System Of Nonlinear Equations Research Articles

Influence of Rotation and Viscosity on Parallel Rolls of Electrically Conducting Fluid

Rayleigh–Bénard convection is a fundamental fluid dynamics phenomenon that significantly influences heat transfer in various natural and industrial processes, such as geophysical dynamics in the Earth’s liquid core and the performance of heat exchangers. Understanding the behavior of conductive fluids under the influence of heating, rotation, and magnetic fields is critical for improving thermal management systems. Utilizing the Boussinesq approximation, this study theoretically examines the nonlinear convection of a planar layer of conductive liquid that is heated from below and subjected to rotation about a vertical axis in the presence of a magnetic field. We focus on the onset of stationary convection as the temperature difference applied across the planar layer increases. Our theoretical approach investigates the formation of parallel rolls aligned with the magnetic field under free–free boundary conditions. To analyze the system of nonlinear equations, we expand the dependent variables in a series of orthogonal functions and express the coefficients of these functions as power series in a parameter ϵ. A solution for this nonlinear problem is derived through Fourier analysis of perturbations, extending to O(ϵ8), which allows for a detailed visualization of the parallel rolls. Graphical results are presented to explore the dependence of the Nusselt number on the Rayleigh number (R) and Ekman number (E). We observe that both the local Nusselt number and average Nusselt number increase as the Ekman number decreases. Furthermore, the flow appears to become more deformed as E decreases, suggesting an increased influence of external factors such as rotation. This deformation may enhance mixing within the fluid, thereby improving heat transfer between different regions.

Tripled Fixed Point Methodologies for Finding a Solution to a System of Nonlinear Integral Equations on Orthogonal Metric Spaces

Tripled Fixed Point Methodologies for Finding a Solution to a System of Nonlinear Integral Equations on Orthogonal Metric Spaces

Enhanced heat transfer in novel star-shaped enclosure with hybrid nanofluids: A neural network-assisted study

The current research is a numerical investigation of the thermo-fluidic transport trend within a new star-shaped enclosure filled with Fe3O4−MWCNT (Multiwall Carbon Nanotubes) suspended hybrid nanoparticles in a base fluid (water), induced by a horizontal magnetizing field. A rectangular rod inside the cavity positioned at center is kept at a high temperature, as the low temperature is chosen for outer corrugated boundary. Formulation yelled in mathematical principles, along with the boundary conditions and some physical assumptions, are formulated as dimensionless PDEs. The finite element method (FEM) assisted by “COMSOL” software, employed for numerical simulations, and the PARDISO solver is used for solving nonlinear system of equations. Also, an artificial neural network (ANN) model is assembled to analyze the effect of these parameters on thermal and flow transport properties of Hybrid nanofluid. The training of this ANN model is done by a dataset generated from numerical simulations, suggesting predictions regarding heat transport under varying parameters. This approach does not only reduce the effort, but also reduces computing time when exploring the thermal behaviors across various datasets. The visualization of velocity and temperature fields through streamlines and isotherms, along with the evaluation of the Nusselt number, illustrate the enhanced heat transfer facilitated by the incorporation of Fe3O4−MWCNT nanoparticles. The results show that increased magnetic field strength reduces fluid velocity due to the generated Lorentz force, which counteracts convective flow. Percentage analysis indicates that hybrid nanofluids (Fe3O4−MWCNT) significantly improve thermal distribution compared to conventional fluids. This study demonstrates the effectiveness of ANN models in forecasting the influence of diverse factors on the flow and heat transport characteristics of hybrid nanofluids. These insights are essential to the design and optimization of heat transfer systems.

On a priori estimate of periodic solutions of a system of nonlinear ordinary differential equations of the second order

The question of a priori estimation of periodic solutions for a system of non-linear ordinary differential equations of the second order with a distinguished main positively homogeneous part is investigated. In this question, the well-known methods for deriving an a priori estimation of periodic solutions for similar systems of first-order ordinary differential equations are not directly applicable.Combining these methods with the idea of qualitative research of singularly perturbed ordinary differential equations, conditions are found that provide an a priori estimation of periodic solutions for the system of second-order equations.The conditions for the a priori estimation are formulated in terms of the properties of the main positively homogeneous part of the system of equations. The existence of periodic solutions is proved to be invariant under a continuous change of the main positively homogeneous part and the conditions of a priori estimation are preserved. Based on the results obtained, in the future, it is possible to investigate the existence of periodic solutions.

A new shifted generalized Chebyshev approach for multi-dimensional sinh-Gordon equation

The numerical treatment of multi-dimensional non-linear sinh-Gordon equations is the focus of this paper. We numerically solve the (1 + 1) and (2 + 1) sinh-Gordon equations using two collocation algorithms. We select the set of basis functions as a set of generalized Chebyshev polynomials (CPs), which we express as orthogonal combinations of CPs. We develop and utilize some formulas related to these polynomials to propose our numerical algorithms. Specific values for the high-order derivatives of the basis functions serve in the derivation of the two presented algorithms. Additionally, we provide an estimation of the basis functions used in the convergence analysis study. We follow the two collocation algorithms to transform the sinh-Gordon equations into non-linear equation systems, which any suitable solver can handle. We provide some examples and comparisons to confirm the effectiveness of our presented algorithms.

An intriguing numerical strategy for Zakharov–Kuznetsov equation through graph-theoretic polynomials

This paper explores graph-theoretic polynomials to find the approximate solution of the (2+1)D Time-fractional Zakharov-Kuznetsov(TF-Z-K) equation. The Zakharov-Kuznetsov equations govern the behavior of nonlinear acoustic waves in the plasma of hot isothermal electrons and cold ions in the presence of a homogeneous magnetic field. Independence polynomials of the Ladder-Rung graph serve as the polynomial approximation for the suggested Independence Polynomial Collocation Method (IPCM). The Caputo fractional derivatives are adopted to determine the fractional derivatives in the TF-Z-K equation. The TF-Z-K equation is converted into a system of nonlinear algebraic equations using the collocation points in IPCM. The Newton-Raphson approach yields the solution of the suggested method by solving the resulting system. We’ve compared a few scenarios with the tangible outcomes to validate the procedure. Quantitative outcomes match the current findings and validate the exactness of IPCM compared t o the recent numerical and semi-analytical approaches in the literature.

A compact combination of second-kind Chebyshev polynomials for Robin boundary value problems and Bratu-type equations

AbstractThis paper presents a new algorithm for resolving linear and non-linear second-order Robin boundary value problems (BVPS) and the Bratu-type equations in one and two dimensions using spectral approaches. Basis functions according to second-kind shifted and modified shifted Chebyshev polynomials that comply with the Robin conditions are created. It has produced operational matrices for its derivatives. The provided solutions are the result of applying the collocation and tau approaches. These methods convert the problem dictated by its boundary conditions into a system of linear or non-linear algebraic equations that may be solved using any suitable numerical solver. Convergence analysis has been provided and it accords with the numerical results. Six numerical problems are provided to investigate and demonstrate the practical utility of the suggested method. The current results show that our method outperforms the previous methods in terms of accuracy which are presented in tables and figures.

A least-change secant algorithm for solving generalized complementarity problem

AbstractIn this paper, we propose a least-change secant algorithm to solve the generalized complementarity problem indirectly trough its reformulation as a nonsmooth system of nonlinear equations using a one-parametric family of complementarity functions. We present local and superlinear convergence results of new algorithm and analyze its numerical performance.

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.

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.

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.

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.

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.

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.

A neural network computational procedure for the novel designed singular fifth order nonlinear system of multi-pantograph differential equations

The current investigations present the numerical solutions of the novel singular nonlinear fifth-order (SNFO) system of multi-pantograph differential model (SMPDM), i.e., SNFO–SMPDM. The novel SNFO–SMPDM is obtained using the sense of the second kind of typical Emden–Fowler and prediction differential models. The features of shape factor, pantograph along with singular points are provided for all four obtained classes of the SNFO–SMPDM. The extensive use of the singular models is observed in the engineering and mathematical systems, e.g., inverse systems and viscoelasticity or creep systems. For the correctness of the proposed novel SNFO–SMPDM, one case of each class is numerically handled by applying supervised neural networks (SNNs) along with the optimization of Levenberg–Marquardt backpropagation scheme (LMBS), i.e., SNNs–LMBS. A dataset using the traditional variational iteration scheme is designed to compare the proposed results of each case of SNFO–SMPDM. The obtained approximate solutions of each class using the novel SNFO-SMPDM are presented based on the training (80%), authentication (10%) and testing (10%) measures to evaluate the mean square error. Fifteen numbers of neurons, and sigmoid activation function are used in this SNN process. To authenticate the competence, and precision of SNFO–SMPDM, the numerical simulations are accessible by applying the relative measures of regression, error histogram plots, and correlation.

Parameter Regression for Porous Electrodes Employed in Lithium-Ion Batteries and Application to Ni0.89Co0.05Mn0.05Al0.01O2

We develop a parameter regression scheme that can be used with battery models of interest to the battery-analysis community. We show that the recent reduced order model (ROM1, 2022 J. Electrochem. 169 070520, DOI: 10.1149/1945–7111/ac7c93), which is based on a perturbation solution, can be used in place of the full system of nonlinear partial differential equations with minimal loss of accuracy for the conditions of this work, which are relevant for electric vehicle applications. The use of the computationally efficient ROM1, cast in the Python programming language, along with a routine native to Python for the nonlinear regression of model parameters through the minimization of the squared differences between experimental results and model calculations, provides a fast method for the overall endeavor. We apply the procedure to examine Ni0.89Co0.05Mn0.05Al0.01O2, a high-capacity material that is of increasing interest with respect to electric vehicles and other products that rely on batteries of high energy density. Difficulties encountered in this work include the large number of parameters governing the battery model, parameter sensitivity in the regression analyses, and the potential for multiple solutions. We close this publication with a discussion of these challenges and open questions with respect to parameter identification.

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.