaneous Equation (NSE) with error-in-variables and quantifying the uncertainties of model residuals, model parameters, and model independent variables. The findings revealed a high accuracy from tree to plot AGB estimation (R2: 0.75, Root Mean Square Error (RMSE): 6.65 Mg/ha, relative RMSE (rRMSE): 5.40%), with the total and relative uncertainties of 16.85 Mg/ha and 16.29%, respectively, with the highest uncertainty (9.73 Mg/ha) observed in tree species classification. The AGB estimation using NSE achieved an R2 of 0.69, with an RMSE of 9.91 Mg/ha and an rRMSE of 10.43% from the plot to landscape level; and the uncertainties caused by model parameters, independent variables, and residuals were 5.52 Mg/ha, 14.56 Mg/ha, and 25.25 Mg/ha, respectively, accounting for 3.46%, 24.09%, and 72.45% of the total uncertainty. This study develops a framework for large-scale AGB estimation of mixed forests based on the individual tree approach and uncertainty quantification of multiscale estimates and provides a foundation for precise forestry, sustainable forest management, and carbon neutrality.

Exponential almost Sure Stabilization of Nonlinear Delay Differential Systems under Stochastic Optimal Control Driven by Ito Brownian Noise

This study investigates the role of Brownian white noise in stabilizing nonlinear optimal control delay differential equations (OCDDES) that are typically unstable in their deterministic form. The technique applied involves the use of Lyapunov sample exponent and a specialized partial differential equation suggested by Mao, (1997). It is demonstrated that if the noise scaling parameters of the stochastically perturbed equation is finite, then the new stochastic optimal control delay differential equation (SOCDDES ) is self - stabilized in an almost sure exponential sense. This phenomenon does not occur in deterministic optimal control delay differential equations where noise is absent.

Controlled Filtered Poisson Processes

Filtered Poisson processes are used as models in various applications, in particular in statistical hydrology. In this paper, controlled filtered Poisson processes are considered. The aim is to minimize the expected time that the process will spend in the continuation region. The dynamic programming equation satisfied by the value function is derived. To obtain the value function, and hence the optimal control, a non-linear integro-differential equation must be solved, subject to the appropriate boundary conditions. Various cases for the size of the jumps are treated and explicit results are obtained.

An improved fractional predictor-corrector method for nonlinear fractional differential equations with initial singularity

An improved fractional predictor-corrector method for nonlinear fractional differential equations with initial singularity

Approximate Analytical Approaches to Nonlinear Differential Equations: A Review of Perturbation, Decomposition and Coefficient Methods in Engineering

Approximate Analytical Approaches to Nonlinear Differential Equations: A Review of Perturbation, Decomposition and Coefficient Methods in Engineering

A General Perturbed Newtonian Framework and Critical Solutions of Nonlinear Equations

A General Perturbed Newtonian Framework and Critical Solutions of Nonlinear Equations

Numerical Study of Williamson Fluid Flow over a Stretching Sheet with Newtonian Heating Embedded in a Porous Medium in Presence of Thermal Radiation and Heat Source/Sink

Williamson fluid has been utilized to explore the aligned magnetic field in the presence of thermal radiation and heat source/sink utilizing Newtonian heating on a stretched sheet soaked in porous medium. The governing partial differential equations are transformed into nonlinear ordinary differential equations by applying similarity transformations. These equations are then solved using MATLAB and the Runge-Kutta fourth-order technique with a shooting technique. Graphs were used to investigate the effects of different factors on dimensionless velocity and temperature. A decrease in the velocity field and an improvement in the distribution of temperature are caused by an enhancement in the Non-Newtonian Williamson fluid parameter and permeability restriction K, whereas the opposite effect is evident when the Aligned angle parameter is increased. When the conjugate factor for Newtonian heating parameter is increased, temperature rises, even as the reverse effect is seen in Prandtl number The coefficient of skin friction is an decreasing function and the Nusselt number is increasing function of Non-Newtonian Williamson fluid parameter and permeability restriction K. while the reverse effect is seen in Aligned angle parameter . The Nusselt number is increased function of heat source/sink parameter , and Newtonian heating parameter . While the reverse effect is seen in Prandtl number .

Solving nonlinear fractional equations and some related integral equations under a measure of noncompactness

Solving nonlinear fractional equations and some related integral equations under a measure of noncompactness

Study of Burger Equation Using <i>q</i>-HAM with Yang-Abdel-Cattani Derivative

In order to solve the generalized Burgers equation, this research work introduces one of the most recent operators in fractional calculus. An easier-to-understand version of the problem can be obtained by using the Yang-Abdel-Cattani fractional operator. The generalized Burgers equation's result is obtained analytically effectively using the q-homotopy analysis method (q-HAM). A visual study is also acquired to demonstrate the operation of the technique. By employing this approach to solve the generalized Burgers problem, this study advances the field of nonlinear differential equations.

Hybridizing evolutionary algorithms and multiple non-linear regression technique for stream temperature modeling

Abstract The present study hybridizes the new-generation evolutionary algorithms and the nonlinear regression technique for stream temperature modeling and compares this approach with conventional gray and black box approaches under natural flow conditions, providing a comprehensive assessment. The nonlinear equation for water temperature modeling was optimized using biogeography-based optimization (BBO) and invasive weed optimization (IWO), simulated annealing algorithm (SA) and particle swarm optimization (PSO). Two black box approaches, a feedforward neural network (FNN) and a long short-term memory (LSTM) network, were also employed for comparison. Additionally, an adaptive neuro-fuzzy inference system (ANFIS) served as a gray box model for river thermal regimes. The models were evaluated based on accuracy, complexity, generality and interpretability. Performance metrics, such as the Nash–Sutcliffe efficiency (NSE), showed that the LSTM model achieved the highest accuracy (NSE = 0.96) but required significant computational resources. In contrast, evolutionary algorithm-based models offered acceptable performance while reducing the computational complexities of LSTM, with all models achieving NSE values above 0.5. Considering interpretability, accuracy and complexity, evolutionary-based nonlinear models are recommended for general applications, such as assessing thermal river habitats. For tasks requiring very high accuracy, the LSTM model is preferred, while ANFIS provides a balanced trade-off between accuracy and interpretability, making it suitable for engineers and ecologists. While all models demonstrate similar generality, this model is developed for a specific location. For other locations, independent models with a similar architecture would need to be developed. Ultimately, the choice of model depends on specific objectives and available resources.

An Upscaling Machine Learning Method Based On Physical Information

Objective: The objective of this study is to investigate a machine learning methodology based on the physics of dynamic fluid flow, with the aim of incorporating the knowledge of physical laws into learning algorithms to estimate precise results with lower computational cost. Theoretical Framework: This section presents the main concepts and theories that underpin the research. Key theories include nonlinear partial differential equations (PDEs), Darcy's Law, and the conservation of mass and energy, providing a solid foundation for understanding the research context. Method: The methodology adopted for this research involves modeling a one-dimensional two-phase fluid flow problem in a porous medium, governed by a first-order hyperbolic nonlinear equation. Data collection was conducted through numerical simulations, using micro and macro grids to evaluate permeability and boundary conditions. Results and Discussion: The results revealed that considering learning only in absolute permeability yielded good estimates of reservoir pressures. However, small discrepancies were observed in the estimation of saturations. The discussion contextualizes these results in light of the theoretical framework, highlighting the identified implications and relationships, as well as the study's limitations. Research Implications: The practical and theoretical implications of this research are discussed, providing insights into how the results can be applied or influence practices in the field of reservoir modeling and fluid flow simulations. These implications may include optimizing oil extraction processes and improving predictive models in reservoir engineering. Originality/Value: This study contributes to literature by presenting an innovative approach that integrates physical laws into machine learning algorithms, resulting in more accurate and efficient estimates. The relevance and value of this research are evidenced by the potential application of the results in optimizing industrial processes and improving computational simulations.

MATHEMATICAL MODELING AND Z-CONTROL MECHANISM TO MITIGATE CHLAMYDIA TRACHOMATIS-INDUCED CONJUNCTIVITIS IN NEWBORNS

This study analyzes a mathematical model employing a Z-control mechanism to prevent neonatal Chlamydial conjunctivitis in newborns of infected mothers. Chlamydia trachomatis transmission during delivery can lead to conjunctivitis and severe complications in the lungs and nasopharynx if untreated. The model, based on nonlinear differential equations, evaluates the basic reproduction number at the disease-free equilibrium point. Numerical simulations demonstrate that Z-control effectively stabilizes chaotic oscillations in the system, reducing the risk of conjunctivitis in newborns. The findings highlight that prenatal testing, early treatment of pregnant women, and newborn screening are critical for preventing and controlling Chlamydia transmission. The implementation of Z-control has proven effective in mitigating this epidemic, significantly reducing the incidence of neonatal Chlamydial conjunctivitis.

Boundary estimates for doubly nonlinear parabolic equations

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation ∂tuq-div|Du|p-2Du=0in the super-critical fast diffusion regime 0<p-1<q<N(p-1)(N-p)+. We show that when solutions vanish continuously at the Lipschitz boundary of a parabolic cylinder ΩT, they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain ΩT, we obtain a power-like decay at the boundary and a boundary Harnack inequality.

A Caputo-based nonlocal arithmetic-mean discretization for solving nonlinear time-fractional diffusion equation using half-sweep KSOR

A Caputo-based nonlocal arithmetic-mean discretization for solving nonlinear time-fractional diffusion equation using half-sweep KSOR

Nonlinear wave dynamics of (1+1)- dimensional conformable coupled nonlinear Higgs equation using modified ( G′G2)-expansion method

Abstract The conformable coupled nonlinear Higgs equation has been analyzed here using the modified \((\frac{G'}{G^2})\)-expansion method. This equation describes the dynamics of conserved scalar nucleons interacting with neutral scalar mesons. Accordingly, new exact solutions have been obtained. Functions derived from trigonometry, hyperbolic operations, and rational functions are used to define the traveling waveform responses. These solutions are illustrated with various interesting figures for different parameters. Additionally, physical interpretations of these results are provided. The solutions uncovered in this study are more thorough, wide-ranging, and, in some cases, novel, as they have not been discovered in previous studies. {\color{red}These solutions may be useful in understanding nonstandard interactions or forces and potentially addressing phenomena beyond the standard model.} The implemented strategies are efficient, align with computer algebraic expressions, and provide a sufficient number of diverse wave solutions.

High order semi-IMEX BDF schemes for nonlinear partial integro-differential equations arising in finance

In this paper, semi-implicit-explicit (Semi-IMEX) and semi-implicit multistep methods are proposed to solve nonlinear partial integro-differential equations (PIDEs), which describe the option pricing models with transaction costs or illiquid markets under Merton's jump-diffusion process. After spatial differential operators are treated by using finite difference methods and the jump integral is computed by using the composite trapezoidal rule, the consistency error and global error bounds for the semi-IMEX and semi-implicit multistep methods for abstract PIDEs are provided when the nonlinear operator satisfies the boundedness and coercivity conditions. A numerical study is carried out for different option pricing models based on the convergence properties of the schemes and the comparison of the different Greek values. Numerical experiments verify our theoretical results and illustrate the intrinsic nature of the proposed option pricing models under jump-diffusion models with transaction costs or illiquid markets.

Application of fractional shifted Vieta-Fibonacci polynomials in nonlinear reaction diffusion equation with variable order time-space fractional derivative

Abstract In this article, an accurate optimization algorithm based on new polynomials namely generalized shifted Vieta-Fibonacci polynomials (GSVFPs) is employed to solve the nonlinear variable order time-space fractional reaction diffusion equation (NVOTSFRDE). The algorithm combines GSVFPs, new variable order fractional operational matrices in the Caputo sense, and the Lagrange multipliers to achieve the optimal solution. First, the solution of the NVOTSFRDE is approximated as a series of GSVFPs with unknown coefficients and parameters. Then, the Lagrange multipliers method is adopted so that the NVOTSFRDE can be transformed into a class of nonlinear algebraic system of equations and we solve these equations using MATLAB and MAPLE software. Solving this system and substituting the coefficients and parameters into the approximation of the guessed functions, the solution of the NVOTSFRDE is obtained. The convergence analysis of the approach are discussed. The accuracy of the algorithm is verified through error analysis and mathematical examples. The accuracy of the new method is higher than that of the exciting method. The reconstruction results demonstrate that the proposed optimization algorithm is efficient for the NVOTSFRDE, and the algorithm is also convergent.

Design and performance evaluation of a novel fractional order filtered PI controller for fractional order time delay systems

This study introduces an innovative algorithm for designing a PI controller filtered with fractional-order derivatives or integrals. The algorithm is applied to two distinct models of fractional-order time delay systems. A new reference model is proposed, utilising a unity feedback structure and incorporating a fractional-order time-delayed Bode transfer function in the forward path. The parameters of this proposed reference model are determined based on the specified phase margin and gain margin requirements. Before solving the design problem, the stability analysis of the proposed reference model is analysed through a theorem. Next, the four nonlinear equations with four unknown parameters are formulated and a solution method is provided in this study. By employing the recently introduced reference model, a robust fractional-order controller is formulated. The effectiveness of this robust controller is validated through simulations that employ various phase and gain margin specifications and through the existing controllers in the literature.