System Of Equations Research Articles

Numerical Technique Based on Operational Matrices of Fractional Integration Using ψ$$ \psi $$‐Shifted Chebyshev Polynomials

ABSTRACTIn this paper, we present a numerical scheme based on a modified form of shifted Chebyshev polynomials to find the numerical solution of a class of fractional differential equations. For this purpose, we work out operational matrices of fractional integration of ‐shifted Chebyshev polynomials obtained from shifted Chebyshev polynomials. Finally, the solution to the problem under consideration is obtained by solving a system of algebraic equations that results from the use of operational matrices of integration. The analysis of integer and non‐integer order differential equations is presented to show the convergence of the solution of fractional order differential equation to the corresponding solution of the integer order differential equation. At the end, we present some linear and non‐linear examples to validate the theoretical analysis. Non‐linear examples are solved using Quasilinearization and proposed numerical technique.

Dynamics of delay epidemic model with periodic transmission rate

We introduce novel epidemic models with single and two strains described by systems of delay differential equations with a periodic time-dependent disease transmission rate, and based on the number of newly infected individuals. Transitions between infected, recovered, and returning to susceptible compartments due to waning immunity are determined by the accompanying time delays. Positiveness, existence and uniqueness of solutions are demonstrated with the help of fixed point theory. Reducing delay differential equations to integral equations facilitates determining the analytical estimation of the equilibrium solutions. When there are two strains, they compete together, and the strain with a larger basic reproduction number dominates in the population. However, both strains coexist, and the magnitudes of epidemic outbreaks are governed by the basic reproduction numbers. The outlines of this work are verified through comparison with seasonal influenza data.

Unraveling autonomic cardiovascular control complexity during orthostatic stress: Insights from a mathematical model

Understanding cardiovascular control mediated by the autonomic system remains challenging due to its inherent complexity. Consequently, syndromes such as orthostatic intolerance continue to evoke debates regarding the underlying pathophysiological mechanisms. This study develops a comprehensive mathematical model simulating the control of the sympathetic branch of the cardiovascular system in individuals with normal and abnormal responses to the head-up-tilt test. We recruited four young women: one control, one with vasovagal syncope, one with orthostatic hypertension, and one with orthostatic hypotension, exposing them to an orthostatic head-up tilt test (HUTT) employing non-invasive methods to measure electrocardiography and continuous blood pressure.Our work encompasses a compartmental model formulated using a system of ordinary differential equations. Using heart rate as input, we predict blood pressure, flow, and volume in compartments representing the veins, arteries, heart, and the sympathetic branch of the baroreflex control system. The latter is modulated by high- and low-pressure baroreceptor afferents activated by changes in blood pressure induced by the HUTT. Sensitivity analysis, parameter subset selection, and optimization are employed to estimate patient-specific parameters associated with autonomic performance. The model has seven sensitive and identifiable parameters with significant physiological relevance that can serve as biomarkers for patient classification.Results show that the model can reproduce a spectrum of blood pressure responses successfully, fitting the trajectory displayed by the experimental data. The controller exhibits behavior that emulates the operation of the sympathetic system. These encouraging findings underscore the potential of computational methods in evaluating pathologies associated with autonomic nervous system control, warranting further exploration and novel approaches.

Optimization of heat transfer in a channel with stretching walls using a magnetized tetra-hybrid nanofluid

Nanoparticles have numerous applications and are used frequently in different cooling, heating, treatment of cancer cells and manufacturing processes. The current investigation covers the utilization of tetra hybrid nanofluid (aluminium oxide, iron dioxide, titanium dioxide and copper) for Crossflow model over a vertical disk by considering the shape effects (bricks, cylindrical and platelet) of nanoparticles, electro-magneto-hydrodynamic effect and quadratic thermal radiation. The present study is devoted to present the mathematical formulation of the tetra-hybrid nanofluid flow in a porous channel with stretching/shrinking walls. The present inspection model is derived from given partial differential equations (PDEs) and then transformed into a system of ordinary differential equations (ODEs) by incorporating similarity variables. The transformed ODEs are solved using the bvp4c methodology, which yields numerical results. From the obtained results it is observed that the maximum amount of [Formula: see text] happens when there is slightly increase in [Formula: see text] with stretched wall that is [Formula: see text]. A high radiation value indicates a considerable contribution from radiative heat transfer, whereas stretching and shrinking increase and reduce the size of the boundary. The combined impact caused an increase in the Nusselt number. The graphical and tabular representations are used to explain the physical behaviour of various model parameters. Previous outcomes are also contrasted with the current outcomes.

On the construction of an optimal network of observation points when solving inverse lin-ear problems of gravimetry and magnetometry

Unique solvability of systems of linear algebraic equations is studied to which many in-verse problems of geophysics are reduced as a result of discretization after applying the method of integral equations or integral representations. Examples of singular and nonsingular systems of vari-ous dimensions that arise when processing magnetometric and gravimetric data from experimental observations are discussed. Conclusions are drawn about methods for constructing an optimal net-work of experimental observation points.

Research on dynamic well control in riserless mud recovery system

In order to promote the development of RMR (Riserless Mud Recovery) system well control technology and ensure the safe operation of deep water drilling operations, the overflow control equation of RMR system considering the lifting force of subsea pump is established and verified, the conventional overflow monitoring method and the overflow monitoring method based on subsea pump parameter variation are compared and analyzed. the dynamic well control process of the RMR system after overflow under the two modes of constant subsea pump displacement and constant inlet pressure of subsea pump is simulated, and the influencing factors of inlet pressure of subsea pump are further analyzed. The results show that although the temperature effect is taken into account in the model established in this paper, the time error of the mud pool increment, the inlet pressure and the displacement of subsea pump after reaching the overflow criterion is less than 22.5% compared with Froyen's model. The conventional overflow monitoring method has fast response and small change, while the overflow monitoring method based on subsea pump parameter change is on the contrary, and the two should be used together. Both the constant inlet pressure of subsea pump mode and the constant subsea pump displacement mode can control overflow, but from the stability of subsea pump, service life and well control safety, the constant displacement mode is better. In the case of large formation pressure, low drilling fluid density, overflow criteria and circulating displacement, the pressure response adjustment speed of the subsea pump inlet is faster, and the back pressure load of the subsea pump inlet can be effectively alleviated by appropriately increasing the circulating displacement and drilling fluid density. This study has enriched the theoretical system of RMR system in overflow control, and has certain guiding significance for the development of deepwater and ultra-deepwater drilling safety technology.

Assessing the influence of nonlinear mooring restoring forces in the optimization and design of FOWT

Abstract The design of mooring systems plays a pivotal role in the success of Floating Offshore Wind Turbine (FOWT) projects. This process naturally involves iterations, as design and analysis are intricately linked. Despite the inherent nonlinearity of mooring system restoring forces, many designers opt for employing an equivalent linear mooring system stifness matrix to expedite the optimization process in early design stages. This article aims to underscore the limitations associated with relying on the equivalent linear model when compared to accounting for nonlinear restoring forces during optimization processes. To illustrate this point, a case study was conducted using the reference semi-submersible platform VolturnUS-S. The study considered intermediate to large water depths and addressed three different mooring confgurations: catenary, semi-taut, and taut lines. The analysis focused on several critical aspects, including ofset watch circles, line tensions (including those at the anchors), and a cost estimate based on the different models. The findings indicate that, for catenary-based mooring systems, the linear model remains a good approximation, leading to no significant loss of accuracy in the context of early design stages. As the lines become tauter, however, the nonlinearities become more pronounced and the errors involved in the linear model can reach unacceptable levels.

Comparison of British Thyroid Association and TIRADS classifications and their impact on the radiological and surgical management of indeterminate thyroid nodules

BackgroundThe British Thyroid Association (BTA) Guidelines for the Management of Thyroid Cancer advocate for fine-needle aspiration cytology for all thyroid nodules graded indeterminate (U3) at ultrasound assessment. This approach raises concerns regarding potential over-diagnosis of low-risk lesions. Conversely, equivalent Thyroid Imaging Reporting and Data Systems (TIRADS) guidelines permit surveillance or discharge of indeterminate thyroid nodules of certain sizes. This service analysis analyses how guideline choice impacts the fine-needle aspiration cytology rate and subsequent surgical management of indeterminate thyroid nodules. MethodsAll patients with an indeterminate (U3) thyroid nodule identified on ultrasound over a 12-month period were included. Indeterminate thyroid nodules were retrospectively rescored using three equivalent TIRADS classifications by three independent reviewers, blinded to the histopathology. Hypothetical differences in fine-needle aspiration cytology rates and surgical activity were then compared. ResultsNinety-six nodules were identified. Retrospective application of TIRADS guidelines resulted in a hypothetical 44.8-55.2% reduction in fine-needle aspiration cytology performed for indeterminate thyroid nodules compared to BTA. A statistically significant increase in rates of surgical activity for indeterminate thyroid nodules was observed between BTA guidance and all retrospectively applied TIRADS guidelines (p<0.001). Of four confirmed thyroid cancers, three would have been unanimously removed. ConclusionUnder BTA guidance, increased fine-needle aspiration cytology rates for indeterminate thyroid nodules resulted in significantly increased surgical activity in our cohort compared to retrospectively applied TIRADS guidelines.

‘Let me tell you a story’: the politics of macroeconomic models

ABSTRACT Various social science literatures suggest that the general character of macroeconomic models reflects their assumptive base. A more specialist literature in the Weintraub-Boumans-Morgan tradition shows how the mathematics of those models moulds together logical implications of particular starting assumptions, insights from generally accepted theoretical propositions, and professionalised common-sense about how the world works. I go one step further in arguing that a process of narrative moulding operates in tandem with this mathematical moulding. A naming strategy provides the mathematical properties of macroeconomic models with economic labels to create the feeling that they are something more than a merely mathematical structure. A storytelling strategy then informs policy-makers of where the solution to the system of equations positions the outer limits of both political desirability and political possibility. Future dedicated research programmes into the narrative dimensions of macroeconomic models can be expected to shed further light on how theory models can masquerade as policy models and substitute models as surrogate models.

Algorithms for solving the inverse scattering problem for the Manakov model

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).

Asymptotic integrability of nonlinear wave equations.

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related to the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory with several examples.

On a new globally optimal method for the design optimization of air coolers coupled with real fans

A new approach for the globally optimal design of air coolers is presented. First, by coupling the air cooler geometric options with commercially available fans, the traditional mismatch emerging from designing the air cooler first and selecting a commercial fan later is removed. Second, the method departs from the traditional LMTD and ε-NTU methods, adopting a model composed of a differential–algebraic system of equations (DAE system) for the air cooler simulation, which is discretized to consider properties variable with temperature. The resultant optimization problem is solved using Set Trimming and Smart Enumeration, which can identify the global optimum through the simulation of only a small fraction of the search space. The comparison of the simulation results with different commercial software provided similar outputs. A comparison with design results obtained using LMTD solutions indicates that the LMTD models may overdesign/underdesign the unit

New error estimates for the conjugate gradient method

The conjugate gradient method is the default iterative method for the solution of linear systems of equations with a large symmetric positive definite matrix A. The development of techniques for estimating the norm of the error in iterates computed by this method has received considerable attention. Available methods for bracketing the A-norm of the error evaluate pairs of Gauss and Gauss-Radau quadrature rules to determine lower and upper bounds. The latter methods require a user to allocate a node (the Radau node) between the origin and the smallest eigenvalue of the system matrix. The determination of such a node generally demands further computations to estimate the location of the smallest eigenvalue. An approach that avoids the need for a lower bound of the smallest eigenvalue is to replace the Gauss-Radau quadrature rule by an anti-Gauss rule; see, e.g., Golub and Meurant (1997), Golub and Meurant (2010), Golub and Strakoš (1994), Meurant (1997), Meurant (1999). However, this approach may sometimes yield inaccurate error norm estimates. This paper proposes the use of pairs of Gauss and associated optimal averaged Gauss quadrature rules to estimate the A-norm of the error in iterates determined by the conjugate gradient method.

Thermal Wave and Pennes’ Models of Bioheat Transfer in Human Skin: A Transient Comparative Analysis

The primary focus of this study is to analyze comparative heat transfer in a two-dimensional (2D) multilayered human skin using thermal waves and Pennes' bioheat transfer models. The model comprises the epidermis, dermis, hypodermis tissue, and inner cells, and aims to understand their response to microwave (MW) power and electromagnetic (EM) frequency. The system of equations involves EM wave frequency and bioheat equations and uses the finite element method (FEM) for solving. It encompasses a range of microwave power levels (4-16 W), frequencies (0.9-4 GHz), and exposure durations (0-180 s). It examines how MW power and frequency affect temperature predictions due to different relaxation times. The results are visually represented, illustrating microwave power dissipation, isothermal profiles within the skin tissue, temperature trends at several locations, relaxation times, specific absorption rate (SAR), and the mean surface temperature of the multilayered dermal cell. Thermal analysis shows that Pennes' equation predicts higher temperatures than the thermal wave model of bioheat transfer (TWMBT). A notable disparity in temperature evolution is observed between the two models, especially in high-frequency transient heating scenarios. The TWMBT forecasts a delay in heat transfer, offering valuable insights into the more realistic short-term thermal behavior that the classical Pennes' model fails to capture. This comparative study underscores the significance of selecting an appropriate bioheat transfer model for precise thermal analysis in biomedical applications, such as hyperthermia treatment and thermal diagnostics. The findings emphasize the potential of the TWMBT to enhance the accuracy of thermal treatments in clinical settings.

Compositional reservoir simulation with a high-resolution compact stencil adaptive implicit method

The adaptive implicit method (AIM) is the mainstream approach for compositional reservoir simulation. In its standard form, AIM uses single-point upwinding to reconstruct the fluxes across the interfaces of the control volume, and this leads to substantial numerical diffusion and loss of accuracy. Previous efforts to improve the accuracy of AIM focused on using high-order schemes to reconstruct the interfacial fluxes; those schemes introduced additional numerical nonlinearities to the system of equations or compromised the accuracy of the Jacobian by neglecting the high-order terms in its construction. In this work, we describe a high-resolution compact-stencil (HRCS) AIM. The new scheme is applied to compositional reservoir simulation. In addition to the mixed implicit/explicit time discretization of standard AIM, the HRCS AIM scheme uses a mixed time and space discretization. Specifically, we blend low- and high-order fluxes according to a well defined rule that uses a high-order reconstruction in the explicit regions of the domain and a low-order reconstruction in the implicit regions. This strategy ensures that additional nonlinearities introduced by the high-order reconstruction do not impact the Jacobian matrix, thus preserving the algebraic structure of standard AIM. The HRCS AIM method is demonstrated using several compositional problems. The results indicate substantial gains in accuracy with a small additional computational cost compared with standard AIM. Additionally, HRCS AIM is more robust and has a smaller computational cost compared with its full high-resolution counterpart.

Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field Equations

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Discussion of S. Ponsioen, S, Jain and G. Haller: “Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems”, Journal of Sound and Vibration 488, 2020, pages 1–23

The authors of the discussed article considered two examples (1) a finite element model of a linear beam with a cubic spring attachment having 10 to 10,000 degrees of freedom, and (2) a finite element model of a geometrically nonlinear Timoshenko beam having 21 degrees of freedom. The authors concluded that Harmonic Balance (HB) becomes intractable for those examples. In this discussion, we falsify this conclusion. We demonstrate that very well-tractable computation effort is obtained when properly using the HB tool NLvib that was also employed in the discussed article. In particular, for (1), the well-known exact condensation to the nonlinear part of the equation system is carried out. For (2), suitable parameters are set for the path continuation and the Newton-type solver. With this, the computation takes from split seconds to about two minutes instead of the many hours reported in the discussed article.

Advanced piezoelectric fluid energy harvesters by monolithic fluid–structure–piezoelectric coupling: A full-scale finite element model

A full-scale finite element model is presented for monolithic fluid–structure interaction (FSI) simulations of thin-walled piezoelectric fluid energy harvesters (PFEHs). Unlike widely used beam/plate-based models, our model employs a solid finite element discretization to precisely represent the complex PFEH designs involving microstructured transducers and non-uniform cantilevers. These features, plus the local FSI effects, are often ignored by simplified models. We applied the Galerkin method to formulate the weak form of the mixed equation system, integrating the flow dynamics, the geometrically nonlinear cantilever, the piezoelectric components, the electrode, and the output circuit within a closed-circuit electro-mechanical coupled system. The coupling of the multiple domains is achieved through boundary-fitted discretization within a monolithic scheme, using shifted-Crank–Nicolson temporal integration. This work explored implementing piezoelectric FSI systems within the FEniCS-based TurtleFSI library, and experimented techniques such as employing penalty functions for achieving electrode components with uniform electric potentials. We investigated various advanced PFEH features, including the baseplate design, the arrangement and microstructure of the piezoelectric components, and their influence on the system's dynamic and energy output behavior. The results confirmed the model's key advantages: full-scale modeling allows the integration of complex base structures and transducer microstructures in PFEH design. Combined with monolithic FSI coupling, it offers greater versatility, supporting a wider range of fluid environments and configurations in both wind and hydropower harvesting. Additionally, the modeling strategy can be intended not only to enhance power output, but also to minimize material usage, reduce mechanical fatigue, and extend the operational lifespan of PFEH systems.

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

On the Periodicity Solutions of Five Systems of Rational Systems of Difference Equations of Order Five

The primary aim of this paper is to explore the behavior of several nonlinear systems of difference equations following T_{n+1}=(E_{n}E_{n-4})/(T_{n-3}(\pm 1 \pm E_{n}E_{n-4})), E_{n+1}=(T_{n}T_{n-4})/(E_{n-3}(pm 1 \pm T_{n}T_{n-4})), and obtain solution expressions for them. Moreover, we utilize MATLAB programming to simulare the dynamics and validate our results.