System Of Nonlinear Equations Research Articles

A new solution to force analysis including Coulomb friction in mechanism joints

The objective of this paper is to propose a novel approximate solution for determining the reactions of joints in mechanical systems, which involves the influence of Coulomb friction. It is widely acknowledged that if Coulomb friction is used in determining an exact solution to the equilibrium equations for a mechanism, then it will involve nonlinear systems of equations. With the abundance of computer tools now available, tasks of this kind, especially numerical computations, are not particularly challenging. However, new analytically tractable approximate methods are still valuable as a straightforward way of solving the problem. Several studies have been carried out in this field to find a simple solution. In this paper, a new approach based on friction circle concept and Babylonian algorithm is developed for various mechanisms, which is exceptionally well-suited for calculating and streamlining the solution process for mechanical systems by eliminating the necessity for iterative steps at each stage of the force analysis.

Exact cosmological solutions for a Chaplygin Gas in anisotropic petrov type D spacetimes in Eddington-inspired-Born-Infeld gravity: Dark Energy model

We consider a Petrov Type D physical metric g, an auxiliary metric q and a Chaplygin Gas of pressure P in Eddington-inspired-Born-Infeld theory. From the Eddington-inspired-Born-Infeld-Chaplygin Gas equations, we first derive a system of second order nonlinear ordinary differential equations. Then, by a suitable change of variables, we arrive at a system of first order linear ordinary differential equations for the non-vanishing components of the pressure P, the physical metric g and the auxiliary metric q. Thanks to the superposition method, we collect an analytical solution for the nonlinear system obtained, which allows to retrieve new exact cosmological solutions for the model considered. By studying the Kretschmann invariant, we see that a singularity exists at the origin of the cosmic time. By the Kruskal-like coordinates, we conclude that this solution is the counterpart of the Friedman-Lemaître-Robertson-Walker spacetime in the Eddington-inspired-Born-Infeld theory. The Hubble and deceleration parameters in both directions of the physical metric g and the auxiliary metric q, as well as their behaviours over time, are also studied. The thermodynamic behaviour of the Chaplygin Gas model is investigated and, as a result, we show that the third-law of thermodynamics is verified. This means that the value of the entropy of the Chaplygin Gas in the perfect crystal state is zero at a temperature of zero Kelvin, which yields a determined value of the entropy and not an additive constant. Finally, we show that the solutions change asymptotically to the isotropic regime of expansion of Dark Energy. With this, we infer that the Chaplygin Gas can show a unified picture of Dark Energy and Dark Matter cooling during the expansion of the Universe.

A new fractional derivative extending classical concepts: Theory and applications

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle’s theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

Vibration suppression of a platform by a fractional type electromagnetic damper and inerter-based nonlinear energy sink

The theory of linear and nonlinear dynamic vibration absorbers is a well-established topic for many years. However, many recent contributions paid attention to the nonlinear vibration absorbers and different practical realizations of corresponding devices. Here, we propose a mechanical system constituted of the inerter-based nonlinear energy sink attached to the main body that is resting on an elastic foundation and is grounded through the fractional type electromagnetic damper. The two-degree-of-freedom system is described via two coupled differential equations with one of them having a fractional-order derivative term and the other one containing cubic stiffness nonlinearity. The incremental harmonic balance (IHB) method is employed to solve the equations and studies the strongly nonlinear periodic responses of the system. Applied approximated solution methodology is validated by the numerical Newmark method adapted to deal with the system of nonlinear fractional-order differential equations. The appropriate and necessary number of harmonics used in the IHB solution is commented and validated. This study can be a first step in understanding the dynamics and giving directions for the future design of vibration-isolating platforms based on inerter-based nonlinear vibration absorbers and electromagnetic dampers.

Fibonacci Wavelet Collocation Method for Solving Dengue Fever SIR Model

The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation method (FWCM). This biological model has been transformed into a system of nonlinear algebraic equations by using the Fibonacci wavelet collocation scheme. Afterward, this system has been solved by using the Newton–Raphson method. Finally, we provide evidence that our results are better than those obtained by various current approaches, both numerically and graphically, demonstrating the method’s accuracy and efficiency.

A spectral collocation scheme for solving nonlinear delay distributed-order fractional equations

The basic purpose of this paper is to implement a variant of the Jacobi–Gauss collocation scheme for solving nonlinear delay (in difference format) fractional differential equations of distributed-order type. At the first stage of the solving procedure, the Legendre–Gauss quadrature rule is implemented to approximate the main problem by a difference delay fractional differential equation of multi-term type. So, one can apply the Jacobi–Gauss collocation approach for localizing the resulting multi-term fractional differential equation with the difference delay factor. Moreover, some special cases of Jacobi–Gauss quadrature rules can be used for approximating the involved integrals in the aforementioned problem. By considering the Lagrange interpolating polynomials, as the numerical solutions, the solution of the main problem can be approximated by solving the associated system of nonlinear algebraic equations in terms of unknown Lagrange multipliers. Convergence analysis of the problem is investigated rigorously under some mild conditions at a spectral rate. Extensive test problems are considered to justify the theoretical analysis.

Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the p-Laplace operator, the ϕ-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator (S(t,u′))′, where ′=ddt and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator (S(t,u′))′ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in C1 spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.

Numerical simulation of hybrid nanofluid flow with homogeneous and heterogeneous chemical reaction across an inclined permeable cylinder/plate

The current study investigates the steady two-dimensional (2D) hybrid nanofluid (Hnf) flow over an inclined permeable plate/cylinder. The Hnf flow has been examined in the context of mixed convection, heterogeneous/homogenous chemical reaction, and permeable medium. The Hnf is prepared by dispersing silver (Ag), and iron ferrite (Fe3O4) nanoparticles (NPs) in water. The current research is motivated by the increasing demand for highly efficient cooling devices in a variety of industries and energy-associated operations. The energy transmission and fluid flow are mathematically specified by using a coupled nonlinear system of partial differential equations (PDEs). The system of PDEs is simplified into a dimensionless form of ODEs, which are then further numerically treated with the MATLAB package based on the finite difference method (bvp4c). It has been noticed that the permeability component develops the heat transfer curve while decreasing the flow rate of the fluid. The impact of heat source/sink increases the energy profile. Moreover, the plate surface demonstrates the dominant behavior of energy transportation than a cylinder with the variance of Ag-Fe3O4-NPs.

A blow‐up result of a nonlinear viscoelastic wave equations system with variable dampings and sources

A blow‐up result of a nonlinear viscoelastic wave equations system with variable dampings and sources

A Fitness Distance Correlation-Based Adaptive Differential Evolution for Nonlinear Equations Systems

A Fitness Distance Correlation-Based Adaptive Differential Evolution for Nonlinear Equations Systems

Derivative free King's-type family for solving system of nonlinear equations

Derivative free King's-type family for solving system of nonlinear equations

A varying-parameter complementary neural network for multi-robot tracking and formation via model predictive control

In this paper, a varying-parameter complementary neural network (VPCNN) is designed and combined with model predictive control (MPC) to solve the multi-robot tracking and formation problems via a leader–follower strategy. First, multi-robot tracking and formation problems are transformed into quadratic programming (QP) problems employing an MPC approach. Second, a nonlinear complementary function approach, i.e., the Fischer–Burmeister function, is used to map the Karush–Kuhn–Tucker conditions of the QP problem with double-ended inequality constraints to a system of nonlinear equations. Finally, the VPCNN is designed to solve the multi-robot tracking and formation problem. The effectiveness of the proposed method is demonstrated by numerical simulations, and the advantage of VPCNN in terms of solution speed is indicated by comparisons with a primal–dual neural network.

Two-Level method for blind image deblurring problems

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h (h≤H). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.

Statistical investigation of heat transfer efficiency on a ternary nanoflow over a stretching surface

Ternary hybrid nanofluids offer significant advantages in heat transfer due to their enhanced thermal conductivity compared to traditional fluids. Investigating their flow behavior over expanding surfaces can contribute to the development of more efficient heat exchangers and cooling systems in the electronics, energy production, and transportation industries. This paper presents a detailed analysis of the flow and thermal transfer characteristics of a ternary hybrid nanofluid (SiO2–Cu–Al2O3/H2O) over a radially stretching surface, utilizing both numerical techniques (Keller Box method) and statistical techniques. Furthermore, the model includes factors like convective heat transport, radiation, internal fluid friction, and suction. Similarity transformations transform the governing equations for fluid flow and heat transfer into a system of nonlinear ordinary differential equations. Then the equations are solved numerically using the Keller Box method with the aid of MATLAB software. Additionally, we have employed statistical techniques such as correlation analysis, probable error estimation, and multivariate regression to validate and ensure the accuracy of the numerical outcomes. Parameter ranges for Biot number, radiation, unsteadiness, Eckert number, and energy generation parameter are 0.1 ≤ Mp ≤ 0.7, 0.1 ≤ Ks ≤ 1.5, 0.2 ≤ Ps ≤ 0.8, 0.1 ≤ EN ≤ 0.4, 0.5 ≤ RN ≤ 2, 1≤ BT ≤ 2.5, 0.1 ≤ Hsc ≤ 0.7 and found to significantly impact heat transfer behavior within these bounds. This study utilizes 3D surface plots to visualize the influence of key parameters on engineering quantities. R squared values show that the data strongly match the regression model. The findings demonstrate excellent concordance between the predicted values and the actual Nusselt number and skin friction measurements. Biot number has a very strong correlation with heat transfer, with minimal chance of error. And the same is evidenced by the correlation matrix and multiple regression analysis. Eckert number also exhibits a negative correlation with the Nusselt number. This translates to a situation where increasing Eckert number directly leads to a decrease in the rate of heat transfer, with no margin for error. Energy generation parameter demonstrates a negative correlation with heat transfer. However, there's a slight possibility of error, with an estimated error percentage of around 0.0017.

Bang-bang optimal control in coherent spin dynamics of radical pairs in quantum biology

Optimal control of the external electromagnetic field input for the maximization of the quantum triplet-singlet yield of the radical pairs in biochemical reactions modeled by Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. Fréchet differentiability and Pontryagin Maximum Principle in Hilbert space is proved and the bang-bang structure of the optimal control is established. A closed optimality system of nonlinear differential equations for the identification of the bang-bang optimal control is revealed. Numerical methods for the identification of the bang-bang optimal control based on the Pontryagin maximum principle are developed. Numerical simulations are pursued, and the convergence and stability of the numerical methods are demonstrated. The results contribute towards understanding the structure-function relationship of the putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic function to inspire novel quantum devices.

Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (M&amp;lt;i&amp;gt;T_3&amp;lt;/i&amp;gt;) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

Innovative thermal management in the presence of ferromagnetic hybrid nanoparticles

In the present work, a simple intelligence-based computation of artificial neural networks with the Levenberg-Marquardt backpropagation algorithm is developed to analyze the new ferromagnetic hybrid nanofluid flow model in the presence of a magnetic dipole within the context of flow over a stretching sheet. A combination of cobalt and iron (III) oxide (Co-Fe2O3) is strategically selected as ferromagnetic hybrid nanoparticles within the base fluid, water. The initial representation of the developed ferromagnetic hybrid nanofluid flow model, which is a system of highly nonlinear partial differential equations, is transformed into a system of nonlinear ordinary differential equations using appropriate similarity transformations. The reference data set of the possible outcomes is obtained from bvp4c for varying the parameters of the ferromagnetic hybrid nanofluid flow model. The estimated solutions of the proposed model are described during the testing, training, and validation phases of the backpropagated neural network. The performance evaluation and comparative study of the algorithm are carried out by regression analysis, error histograms, function fitting graphs, and mean squared error results. The findings of our study analyze the increasing effect of the ferrohydrodynamic interaction parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} to enhance the temperature and velocity profiles, while increasing the thermal relaxation parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} decreases the temperature profile. The performance on MSE was shown for the temperature and velocity profiles of the developed model about 9.1703e−10, 7.1313ee−10, 3.1462e−10, and 4.8747e−10. The accuracy of the artificial neural networks with the Levenberg-Marquardt algorithm method is confirmed through various analyses and comparative results with the reference data. The purpose of this study is to enhance understanding of ferromagnetic hybrid nanofluid flow models using artificial neural networks with the Levenberg-Marquardt algorithm, offering precise analysis of key parameter effects on temperature and velocity profiles. Future studies will provide novel soft computing methods that leverage artificial neural networks to effectively solve problems in fluid mechanics and expand to engineering applications, improving their usefulness in tackling real-world problems.

Multiple slip nonlinear radiative bioconvective flow of nanofluid with variable thermal conductivity

Owing to the improved thermal applications of nanomaterials, various applications have been suggested in various era of engineering including heat transfer devices, chemical processes, thermal management devices, electronic cooling etc. The aim of current investigation is to presents the bioconvective flow of nanofluid with applications of multiple slip effects. The motivations for considering the slip effects for heat and mass transfer analysis are associated to its significance in the petroleum sciences, extrusion processes, oil recovery and plasma physics. The nonlinear thermal radiation effects and chemical reaction consequences have been attributed. In contrast to traditional investigations, the thermal conductivity of nanofluid have been assumed to be temperature dependent. Following the assumptions of dimensionless variables, a system of nonlinear differential equations is obtained. The shooting technique is employed for computing the numerical simulations. The accuracy of solution is ensured. The physical assessment of problem is incorporated in view of involved parameter. It has been claimed that interaction of slip effects enhances the heat and mass transfer effects. Change in microbes density slip parameter leads to improvement in the microorganisms profile. Moreover, local Nusselt number and Sherwood number reduces for slip parameters. Current results present significance in heat transfer systems, thermal engineering, chemical engineering, fertilizers, biofuels etc.

Parsimonious streamflow forecasting system based on a dynamical systems approach

Operational hydrologic forecasting often relies on the use of physically-based computer models at the catchment scale. However, our understanding of streamflow generation is still evolving and incomplete, and oftentimes linearization assumptions must be made to make physically-based hydrologic and hydraulic models tractable. In addition to uncertainty in the physical runoff-generation mechanisms, many physically-based approaches do not formally assimilate observed streamflow data. Streamflow is their output, deterministically simulated from observed inputs (such as precipitation, temperature, etc.). Divergences between observed and simulated streamflow may be due to input uncertainty, calibration issues, or problems with the inherent physical process representation. Thus, data-driven approaches that are responsive to observed streamflow by including assimilation as an essential feature are promising in operational hydrology. In this study, we use a novel forcings-weighted k-Nearest Neighbor (kNN) nonlinear timeseries forecasting method developed in the context of the modeling of nonlinear dynamical systems. The rainfall-runoff transform is assumed to be a low-dimensional system of (unknown) nonlinear differential equations; the deterministic state space is reconstructed based on the streamflow timeseries; and a forecast is generated based on how “similar” the current hydrograph is to historical events encoded within the reconstructed state space, weighted by the similarity of forcings. We tested the model in a long-term test period with no real data assimilation, and in forecast mode with daily data assimilation. When used for forecasting, we also illustrate two different uncertainty quantification approaches: forecast postprocessing and internal-to-the-model ensemble generation. The method is conceptually and computationally simple, but offers promising results. We tested the model in three different scenarios. First, investigating a long-term test period with no real data assimilation in a rainfall-dominated research catchment at the Coweeta Hydrologic Lab in North Carolina, USA. The simulation exhibited an NSE of 0.73. Next, we included a conceptual snowmelt model to drive the long term simulation and studied boreal-temperate transition zone catchments in northern Minnesota, USA. The study area included a small (<0.1 mi2) reference research catchment at the Marcell Experimental Forest, and a large river basin (>19 k mi2) with incomplete forcings information. The kNN method yielded NSEs from 0.28 to 0.50. And finally, we conducted a pseudo-operational forecasting test for a large river basin in northern Minnesota, USA with a focus on performance forecasting a flood-of-record on the Rainy River. The test involved reforecasting, data assimilation, and implementation of ensemble and postprocessing bias correction uncertainty-quantification schemes. The technique demonstrated good performance capturing a flood-of-record on a large catchment (−5.7 to −3.3 % crest error with 1-day of lead time). For the entire test period in forecast mode, the 1-day lead time NSE was 0.99, going down to 0.79 for the 7-day lead time, showing its general utility as an operational forecasting tool.

Robust anti-disturbance interval type-2 fuzzy control for interconnected nonlinear PDE systems via conjunct observer

This paper investigates a robust anti-disturbance interval type-2 (IT2) fuzzy control for interconnected nonlinear partial differential equation (PDE) systems subject to parameter uncertainties by the conjunct observer. First, an IT2 fuzzy model is adopted to remodel the target system. Second, a state observer with mismatched premise variables is constructed to solve the problem that the original system and the observer do not share a uniform premise variable. Moreover, a disturbance observer is designed to estimate the unknown external disturbances, which can be modeled by exogenous PDE systems. Then, utilizing the conjunct observation information, an anti-disturbance IT2 fuzzy control strategy is proposed to attenuate the effect of disturbances on the system performance while ensuring that the closed-loop system is stable. Finally, simulation results verify the effectiveness of the proposed method.