System Of Nonlinear Equations Research Articles

A Jacobian smoothing inexact Newton method for solving the nonlinear complementary problem

In this paper, we propose a Jacobian smoothing inexact Newton-type algorithm for solving the nonlinear complementarity problem by reformulating it as a system of nonlinear equations. We show that the algorithm converges up to q-quadratically and present numerical experiments that show its good local performance. In order to compare in terms of time our algorithm with other known algorithms, we introduce a new normalized measurement that we call time index.

A residual-based weighted nonlinear Kaczmarz method for solving nonlinear systems of equations

A residual-based weighted nonlinear Kaczmarz method for solving nonlinear systems of equations

Enhanced spectral collocation Gegenbauer approach for the time‐fractional Fisher equation

In this study, we utilize a specific combination of orthogonal Gegenbauer polynomials as basis functions that satisfy homogeneous boundary conditions in the spatial variable and homogeneous initial conditions in the temporal variable. We derive an expression for the fractional derivative of the temporal basis and another expression for the second‐order derivative of the spatial basis. Subsequently, we discretize the nonlinear Fisher problem using the spectral collocation method to transform the problem into the solution of a nonlinear system of algebraic equations. We then solve this resulting system using Newton's method with an initial guess approaching zero. Furthermore, we conduct an error analysis of the proposed method and assess its applicability through three test problems, providing comparisons for validation.

Analog quantum simulation of partial differential equations

Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular—Schrödinger’s equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs—both classical and quantum—are digital (they must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrödingerisation, we show how to directly map D-dimensional linear PDEs onto a (D+1) -qumode quantum system where analog or continuous-variable (CV) Hamiltonian simulation on D + 1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker–Planck equation, Black–Scholes equations, wave equation and Maxwell’s equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or CV framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations

The symmetric successive overrelaxation (SSOR) and symmetric accelerated overrelaxation (SAOR) are conventional iterative methods for solving linear equations. In this paper, novel approaches are presented by combining a splitting–linearizing method with SSOR and SAOR for solving a system of nonlinear equations. The nonlinear terms are decomposed at two sides through a splitting parameter, which are linearized around the values at the previous step, obtaining a linear equation system at each iteration step. The optimal values of parameters are determined to minimize the reciprocal of the maximal projection, which are sought in preferred ranges using the golden section search algorithm. Numerical tests assess the performance of the developed methods, namely, the optimal splitting symmetric successive over-relaxation (OSSSOR), and the optimal splitting symmetric accelerated over-relaxation (OSSAOR). The chief advantages of the proposed methods are that they do not need to compute the inverse matrix at each iteration step, and the computed orders of convergence by OSSSOR and OSSAOR are between 1.5 and 5.61; they, without needing the inner iterations loop, converge very fast with saving CPU time to find the true solution with a high accuracy.

Analytical Investigation of Thermal Radiation Effects on Electroosmotic Propulsion of Electrically Conducting Ionic Nanofluid with Single-Walled Carbon Nanotube Interaction in Ciliated Channels

This study examines the behavior of single-walled carbon nanotubes (SWCNTs) suspended in a water-based ionic solution, driven by the combined mechanisms of electroosmosis and peristalsis through ciliated media. The inclusion of nanoparticles in ionic fluid expands the range of potential applications and allows for the tailoring of properties to suit specific needs. This interaction between ionic fluids and nanomaterials results in advancements in various fields, including energy storage, electronics, biomedical engineering, and environmental remediation. The analysis investigates the influence of a transverse magnetic field, thermal radiation, and mixed convection acting on the channel walls. The novel physical outcomes include enhanced propulsion efficiency due to SWCNTs, understanding the influence of thermal radiation on fluid behavior and heat exchange, elucidation of the interactions between SWCNTs and the nanofluid, and recognizing implications for microfluidics and biomedical engineering. The Poisson–Boltzmann ionic distribution is linearized using the modified Debye–Hückel approximation. By employing real-world approximations, the governing equations are simplified using long-wavelength and low-Reynolds-number approximation. Conducting sensitivity analyses or exploring the impact of higher-order corrections on the model’s predictions in recent literature might alter the results significantly. This acknowledges the complexities of the modeling process and sets the groundwork for further enhancement and investigation. The resulting nonlinear system of equations is solved through regular perturbation techniques, and graphical representations showcase the variation in significant physical parameters. This study also discusses pumping and trapping phenomena in the context of relevant parameters.

Stochastic electromechanical bidomain model *

We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.

MONTE CARLO AND NUMERICAL METHODS TO SOLVE THE TIME SERIES MODEL

In this paper we will solve the nonlinear system of equations in the parameters of the time series model by Monte Carlo methods and by numerical methods. When we identify the variance and the inter-covariances of time series, we obtain, dividing by variance, a quadratic nonlinear system of equation that does not contain the variance of wihite noise. We use only the autocorrelation function. AMS Subject Classification: 62M10, 91B84, 65C05.

Oscillations in Neuronal Activity: A Neuron-Centered Spatiotemporal Model of the Unfolded Protein Response in Prion Diseases.

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S's). The model takes the form of a coupled system of nonlinear reaction-diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression.

Non-local second-order boundary value problems with exponential nonlinearity

In this article, the Numerical treatment (Impilict-Explicite) was applied, for solving problems- related to the nonlocal second-order boundary value problem with exponential nonlinearity. The finite difference method of treatment of the BVP is used as implicit technique within the smallest available subdomain, . A reduced system of nonlinear algebraic equations corresponding to standard BVP is generated within the smallest domain, . A reduction of the computational work is introduced through reducing the size of the nonlinear algebraic system. A marching explicit approach is reemployed to generate the solution in , the complement of the interval with respect to the overall domain. Simple numerical experiments confirming the applicability of the treatment are introduced.

On a fully coupled nonlocal multipoint boundary value problem for a dual hybrid system of nonlinear q-fractional differential equations

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann–Liouville-type [Formula: see text]-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on [Formula: see text]-fractional differential equations. Some interesting special cases are also discussed.

Existence and Hyers–Ulam Stability of Stochastic Delay Systems Governed by the Rosenblatt Process

Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided.

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.

Contribution to the determination of the effect of magnetic storms on the electric power transmission system

Abstract When a magnetic storm hits a power transmission system, quasi-stationary geomagnetically induced currents (GIC) are generated in the high-voltage part of the system. These currents cause semi-saturation of the magnetic circuits of power transformers, which induces current overload in their high-voltage windings and subsequently thermal overload, which can lead to system failures. This rather complex phenomenon was described in [11] by a system of nonlinear differential equations and subsequently solved. This very challenging method is replaced in the present work by a simple approach. It allows not only predicting the imminent danger of system collapse, but gives transformer designers valuable information on how they can counteract this danger.

On convergence of waveform relaxation for nonlinear systems of ordinary differential equations

On convergence of waveform relaxation for nonlinear systems of ordinary differential equations

Application of Artificial Neural Networks Modelling for Analyzing Non-Newtonian Fluid Flow Over Porous Media with an Induced Magnetic Field

The goal of this work is to better comprehend how enhanced thermal transfer, which really increased the induced magnetic field of hybrid nanofluid, may alleviate the growing heat density of tiny and other technological processes. Non-Newtonian fluid, like Oldroyd-B fluid is encrypted for a given physical model. Furthermore, copper and alumina hybridize in sodium alginate to produce a homogeneous, very diluted combination. The non-linear PDE system of equations is transformed into an ordinary differential system by applying the appropriate self-similarity variables. For this stage of the work, we built a mid-rich integrated ANN-LMBP that is envisioned with non-linear thermal radiation and heat production. Graphical depictions are used to demonstrate the interplay between various skewing. The entropy production of the Oldroyd-B fluid slopes with temperature ratio for all comparing scenarios, while Bejan number exhibits a declination with increase in Brinkman number. For uses including thickening, gelling, emulsifying, stabilizing, and texturizing, the predictions come true. Since sodium alginate is a standard component that is utilized in many cuisines today, including ice cream, jelly, acidic milk drinks, sauces, fast noodles and beer.

Comparative Study of Convective Oldroyd-B Nanofluid and Hybrid Nanofluid Flow, Heat and Mass Transfer Analysis Over Stretching Sheet with Cattaneo-Christov Heat Flux Model

The current analysis appraises the flow, heat and mass transfer characteristics of Oldroyd-B nanofluids SWCNT-Water and hybrid nanofluids SWCNT/MWCNT-Water over the stretching sheet. Thermal radiation, modified Fourier heat flux and convective boundary conditions are also used to evaluate the flow. Graphs are used to compare the characteristics of nanofluid SWCNT-Water and hybrid nanofluid SWCNT/MWCNT-Water nanofluids in terms of concentration, temperature, and velocity. The flow mechanism is modelled in the form of nonlinear system of partial differential equation. These PDEs are converted into Ordinary differential equations by using the similarity variable techniques. The Finite Element Method is pre-owned to solve the obtained ODEs. Plots are used to show the results of temperature, concentration and velocity for both nanofluids for various values of the relevant parameters. The dimensionless skin friction coefficient, Nusselt number and Sherwood number values are scrutinized and revealed in tabular form. The important findings of this study reveal that the scattering of temperature and concentration of both SWCNT-Water nanofluid and SWCNT/MWCNT-Water hybrid OBF nanofluid elevates with amplifying values of thermal retardation parameter. With increasing values of the power low index number the sketches of temperature and velocity are both attenuate in the fluid region.

Stabilization of SIMPLE-like RANS solvers for computing accurate gradients using the complex-step derivative method

The Reynolds-averaged Navier-Stokes (RANS) approach in Computational Fluid Dynamics (CFD) is increasingly vital for aerodynamic design across wind turbines, gas turbines, aircraft, and rotorcraft. Enhancing the process through CFD-based design optimization requires numerous design variables, favoring gradient-based methods for better scalability than gradient-free methods. This research uses numerical differentiation techniques, particularly the complex-step derivative method, to compute gradients. Challenges arise during aerodynamic shape optimization when unconventional shapes disrupt RANS solver assumptions, causing convergence failures. These failures undermine optimization, prompting the need to enhance solver convergence for robust optimization. The modified-Boostconv method is a residual recombination method bolstering unstable eigenvalues to stabilize convergence of iterative solvers for nonlinear systems of equations. This study extends the modified-Boostconv method to combine it with the complex-step derivative technique, creating robust optimizations via RANS-CFD with accurate gradients, even for these numerically unstable cases. The main issue encountered during the complexification of the modified-Boostconv method is how to correctly ensure that the model reduction leading to the least-squares problem satisfies the complex-step derivative method. We test whether the dot product operation involved in the model reduction should be a Hermitian or non-Hermitian inner product. The problem is first tested for a simple analytical case using the logistic equation in combination with the modified-Boostconv method. It shows that the non-Hermitian inner product should be used. This is also confirmed with a similar study using the RANS solver.

Optimal Uniform Strength Design of Frame and Lattice Structures

This paper provides a procedure to obtain the uniform strength of frame and lattice structures. Uniform strength condition is achieved by performing the shape optimization of all beam elements of the structure. The beam shape which guarantees uniform strength is analytically deduced from the one-dimensional Timoshenko model. The optimization problem presents itself as the search for the zeros of the objective-functions vector, which is a non-linear system of equations representing the kinematic-congruence and forces balance at every node of the structure. The analytical formulation of the optimization problem allows to construct the objective-functions vector without the use of external structural computation, i.e. not recurring to any Finite Element Analysis to accomplish iterations. This latter feature entails a great advantage in terms of computing time required to perform optimization. The proposed analytical formulation allows to directly insert the uniform strength condition into the objective-functions vector, transforming the optimization into an unconstrained problem. Some examples are shown in which the performance of the optimization procedure is discussed in terms of robustness and rate of computational complexity while increasing the degrees of freedom of the structure. The reliability and the quality of the optimization are verified through Finite Element Analysis.