Nonlinear Algebraic Equations Research Articles

An efficient approach for solving a class of fractional anomalous diffusion equation with convergence

Abstract This article presents a study on Fractional Anomalous Diffusion (FAD) and proposes a novel numerical algorithm for solving Cupoto’s type fractional sub-diffusion equations. to convert the fractional model into a set of nonlinear algebraic equations. These equations are efficiently solved using the Levenberg-Marquardt algorithm. The study provides the error analysis to validate the proposed method. The effectiveness and accuracy of the method are demonstrated through several test problems, and its performance and reliability are compared with other existing methods in the literature. The results indicate that the proposed method is a reliable and efficient technique for solving fractional sub-diffusion equations, with better accuracy and computational efficiency than other existing methods. The study’s findings have important implications for researchers working in the field of fractional calculus. They could provide a valuable tool for solving sub-diffusion equations in various applications, including physics, chemistry, biology, and engineering.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

Advancements in numerical modeling of ideal multicomponent isotope separation Cascades: A comparative study

The complexity of operational conditions in stable isotope separation cascades necessitates the utilization of numerical methods for cascade design and performance analysis. This study focuses on solving a system of nonlinear algebraic equations derived from the mass and energy balance equations, which is a fundamental challenge in analyzing and designing cascades and developing numerical methods. Two existing numerical algorithms were selected from the literature and modified for quasi-ideal cascade systems. The performance of these algorithms was examined and compared, with a particular emphasis on influential parameters such as the number of stages, the number of isotopes, and the separation factor. The first algorithm solves the concentration and flow equations separately, while the second algorithm solves all equations simultaneously. The results indicate that the first algorithm exhibits significant superiority over the second algorithm. In most cases, the second algorithm either fails to converge or exhibits a slower convergence speed. Additionally, the results demonstrate that increasing the number of stages impacts the convergence time, while an increase in the separation factor leads to greater convergence complexity.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

Analyzing the Effects of Polymeric Dielectric Materials on Micro Capacitive Pressure Sensors: A Model Incorporating Displacement-Dependent Porosity

Recently, the extensive utilization of porous polymeric materials to amplify the sensitivity of capacitive devices is noticeable. The absence of an effective mathematical model for studying these devices has spurred the development of a comprehensive mathematical model in the current work. This model is formulated to analyze the static and dynamic behavior of systems incorporating a porous polymer dielectric material within the gap between flexible and fixed microplates. The derived nonlinear governing equations encompass the effects of electrostatic force, von-Karman nonlinear strains, and displacement-dependent porosity. Employing spatial decomposition, the resulting nonlinear algebraic equations and ordinary differential equations are leveraged to study the static and transient dynamic behavior, as well as the frequency response of the sensor using a learning approach. Two scenarios are investigated to assess the impact of various geometrical and physical parameters on sensor sensitivity one with a polymeric material and another without, each with distinct parameter values. The results reveal that the inclusion of a polymeric dielectric material increases electrostatic force but concurrently elevates the equivalent stiffness of the structure. The effectiveness of using a polymeric dielectric material is contingent upon the specific geometrical and physical properties of the sensor. Moreover, the obtained results in simplified cases are compared to existing numerical and experimental data, demonstrating a high degree of agreement. This work significantly contributes to advancing the understanding of sensors incorporating porous polymer dielectric materials and underscores their potential for enhanced sensitivity across diverse applications.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Research on Wireless Multimedia Big Data Transmission Model Based on Non-linear Algebraic Equation with the Adaptive Scheduling Algorithm

With the rapid development of computer technology, image processing and automatic book recognition has become a very important and practical field, and this discipline has a broad prospect and great potential. The digital vision system is a multi-plane information analysis method based on the human brain, which is applied to library identification problems by studying several regular changes in the distribution of objects themselves in two-dimensional space and three-dimensional position, in order to improve book quality and reduce costs. In addition, digital reading devices have become an essential part of people's daily learning life, and multimedia retrieval methods have also undergone radical changes, image processing is becoming more and more complex, and algorithms must become more efficient and reliable as the amount of computing increases to a certain level, so as to adapt to this fast and powerful data storage and query needs. Therefore, the trend is to improve the speed and reliability of computer systems. In this research, we provide a nonlinear model of optimal allocation employing an ant colony management strategy, a genetic operator, and an adaptive scheduling algorithm based on topology (dB) to discover the best scheduling scheme and test its feasibility via simulation. It proposes a viable idea for tackling the practical wireless multimedia massive data transmission model problem by applying the method to the real-time processing process in other sectors with high importance and resilience.

Uniform extension–torsion of helical birods

An analytical formulation is presented to study uniform extension–torsion behavior of helical birods. The uniformity in extension and compression reduces the set of governing nonlinear differential equations of birod theory to a set of nonlinear algebraic equations for the relevant unknown variables of a birod. Upon normalizing the algebraic equations, we note that the birod’s coupled extension–torsion behavior in its natural state can be tuned by changing the birod’s constituent strand’s intrinsic curvature and twist, Poisson’s ratio and the birod’s stiffness parameters corresponding to inter-strand separation and rotation. Analytical expressions of the birod’s extensional stiffness, torsional stiffness, extension–torsion coupling stiffness, effective Poisson’s ratio, overwinding/unwinding effect etc. in the birod’s natural state are obtained. In particular, we show that depending on the parameters chosen, a torsionally relaxed birod can undergo non-intuitive slow overwinding followed by rapid unwinding during its stretching as observed during pulling of a torsionally relaxed DNA.

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

WR21 marine gas turbine thermodynamic simulator for ship propulsion studies

An original modular mathematical model, based on physical laws, for steady state and dynamics performance simulation of the WR21 gas turbine is presented in the paper. The model, developed in Matlab-Simulink language, is organized in modular form. Each gas turbine component (i.e. compressor, turbine, combustor, heat exchanger) is modelled in a specific simulator block, that simulates the performance of its pertinent component by means of steady state performance maps and/or correlations, time dependent momentum, energy and mass equations, nonlinear algebraic equations. The great application flexibility of the component simulator modules, allows to modelling any gas turbine layout (i.e. axial or radial flow compressor and turbine, rotating shafts number, heat exchangers). The WR21gas turbine, developed mainly for marine propulsion application, is a three shafts gas turbine with thermal regeneration. It is characterized by a high thermodynamic efficiency, which remains high up to 30% of the design power. The WR21 simulator results are validated by comparison with manufacturer or literature data. The WR21 gas turbine main performances are compared to those of the LM 2500 gas turbine, obtained by a previously author paper. The LM 2500 is the gas turbine currently most used in naval propulsion plants. In the article, a comparison between the steady state working data of the two gas turbines is reported in tabular and graphical form and commented.

On the update of algebraic states during state estimation of differential–algebraic equation (DAE) systems

This manuscript presents a discussion on the algebraic state update step performed during recursive filtering of the differential–algebraic equation (DAE) systems. Existing DAE state estimation approaches follow a two-step state update procedure at each sampling instant. In particular, they first estimate the differential states using the Kalman update, and then update algebraic states by explicitly solving the algebraic equations. Specifically, for the case of DAE systems involving linear algebraic equations though the differential equations are nonlinear, we show that when appropriately initialized, this two-step state update procedure is not needed. It can instead be replaced with a one-step state update procedure that computes the differential and algebraic state estimates simultaneously through the Kalman update. The satisfaction of algebraic equations is guaranteed by this one-step update without it being explicitly enforced. Towards this end, we show that the error covariance matrix of augmented states, when properly initialized, satisfies a null-space property after prediction and update step at each sampling instant. This property ensures that the state estimates obtained using the proposed one-step update approach, satisfy the algebraic equations. This holds for both analytical linearization based extended Kalman filtering and statistical linearization based sigma-point filtering approaches. We also propose a heuristic-based update procedure for state estimation of DAE systems that involve nonlinear algebraic equations. This procedure draws out inferences from the case of DAE systems involving linear algebraic equations and is based on the analysis of algebraic equations residuals obtained from the updated differential and algebraic state estimates with a one-step state update. The efficacy of the proposed state update procedures is demonstrated by performing simulation studies on a benchmark drum boiler system case study. Results demonstrate that the proposed update procedures satisfactorily estimate the differential and algebraic states of a DAE system when compared to the traditional two-step update procedure.

A New Iterative Broyden Legendre Wavelet Galerkin FEM Applied to Unsteady State Model of Two-Dimensional Elliptic Fin

Abstract The novelty of this paper is the investigation of numerical study of a mathematical model which deals with time dependent heat flow in elliptic fin ( dry, wet and partially wet). In this paper, we developed a nonlinear model of second-order heat equations in unsteady state condition. A new iterative Broyden Legendre Wavelet Galerkin Finite Element Method is used for the solution. The central difference approximation used for discretization of second order derivatives and then utilization of Hadamard, Khatri Rao and Face splitting matrices product with Legendre Wavelet Galerkin Method transfers our main problem in to system of non-linear algebraic equations. The iterative Broyden Method provides the solution for this system. In a particular case, present solution is compared with the exact solution and is approximately the same. Effect of different parameters such as Biot number, Latent heat, Kirpichev number, Fin thickness, μ, η and ξ on the temperature distribution are discussed in detail.

Profile of the Effectiveness Factor under Optimal Operating Conditions for the Conversion of Ortho-Xylene to Phthalic Anhydride in a Fixed-Bed Tubular Reactor

The objective of this research is to find the effectiveness factor of the catalyst particles for the most favorable conditions of the phthalic anhydride production in a fixed bed reactor, with the aim of achieving the highest rate of phthalic anhydride production compared to other secondary products and analyzing the areas of lower effectiveness for the modification of the reactor design. Initially, the material and the energy balances in the catalytic bed are solved to obtain the concentration and temperature profiles based on the radius and length of the reactor, using polymath software(Polymath® v6.2 Software Minitab 19 Matlab 2019) with the data from literature. Once the profiles reproducibility was verified using the initial data (inlet temperature, pressure in the reactor, reactor wall temperature, reactor radius and mass flow rate) the experimental design 35 carry out, which generates 243 “experiments”, whose response variable (phthalic anhydride concentration) was obtained using Matlab. Subsequently, the variables were analyzed using the Minitab 18® that, through the response surface analysis method, allowed us to obtain the optimal values of the tested variables. Then, Subsequently, material and energy balances coupled with Fourier and Fick’s laws, along with the effectiveness factor equation, were applied, resulting in the generation of 9 coupled differential equations. Upon implementing the finite difference method, this yielded 90 nonlinear algebraic equations, which were solved using the Polymath software. A total of 78 particles were preselected based on their radial and axial positions to determine the effectiveness factor profile, with values ranging from 0.83 to near unity. The lower values correspond to the points with higher temperature, as evidenced by the calculations performed.

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

A spectral iterative algorithm for solving constrained optimal control problems with nonquadratic functional

The numerical approximation of the solution to Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) in constrained optimal control problems (COCPs) with nonquadratic functional is studied in this investigation. Discretizing both of the space and time of HJB PDE by the Legendre collocation method, a nonlinear system of algebraic equations is obtained to find the expansion coefficients of the approximate solution. However, solving the system of nonlinear algebraic equations is generally relatively difficult and time-consuming. To improve the implementation of the method, we combine the Legendre collocation method with the policy iteration (PI) algorithm. This process leads to a sequence of linear systems of algebraic equations to find the expansion coefficients. The convergence results for the proposed method are provided. Moreover, it is shown that the PI algorithm for a class of constrained optimal control problems with odd nonquadratic functionals is mathematically equivalent to the quasi-Newton's iteration. Finally, the efficiency and accuracy of the method are demonstrated by solving a number of examples.

Finite element simulations of double diffusion in a staggered cavity filled with a power-law fluid

Double diffusion refers to a phenomenon where two different components of a fluid (such as heat and mass) exhibit distinct diffusive behaviors. In this study, we employ finite element-based numerical simulations to investigate the phenomenon of double diffusion in a non-Newtonian fluid within a staggered cavity. Mathematically, this system can be understood by coupling the two-dimensional continuity, momentum, energy, and concentration equations. Since the governing equations have been written in a dimensionless form, Galerkin's finite element method is used to find a solution. The velocity profile and temperature are calculated in a function space of quadratic polynomials (P2), while the pressure is calculated in a linear (P1) finite element function space. Discrete systems of nonlinear algebraic equations are resolved through the implementation of Newton's method with appropriate damping and PARDISO solver in the inner loops for solving the sparse linear systems. In this work, the data are presented graphically in the form of streamlines, isotherms, iso-concentration, average Nusselt numbers, average Sherwood numbers, and kinetic energy distribution. Code validation and grid independence study are also provided. Moreover, convective mass transfer is significantly correlated with the Lewis number, as demonstrated by the results. As the power-law index increases, convection also exhibits enhanced as a means of transmitting heat and mass.

On the Qualitative Properties of a Solution to a System of Infinite Nonlinear Algebraic Equations

On the Qualitative Properties of a Solution to a System of Infinite Nonlinear Algebraic Equations

Stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate

In this paper, we analyze how Eyring fluid film behavior on a vertically upward moving slippery flat plate can help predictive models in engineering, notably in coating and lubrication operations. This paper deals with the stagnant points and uniform film analysis of Eyring fluid film flow on a vertically upward moving slippery flat plate. The formulated ordinary differential equation is solved for exact analytic solution. Exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant points are derived. A highly nonlinear algebraic equation is derived for film thickness. Equation is solved by Newton’s method using Maple code. The thickness of film widens with increasing relaxation time, constant plate velocity, and fluid density, while it decreases with increasing fluid viscosity and constant slip parameter. The analysis delineates that the positions of stagnant points tend to relocate closer the slippery plate as the Stokes number, Deborah number, and constant slip parameter increase. A high Deborah number enhances drainage, causing stagnant points to relocate closer to the slippery plate and the development of a stable elastic layer adjacent to the plate. For stagnant points and fluid film thickness, a comparison between the Eyring fluid and existing studies (EPTT, LPTT, UCM, and Newtonian fluid) is also provided. The validity of this paper with the existing studies is also presented by reducing Eyring fluid model to Newtonian model. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

Application Of Multipoint Secant-Type Method ForFinding Roots 0f Nonlinear Equations

In this paper, we introduce a family of pk-order iterative schemes for finding the simple root of a nonlinear algebraic equation of the function fx=0 by using the divided difference approximation. The proposed method uses one evaluation of the function per iteration and can achieve convergence order pk. The error equation and asymptotic convergence constant are proved theoretically and numerically. Numerical examples are included to demonstrate the exceptional convergence speed of the proposed method and thus verify the theoretical results

Mapped Gegenbauer functions for solving Hammerstein generalized integral equations with Green’s kernels on the whole line

This paper introduces a collocation method for numerically solving Hammerstein generalized integral equations on the whole line. The approach utilizes mapped Gegenbauer functions to transform the Hammerstein integral equation into a system of nonlinear algebraic equations. Additionally, convergence results are obtained using the standard L2-norm. Finally, the effectiveness of the proposed method is demonstrated through several numerical experiments, which also serve to illustrate our theoretical findings.