System Of Nonlinear Equations Research Articles

Investigation of temperature jump, first and second-order velocity slip effects on blood-based ternary nanofluid flow in convergent/divergent channels

Purpose High surface area-to-volume ratios make nanoparticles ideal for cancer heat therapy and targeted medication delivery. Moreover, ternary nanofluids (TNFs) may possess superior thermophysical properties compared to mono- and hybrid nanofluids due to their synergistic effects. In light of this information, the objective of this article is to examine the blood-based TNF flow within convergent/divergent channels under velocity slip and temperature jump. Design/methodology/approach Leading partial differential equations corresponding to the problem are transformed into a system of nonlinear ordinary differential equations by using similarity variables. The bvp4c code that uses the finite difference method is used to obtain a numerical solution. Findings The effect of nanoparticles may change depending on the characteristics of flow near the wall. The properties and proportions of the used nanoparticles become important to control the flow. When TNF was used, an increase in the Nusselt number between 4.75% and 6.10% was observed at low Reynolds numbers. At high Reynolds numbers, nanoparticles reduce the Nusselt number and skin friction coefficient values under some special flow conditions. Importantly, the effects of second-order slip on engineering parameters were also investigated. Furthermore, the Nusselt number increases with increasing shape factor. Research limitations/implications Obtained results of the study can be beneficial in both nature and engineering, especially blood flow in veins. Originality/value The main innovations of this study are the usage of blood-based TNF and the examination of the effect of shape factor in convergent/divergent channels with second-order velocity slip.

Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits

Abstract In the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highlights the importance of accurately determining the complex roots of equations. This article presents a study in which the complex roots of a system of equations are identified through an approach that utilizes homotopy continuation, with the curve being traced using a hyperspherical path tracking technique. Furthermore, this article details five case studies on electrical networks where this method is applied to solve systems of equations containing imaginary coefficients to find mesh currents. The path tracking shows the behavior of system equation in each case study. Finally, an analysis of the precision of the solutions obtained in these case studies is provided, demonstrating an accuracy of up to 15 SDs in a single iteration during the refinement stage.

Mathematical modeling of the thermal regime in solar photovoltaic thermal panel

Design of a combined solar photovoltaic thermal panel (PV/T) for the simultaneous generation of electrical and thermal energy was proposed in this study. The basis of the new design is a traditional solar panel with poly-Si solar cells. A flat channel with a heat transfer fluid is added to the front size of such a panel. This channel is bounded by cover glass. A non-stationary mathematical model was developed for determination of temperature regime in the PV/T panel. This model consists to the system of nonlinear ordinary differential equa-tions, which describes mutual influence of external and internal heat flows and temperatures. A Math-software was created based on the developed mathematical model. The numerical studies were conducted in in real-time mode for selected geographical location of the PV/T panel. Heat flux density from the Sun, wind speed and ambient temperature were determined based on data from open worldwide climate databases. As result of computer modeling, the typical temperature distributions in each layer of the PV/T panel during daylight hours were founded. It was determined that the heat transfer fluid moving in a transparent channel from the front side of the solar panel does not cool the solar cell. This heat transfer fluid ensures only their thermal stability at the corresponding value of the specific mass flow rate. With an increase of the specific mass flow rate of the heat transfer fluid, the growth of solar cells tem-perature is observed under unchanged environmental conditions. An the same time, the pro-posed design of the PV/T panel ensures a significant increase of the heat transfer fluid tem-perature. This makes it possible to use it in low-potential heat generation systems. This leads to an increase in the economic efficiency of solar panels, economy of occupied areas, optimi-zation of system of production, consumption and storge of thermal and electrical energy.

Multithreaded and GPU-Based Implementations of a Modified Particle Swarm Optimization Algorithm with Application to Solving Large-Scale Systems of Nonlinear Equations

This paper presents a novel Graphics Processing Unit (GPU) accelerated implementation of a modified Particle Swarm Optimization (PSO) algorithm specifically designed to solve large-scale Systems of Nonlinear Equations (SNEs). The proposed GPU-based parallel version of the PSO algorithm uses the inherent parallelism of modern hardware architectures. Its performance is compared against both sequential and multithreaded Central Processing Unit (CPU) implementations. The primary objective is to evaluate the efficiency and scalability of PSO across different hardware platforms with a focus on solving large-scale SNEs involving thousands of equations and variables. The GPU-parallelized and multithreaded versions of the algorithm were implemented in the Julia programming language. Performance analyses were conducted on an NVIDIA A100 GPU and an AMD EPYC 7643 CPU. The tests utilized a set of challenging, scalable SNEs with dimensions ranging from 1000 to 5000. Results demonstrate that the GPU accelerated modified PSO substantially outperforms its CPU counterparts, achieving substantial speedups and consistently surpassing the highly optimized multithreaded CPU implementation in terms of computation time and scalability as the problem size increases. Therefore, this work evaluates the trade-offs between different hardware platforms and underscores the potential of GPU-based parallelism for accelerating SNE solvers.

Learning dynamical systems from noisy data with inverse-explicit integrators

We introduce the mean inverse integrator (MII), a novel approach that improves accuracy when training neural networks to approximate vector fields of dynamical systems using noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge–Kutta methods. We show that the class of mono-implicit Runge–Kutta methods (MIRK) offers significant advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained by inserting the training data in the MIRK formulae. This allows the use of symmetric and high-order integrators without requiring the solution of non-linear systems of equations. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with extensive numerical experiments using data from chaotic and high-dimensional Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, providing theoretical support for the favorable performance of MII and symmetric MIRK methods.

Fractional optimal control of systems using Bernoulli wavelets

In order to provide a new method for partial optimal control of systems, this work uses Bernoulli wavelets and uses Caputo's formula to solve partial optimal control problems (FOCPs) with inequality constraints. For the new optimization technique, the work uses partial order Bernoulli wave functions (F-BWFs) as basic functions. The answer using this method is expressed in terms of F-BWF where the coefficients have not yet been found. To simplify FOCPs in a system of nonlinear algebraic equations the procedure entails transforming inequality constraints into equality requirements by using operational matrices of fractional integration and F-BWFs and using the multipliers technique Lagrange. Numerical examples confirm the validity of the proposed strategy and demonstrate its correctness and efficiency when compared to the analytical or approximate answers provided by other methods.

Expected utility maximization for unobservable Markov-modulated jump-diffusion process with constraint on wealth

This article considers a portfolio selection problem under the criterion of expected utility maximization for a jump-diffusion model, where the drifts of the risky assets are modulated by a continuous hidden Markov chain. Suppose that the utility function is a general concave one and there is a constraint on the wealth process. Due to this constraint, we turn to solve this problem by using the technique of martingale method. Besides, based on the filtering theory, we reduce the original partial information problem into a corresponding completely observable one and closed-form expressions of the optimal results are derived, which is the solution to a system of nonlinear equations. For the specific examples, we present optimal results for the exponential and power utility. Furthermore, some sensitivity analyses of the constraint are provided.

A Model Study of The Impact of Enlightenment Rate on The Dynamics of Typhoid Fever

In this study, a mathematical investigation of the effect of enlightenment rate on the spread of typhoid fever is considered, using a system of nonlinear first order ordinary differential equations and a MATLAB ODE45 numerical scheme. The result shows that a decrease in vaccination rate significantly increases the size of the susceptible () class and the response functions, or thereby heightening the tendency for the disease to be endemic. If then the typhoid infection will spread throughout the population, but if then the infection will not be able to take hold and will eventually die out. Furthermore, it was shown that, the basic reproduction number has the tendency to reveal faster if a disease will result in an epidemic than the effective reproduction number. Finally, as enlightenment rates approach zero, it was observed that, over time, the disease will result in an epidemic. It is therefore recommended that enlightenment rate of the exposed, enlightenment rate to go for treatment and enlightenment rate to go for vaccination should be taken seriously by policy makers in order to stem the spread of typhoid.

Real-World Applications of Analytic Travelling Wave Solutions of a (3+1)-Dimensional Hirota–Satsuma–Ito-Like System Via Polynomial Complete Discriminant System and Elementary Integral Technique

Models reveal the dynamic character of wave motion, delineated in shallow waters alongside fluid dynamics; an example is the Hirota–Satsuma–Ito model equation. Therefore, this paper showcases the detailed analytical investigations of a (3+1)-dimensional Hirota–Satsuma–Ito-like system. In order to reduce the equation to a nonlinear ordinary differential system of equations, a traveling plane wave transformation is engaged. Thereafter, the direct integration technique is adopted to solve the model, thus culminating in obtaining Jacobi elliptic integral function solutions. Moreover, to attain more various solitonic solutions of diverse structures, a standard approach called the polynomial complete discriminant system and elementary integral technique is engaged. This provides exact traveling wave solutions of diverse known functions in the form of periodic, trigonometric, dark, mixed bright, and topological kink, as well as singular soliton solutions. These are found to appear in the form of Jacobi elliptic, trigonometric, as well as hyperbolic functions. Furthermore, some of these solutions are further examined by investigating their wave nature via numerical simulations.

Strict Stability of Fractional Differential Equations with a Caputo Fractional Derivative with Respect to Another Function

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a strict stability in the couple. We illustrate both definitions with several examples and, in these examples, we show that the applied function in the fractional derivative has a huge influence on the stability properties of the solutions. In addition, we use Lyapunov functions and their CFDF to obtain several sufficient conditions for strict stability.

Algorithm for determining the operating parameters of an electrical network in the problem of optimal reconfiguration in real time

Relevance. The need to develop technical solutions to reduce electrical energy losses in medium-voltage distribution electrical networks of oil and gas fields. Aim. Development of an algorithm for determining the parameters of the operating mode of an electrical network, applicable in a system of optimal control of the network configuration in order to reduce the level of electricity losses in the network. Methods. Mathematical modeling methods and combinatorial optimization methods. The nodal voltage method and the Newton–Raphson method are used to calculate the network operating mode. The mode is corrected using a sensitivity matrix. Results and conclusions. The authors have developed an algorithm for determining the parameters of the electric network operating mode when determining its optimal configuration. Optimization of the optimal configuration is performed based on the criterion of minimum active power losses in lines, taking into account the limitations on the power reserve of the feeding transformer substations and the number of normal breaks on one line. The developed algorithm is based on solving nonlinear systems of equations of nodal voltages using the Newton–Raphson method, as well as on correcting the parameters of the previously calculated operating mode using the sensitivity matrix. The optimal network configuration is determined by solving the combinatorial optimization problem using the branch and bound method. The operation of the algorithm is illustrated by the example of a network section containing two overhead power lines with two power sources and three feeding transformer substations. Based on the results of the analysis of all possible network configurations determined by combinations of normal breaks, it is concluded that the use of the algorithm will reduce the level of active power losses by up to 23.8%, which will increase the efficiency of the electric grid complex.

Asymptotic analysis of mathematical model describing a new treatment of breast cancer using AZD9496 and palbociclib

IntroductionCancer is a collective name for a group of diseases consisting of dozens of different types of malignant tumors, characterized by rapid and uncontrolled proliferation of cells in the body. Cancer can start almost anywhere in the human body such as the breast, prostate, colorectal, brain, bones, lungs, bladder etc. The main differences between the different types of cancer are related to the organ in which the tumor develops and the type of cells that compose the tumor.MethodThis paper focused on the breast cancer. Breast cancer is a malignant tumor that originates in the breast tissue. It is the most common malignant tumor in women. There are several types of breast cancer, but in all types early diagnosis and treatment is crucial. In this study, the treatment of breast cancer involving a combination of two drugs was investigated: the oral estrogen receptor inhibitor AZD9496 and the CDK4/6 protein inhibitor Palbociclib. The mathematical model that described the interaction between the cancer cells, the treatment, and the immune system cells includes a system of nonlinear ordinary differential equations of the firs order. In general, dynamic variables of a given system change each at a different rate. And it is not possible to know from the mathematical model which variable is fast and which is slow. Therefore, in order to reveal the hierarchy of the system of equations ,a numerical algorithm called the singularly perturbed vector field (SPVF) was applied. This algorithm transform the mathematical model to a new coordinate system in which the rate of change of each dynamic variable of the system can be known.Results and DiscussionAfter writing the mathematical model in new coordinates, the equilibrium point was obtained analytically. The stability of the equilibrium points is investigated, which is essential from a practical perspective. Investigating the stability of the equilibrium points allows determination of when the tumor does not continue to develop and thereby allows adjustment of treatment continuation.

Polynomial reduction from syndrome decoding problem to regular decoding problem

Abstract The regular decoding problem asks for (the existence of) regular solutions to a syndrome decoding problem (SDP). This problem has increased applications in post-quantum cryptography and cryptanalysis. Recently, Esser and Santini explored in depth the connection between the regular (RSD) and classical syndrome decoding problems. They have observed that while RSD to SDP reductions are known (in any parametric regime), a similar generic reduction from SDP to RSD is not known. In our contribution, we examine two different generic polynomial reductions from a syndrome decoding problem to a regular decoding problem instance. The first reduction is based on constructing a special parity check matrix that encodes weight counter progression inside the parity check matrix, which is then the input of the regular decoding oracle. The target regular decoding problem has a significantly longer code length, that depends linearly on the weight parameter of the original SDP. The second reduction is based on translating the SDP to a non-linear system of equations in the Multiple Right-Hand Sides form, and then applying RSD oracle to solve this system. The second reduction has better code length. The ratio between RSD and SDP code length of the second reduction can be bounded by a constant (less than 8).

Thermal inspection on MHD Prandtl fluid flow past a stretching sheet in the presence of heat generation/absorption and suction/injection

Abstract For the past several years, researchers have been investigating the thermal boundary layer behavior under various assumptions due to contemporary requirements. From the existing literature, it has been noticed that no investigation has been conducted to examine the thermal and mass transport of a magnetohydrodynamic (MHD) flow of a Prandtl fluid past a stretching sheet with heat generation/absorption and suction/injection. To address this gap, this study looks at how the changed thermal flux, heat generation/absorption, and suction/injection affect the convective flow of MHD Prandtl fluid over a stretching surface. The governing system of nonlinear partial differential equations (PDEs) has been reduced to a system of ordinary differential equations (ODEs) through appropriate transformations. The desired numerical solutions of the resulting ODEs have been obtained using the bvp5c MATLAB package. We have analyzed and presented the effects of the physical parameters on the velocity, temperature, and concentration fields using graphs and tables. Also, the drag coefficient, heat, and mass transport rates were examined. Additionally, an entropy analysis has been conducted to explore the reusability of heat production. It has been observed that intensifying the magnetic and porosity parameters reduces the flow velocity while increasing the Eckert number, heat generation parameter, and the Biot number significantly increases the temperature. The concentration drops with an increase in the Schmidt number and the chemical reaction parameter. This study holds significant implications for industries that rely on drug delivery, heat exchanger technology, polymerization, and refining processes.

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

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

Estimation of stationary and non-stationary moving average processes in the correlation domain.

This paper introduces a novel approach for the offline estimation of stationary moving average processes, further extending it to efficient online estimation of non-stationary processes. The novelty lies in a unique technique to solve the autocorrelation function matching problem leveraging that the autocorrelation function of a colored noise is equal to the autocorrelation function of the coefficients of the moving average process. This enables the derivation of a system of nonlinear equations to be solved for estimating the model parameters. Unlike conventional methods, this approach uses the Newton-Raphson and Levenberg-Marquardt algorithms to efficiently find the solution. A key finding is the demonstration of multiple symmetrical solutions and the provision of necessary conditions for solution feasibility. In the non-stationary case, the estimation complexity is notably reduced, resulting in a triangular system of linear equations solvable by backward substitution. For online parameter estimation of non-stationary processes, a new recursive formula is introduced to update the sample autocorrelation function, integrating exponential forgetting of older samples to enable parameter adaptation. Numerical experiments confirm the method's effectiveness and evaluate its performance compared to existing techniques.

Reduced-Order Model for Cell Volume Homeostasis: Application to Aqueous Humor Production

The ability of a cell to keep its volume constant irrespective of intra- and extracellular conditions is essential for cellular homeostasis and survival. The purpose of this study is to elaborate a theoretical model of cell volume homeostasis and to apply it to a simulation of human aqueous humor (AH) production. The model assumes a cell with a spherical shape and only radial deformation satisfying the property that the cell volume in rest conditions equals that of the cell couplets constituting the ciliary epithelium of the human eye. The cytoplasm is described as a homogeneous mixture containing fluid, ions, and neutral solutes whose evolution is determined by net production mechanisms occurring in the intracellular volume and by water and solute exchange across the membrane. Averaging the balance equations over the cell volume leads to a coupled system of nonlinear ordinary differential equations (ODEs) which are solved using the θ-method and the Matlab function ode15s. Simulation tests are conducted to characterize the set of parameters corresponding to baseline conditions in AH production. The model is subsequently used to investigate the relative importance of (a) impermeant charged proteins; (b) sodium–potassium (Na+/K+) pumps; (c) carbonic anhydrase (CA) in the AH production process; and (d) intraocular pressure. Results suggest that (a) and (b) play a role; (c) lacks significant weight, at least for low carbon dioxide values; and (d) plays a role for the elevated values of intraocular pressure. Model results describe a higher impact from charged proteins and Na+/K+ ATPase than CA on AH production and cellular volume. The computational virtual laboratory provides a method to further test in vivo experiments and machine learning-based data analysis toward the prevention and cure of ocular diseases such as glaucoma.

On the Efficacy of Sparse Representation Approaches for Determining Nonlinear Structural System Equations of Motion

Abstract A sparsity-based optimization approach is presented for determining the equations of motion of stochastically excited nonlinear structural systems. This is done by utilizing measured excitation-response realizations in the formulation of the related optimization problem, and by considering a library of candidate functions for representing the system governing dynamics. Note that a novel aspect of the approach relates to treating, also, systems endowed with fractional derivative elements. Clearly, this is of significant importance to a multitude of diverse applications in engineering mechanics taking into account the enhanced modeling capabilities of fractional calculus. Further, the fundamental theoretical and computational aspects of various representative, state-of-the-art, numerical schemes for solving the derived sparsity-based optimization problem are reviewed and discussed. A Bayesian compressive sampling approach that exhibits the additional advantage of quantifying the uncertainty of the estimates is considered as well. Furthermore, comparisons and a critical assessment of the employed numerical schemes are provided with respect to their efficacy in determining the nonlinear structural system equations of motion. In this regard, two illustrative numerical examples are considered pertaining to a nonlinear tuned mass-damper–inerter vibration control system and to a nonlinear electromechanical energy harvester, both endowed with fractional derivative elements.

Two-fluid flow of blood in a curved stenotic artery under pulsating condition.

The present article focuses on the analysis of the two-phase flow of blood via a stenosed artery under the influence of a pulsatile pressure gradient. The core and plasma regions of flow are modeled using the constitutive relations of Herschel-Bulkley and the Newtonian fluids, respectively. The problem is modeled in a cylindrical coordinate system. A modest stenosis assumption is used to simplify the non-dimensional governing equations of the flow issue. An explicit finite difference approach is used to solve the resultant nonlinear system of differential equations while accounting for the provided boundary conditions. After the necessary adjustments have been made to the crucial non-dimensional parameters, an analysis of the data behind the huge image, such as axial velocity, temperature field, concentration wall shear stress, flow rate, and flow impedance, is conducted. The current study shows that the curvature of blood vessels plays a significant role in influencing blood velocity. Specifically, a unit increase in the curvature radius results in a 24% rise in blood velocity.

Some Common Fixed-Point Theorem on Complex-Valued Metric Spaces and Its Application to Solve Urysohn Integral Equations

In this paper, a new and generalized contraction principles are proved on complex-valued metric space. By adopting a suitable hypothesis on sequence converging in complex-valued metric space new contractions are established for proving the common fixed-point theorem. Moreover, a rational contractive condition is improved in the complex-valued metric spaces. The obtained results through theoretical study are verified by solving the solution of the nonlinear system of Urysohn integral equations.