Unknown Coefficients Research Articles

An innovative approach to approximating solutions of fractional partial differential equations

Abstract The RPS-M (residual power series method) is a valuable technique for solving F-PDEs (fractional partial differential equations). However, the derivative of the residual function to obtain the coefficients of the series is required in RPS-M. This makes the application of the classical RPS-M limited to a certain extent due to the complexity of the derivation of the residual function for higher iterations. To overcome this obstacle, in this study, we present a simplified version of this approach with the help of Laplace transform that requires less computation and offers higher accuracy. This modified method does not require derivation as well as limit of the residual function to estimate the unknown coefficients of the series solution. To demonstrate its effectiveness, we apply the proposed method to nonlinear F-PDEs to obtain their semi-analytical solution. The obtained solutions exhibit excellent agreement when compared to results obtained using other established approaches. We have also provided the convergence analysis of the obtained solution. Furthermore, by comparing the outcomes for various values of the non-integer order \(\sigma\), we observe that as the value approaches an integer order, the solution converges towards the exact solution.

New bandgap analysis method for metamaterial structures using variational principle

This paper presents an efficient bandgap analysis method for metamaterial structures based on variational principle and linear expression technique. This is in view of the challenges in constructing a displacement function that satisfies the Bloch boundary when applying the traditional energy method to analyze the dispersion relations of metamaterial structures. Also, the inclusion of the wave numbers in the displacement function leads to inefficient computation. While there have been methods, such as virtual springs, have been proposed to deal with such problems, issues remain to be resolved. The main idea of the current paper is the solution of linear uncorrelated coefficients by processing the constraints. In this way, the unknown coefficients in the equations of motion can be linearly expressed, thereby allowing the equations to become variational. Both Gaussian elimination and null-space techniques are used to implement the method. The solution procedures of the two methods are detailed and demonstrated by using one-dimensional and two-dimensional metamaterial structures. The accuracy of the method is verified by comparing the results with those from using finite element method. The calculation efficiency is compared with that of the traditional energy method in terms of both the matrix dimension and the number of the points of wave numbers. The results show that the method has a much higher calculation efficiency than that of the traditional energy method. The method proposed in this paper is free of convergence problems in dealing with boundary problems. Furthermore, in the discussion of combined periodic structures, it is shown that the method still has good geometrical applicability in structures containing multiple boundary conditions, showing great potential in solving complex engineering problems.

Approximate analytical solutions to nonlinear oscillations via semi-analytical method

A modified algebraic method is proposed in this study as an effective semi-analytical method for obtaining analytical solutions for two types of nonlinear differential equations. The trial solution is assumed to be a polynomial, and the algebraic approach is used to find the unknown coefficients. To enhance the solution, first, we apply the Laplace transformation on the series solution and then create the Padé approximations of the resultant equation. Finally, for the nonlinear problems under discussion, the inverse Laplace transformation is used to obtain periodic solutions.

Aerodynamic Modeling and Design Optimization of a Variable-Camber Piezocomposite Rotor

Multirotor vehicles typically use either complex mechanisms to vary blade pitch angle or speed control to vary thrust and torque to attain stability and maneuverability. Alternatively, thrust and torque can be varied by morphing the blades without the use of complex mechanisms and without varying the rotor speed. This method could be faster to change thrust, more efficient, and it may require fewer mechanical components. This paper presents an experimentally validated aerodynamic model with an empirical static-aeroelastic correction to predict the thrust and torque coefficient of a mechanism-free rotor with Macro-Fiber Composite actuators. A variable-camber piezocomposite rotor prototype is experimentally tested on a static-thrust setup. The experiment is used to identify the so-called unknown coefficients of the model (e.g., related to the shape of the blade induced by the piezoelectric device) and to identify a correction to account for flow-induced deformations. Using the identified model, several parametric analyses are conducted to predict thrust and torque coefficients as a function of rotor speed and the electrical excitation of the piezocomposite actuators. In addition, the model is used for design optimization, leading to several optimal designs based on a limited parameter space. Three different objective functions are used: maximizing thrust, minimizing torque, and maximizing thrust-to-torque ratio. The operational aerodynamic characteristics of these designs are also presented.

Global adaptive practical tracking control for high‐order uncertain nonstrict feedback nonlinear systems with unknown control coefficients

SummaryIn this article, the problem of global adaptive practical tracking for high‐order uncertain nonstrict feedback nonlinear systems with unknown control coefficients is studied. To avoid the algebraic loop problem associated with the nonstrict feedback condition and guarantee the controllability of the tracking error, a novel dual dynamic gain scaling method is introduced to compensate nonlinearities and the tracking error simultaneously. Besides, by incorporating the sign functions into the design of adding a power integrator, a general approach for the handing of unknown control coefficients and the direction design of the controller is developed. The presented control scheme can ensure that all system states are globally bounded without constraints on state variables while the reference signal is tracked with expected precision. Three simulation examples, including a practical application, are provided to illustrate the validity of the control scheme.

A preserving accuracy two-grid reduced-dimensional Crank-Nicolson mixed finite element method for nonlinear wave equation

In this paper, we mainly resort to a proper orthogonal decomposition (POD) method to study the reduced-dimension of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the nonlinear wave equation and build a preserving accuracy reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method for the nonlinear wave equation. For this purpose, we first build a new TGCNMFE method for the nonlinear wave equation and demonstrate the existence, unconditional stability, and error estimates for the TGCNMFE solutions. From there, we build a preserving accuracy RDETGCNMFE method for the nonlinear wave equation by using the POD method to decrease the dimension of unknown TGCNMFE solution coefficient vectors and employ the matrix analysis to analyze the existence, unconditional stability, convergence, and errors for the RDETGCNMFE solutions. We finally make use of numerical experiments to verify our theoretical results and to show the advantages of RDETGCNMFE method.

Dynamic inverse plane characteristics of circular holes in a strip-shaped medium under line source excitation

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of circular holes in the strip domain under line source excitation were studied. The total wave field in the strip domain is constructed using the concept of infinite mirroring, which eliminates the mutual impact of the wave field under the action of the upper and lower limits of the strip domain. To determine the overall wave field, solve the wave field's unknown coefficients in accordance with the boundary conditions. The right number of mirror images is found using the dynamic stress distribution of a circular hole in the band-shaped domain as a specific calculation example. The effect of altering the incident frequency and band-shaped domain width on the dynamic stress concentration coefficient surrounding the circular hole is also explored. As a consequence of the findings, the dynamic stress concentration coefficient around the circular hole in the zone domain at low frequencies is substantially bigger than it is at medium and high frequencies. Under identical conditions, the dynamic stress concentration coefficient will exhibit periodic extreme values and converge as the distance increases between the lower boundary and the circular hole’s center.

Derivative block methods for the solution for fourth-order boundary value problems of ordinary differential equations

The application of block methods in the study of dynamical system is one area which needs more research. Therefore, this work presents block methods and its application to solve some problems in solid and fluid mechanics. The method was developed directly using the approaches of collocation and interpolation, and using power series polynomial as a trial solution. First, the system of linear equations is solved to obtain the unknown coefficients. The coefficients gotten are then substituted into the approximate solution to obtain continuous scheme. The continuous scheme, its first, second and third derivatives are evaluated at all the grid points to generate the block methods. The derived methods were applied to solve fourth-order boundary value problems of ordinary differential equations arising from beams and chemical problems. The results demonstrate the reliability and efficiency of the proposed method.

Study on the turbulent Prandtl number model for liquid metal flow and heat transfer in a narrow rectangular channel

In order to improve the core power density of liquid metal reactors, plate-type fuel assemblies are typically used, which have obvious advantages in heat transfer capacity. For the low Prandtl number fluid, the RANS turbulence model-based CFD method needs to be modified by inserting the turbulent Prandtl number model. This paper investigated the turbulent Prandtl number model for the turbulent heat transfer of liquid metal flow in a narrow rectangular channel. First, the existing turbulent Prandtl number models were summarized and classified as the local turbulence Prandtl number model and the global turbulence Prandtl number model. They were evaluated based on the experimental data. The results showed that the prediction results of each model are not ideal. Then, a typical numerical example was used to analyze the temperature field and flow field in the narrow gap of a narrow rectangular channel. It was shown that the entire flow region only includes the near-wall region and the transition region, and the transition region accounts for the main proportion. A semi-empirical local turbulence Prandtl number model was obtained by solving the turbulent transport equation, and the unknown coefficients were determined iteratively with the experimental data. The error was controlled within ±20%. Finally, the applicability of the newly proposed turbulent Prandtl number model was explored, and the results showed that it has good applicability for the different runner sizes and different liquid metal types. This study can provide a reference for the numerical simulation of turbulent heat transfer of metal liquid in the narrow rectangular channel.

Adaptive Projection and Fuzzy Tracking Design for Unknown Control Coefficients and References.

This article addresses the tracking control problem of nonlinear pure-feedback systems, where the control coefficients and the dynamics of the references are unknown. Fuzzy-logic systems (FLSs) are used to approximate the unknown control coefficients and at the same time the adaptive projection law is designed to allow each fuzzy approximation to cross zero, which yields that the proposed method avoids the assumption of using Nussbaum function, that is, the unknown control coefficients never cross zeros. Another adaptive law is designed to estimate the unknown reference and then it is intergraded into the saturated tracking control law to achieve the uniformly ultimately bounded (UUB) performance of the resulting closed-loop system. Simulations show the feasibility and effectiveness of the proposed scheme.

Hydrodynamic performance of vertical cylindrical wave energy absorbers in front of a vertical wall

This paper studies the hydrodynamic performance of vertical cylindrical absorbers in front of a vertical wall. All the absorbers are independent of each other and restricted to only the heave motion. Based on a linear potential flow theory, an analytical solution is developed for the problems of wave diffraction and radiation by absorbers. In the solving procedure, the hydrodynamic problem is first transformed into an equivalent problem in an open water domain using the image principle. The number of absorbers in the equivalent problem is twice that in the real problem, and the plane layout is symmetric about the original vertical wall. The velocity potential of the fluid domain is obtained using the method of variable separation, and the unknown expansion coefficients in the velocity potential are determined by the matched boundary conditions. The heave excitation force, added mass, radiation damping, motion response, and energy capture width of the absorbers are calculated. Case studies are presented to show the effects of the wall reflection and hydrodynamic interaction on the energy extraction performance of the wave energy converter (WEC) system. Subsequently, the WEC performance under the action of irregular waves is analyzed by considering an incident wave spectrum, and the mean annual absorbed power of the device is estimated by considering the wave data statistics at the actual sites. The results indicate that when the wave motion resonates with the absorber motion, the energy extraction performance of the absorbers is significantly improved. The performance of the absorbers can be effectively improved when the structures are close to the antinodes of a standing wave field. By designing a reasonable plane layout, the hydrodynamic interaction can play a constructive role in the performance of the WEC system.

Adaptive event-triggered tracking control for nonlinear systems with prescribed performance: A time-domain mapping approach

This paper proposed an event-triggered adaptive fuzzy prescribed time control strategy for a class of nonlinear systems with uncertain external disturbances and unknown virtual control coefficients under prescribed performance. A novel time-domain mapping technique is proposed to transform the prescribed time tracking control problem into an asymptotic convergence problem. An observer is designed to estimate for the external disturbance and fuzzy approximation error of fuzzy logic systems (FLSs). The problem of “explosion of complexity” is addressed by applying dynamic surface control (DSC) technique to backstepping. An adaptive fuzzy event-triggered strategy is designed using the Lyapunov stability theorem. This strategy ensures that, after time-domain mapping, the tracking error asymptotically converges to zero. Additionally, there is no Zeno behavior and all system signals are bounded. Finally, the effectiveness of the proposed scheme is verified through three simulation examples.

Learning the flux and diffusion function for degenerate convection-diffusion equations using different types of observations

In recent years, there has been an increasing interest in utilizing deep learning-based techniques to predict solutions to various partial differential equations. In this study, we investigate the identification of an unknown flux function and diffusion coefficient in a one-dimensional convection-diffusion equation. The diffusion function is allowed to vanish on intervals implying that solutions generally possess low regularity, i.e., are discontinuous. Therefore, solutions must be interpreted in the sense of entropy solutions which combine a weak formulation with an additional constraint (entropy condition). We explore a methodology that utilizes symbolic neural networks (S-Nets) in combination with an entropy-consistent discrete numerical scheme (ECDNS). Different types of observation data are explored. Extensive experiments in this paper demonstrate that the proposed method is a robust tool to identify the unknown flux and diffusion function. The flux and diffusion functions are restricted to be rational functions.

Adaptive asymptotic tracking control of uncertain fractional-order nonlinear systems with unknown control coefficients and actuator faults

For uncertain fractional-order nonlinear systems (UFONS) with unknown control coefficients and intermittent actuator faults, the asymptotic tracking control problem is investigated in this paper. Firstly, to weaken the influence of virtual control coefficients and intermittent actuator faults, a smooth fractional-order projection operator-based adaptive compensation mechanism is presented. Additionally, a fractional-order nonlinear filter is constructed to replace the fractional-order derivative of virtual control functions approximately, which not only avoids the issue of complexity explosion existed in backstepping control frame, but fully compensates the effects of boundary errors caused by the employed filter in spite of the unknown virtual control coefficient. By constructing a fractional Lyapunov function from the property of projection operator, it is proved that all signals in the closed-loop system are bounded, and the asymptotic tracking control object is achieved. Definitively, a simulation study is presented to verify the availability of the presented method.

Impact of Two Vertical Porous Barriers in Reflection of Water Waves and Mitigation of Wave Forces on a Rigid Floating Structure with Consideration of Uniform Current Over a Porous Sea-Bed

This work examines an oblique water wave scattering by a rigid floating structure with the consideration of uniform current and no current, in the presence of two porous barriers with distinct height and porous properties placed in front of the structure. The objective is to study the associated reflection and to find means to mitigate the waveload on the floating structure. These barriers are considered at some distance apart from each other. It is assumed that the train of water waves interacts with the porous barriers and the floating structure due to which it undergoes partial reflection. The fluid domain is considered in terms of five distinct sub-regions in each of which the solution of the relevant boundary value problem in terms of the velocity potential is obtained through the employment of the techniques of separation of variables and eigenfunction expansion. The resulting system of equations is obtained by utilizing the matching conditions corresponding to the continuity of mass flux and pressure across each virtual vertical boundary. The roots of the dispersion relation are evaluated and utilized while solving the system of linear equations through a standard matrix technique which gives the unknown coefficients. For understanding the role of the porous barriers, the reflection coefficient and the hydrodynamic forces are evaluated and analyzed. In this context, the impact of the space between these two porous barriers, their height, and the values of the porous-effect parameters are thoroughly examined. It is observed that placing of these barriers at an appropriate distance from each other can substantially mitigate both horizontal force and vertical force acting on the floating structure. This study reveals a periodic nature of the reflection coefficient corresponding to an increase in the distance between the barriers. Further, another important observation is that the reflection coefficient results in an increment as the height of these porous barriers is increased. It, as a result, produces a reduction of the wave forces acting on the floating structure. The study of the oblique wave scattering by the floating structure is extended to the case of the presence of a uniform current. In this case, an enhancement in the reflection coefficient is observed, and subsequently, horizontal and vertical forces on the floating structure decrease. The impact of some other parameters on the reflection coefficient and the forces is also studied, and some observations are recorded. The results are validated against two established results which shows a very close match.

Determination of channel curvature in the map forecast model
 of river meandering

The article is a development of the author's previous work aimed at predicting channel displacements during the meandering of lowland rivers. The hypothesis about the instability of a straight river bed is accepted as the main hypothesis about the causes of meandering. To construct a meandering model, a statement is used according to which the river bed shifts from its current position along the normal to the dynamic axis of the flow in the direction from the center of its curvature in proportion to the value of the curvature of the channel in a given section, and the proportionality coefficient is unknown in advance and must be determined in the process of cartographic (hydrological) monitoring. Since the curvature of a river channel is ex-pressed through the second derivative of a curve that approximates the outline of the bank, the important question is how to accurately calculate this curvature from the coordinates of bank points. A way to do that is proposed in the article. This way is based on approximating the coast-line with a 2nd order spline in the form of a circle most closely adjacent to the coastline. Since hydrological monitoring makes it possible to find normal channel displacements over a certain period of time, then knowing the curvature of the coastline in all sections of the channel, it is possible to calculate the unknown coefficient of proportionality of this displacement to the cur-vature. The obtained values of the coefficient of proportionality are proposed to be used for pre-dictive mapping of channel dynamics for future time intervals.

INVERSE BOUNDARY PROBLEM FOR THE EQUATION OF FORCED VIBRATIONS OF A CANTILEVER BEAM WITH INTEGRAL CONDITIONS

The paper studies the solvability of an inverse boundary value problem with an unknown time-dependent coefficient for the equation of forced vibrations of a cantilever beam with the integral. Bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending are described by a fourth-order differential equation. The purpose of the work is to determine the unknown coefficient and solve the problem under consideration. THIS problem under consideration is reduced to an auxiliary equivalent problem. Next, the existence and uniqueness of a solution to the equivalent problem is proved using the contraction mapping principle. As a result, using equivalence, the uniqueness of the existence of the classical solution is proved.

Estimating the strong coupling from τ decay using accelerating series convergence

We apply the Euler transformation to accelerate the convergence of the QCD perturbative series with the aim to determine the strong coupling αs in terms of the total τ-decay rate rτ. The variation of the result with the order of the QCD perturbation theory is small and comparable with the uncertainties of rτ. We also present an estimate of a range of the yet unknown 5th and 6th order coefficients k5 and k6 of the Adler function.

Collaborative modeling-based improved moving Kriging approach for low-cycle fatigue life reliability estimation of mechanical structures

To effectively estimate the reliability level of low-cycle fatigue (LCF) life of mechanical structures, a novel method of collaborative modeling-based improved moving Kriging (IMKCM) approach is proposed by fusing improved moving Kriging method and collaborative modeling strategy. In this method, the improved moving Kriging model is developed by the Kriging model, moving least squares method and artificial hummingbird algorithm, in which the Kriging model is selected as the basis function, the artificial hummingbird algorithm is used to define the optimal radius of compact support region and determine the effective modeling samples, and the moving least squares method is adopted to resolve the unknown coefficients; the collaborative modeling strategy is developed from the decomposition and coordination method, in which the mathematical models of the decomposed sub-objectives are synchronously built by the cell array theory, and the mathematical model between the objective and the sub-objectives is established by the improved moving Kriging model. Furthermore, a numerical example (i.e., nested nonlinear function) and an engineering example (i.e., turbine blisk LCF life reliability estimation) are employed to verify the effectiveness of the IMKCM approach. The results show that the proposed IMKCM method holds excellent abilities in modeling features and simulation performance.

On fluorophore imaging by nonlinear diffusion model with dynamical iterative scheme

Fluorescence imaging aims at recovering the absorption coefficient of fluorophore in biological tissues. Due to the nonlinear dependence of excitation and emission fields on the unknown coefficient, such an inverse problem with the boundary measurement as inversion input is nonlinear. We reformulate this inverse problem as an optimization problem for a non-convex cost functional consisting of the unknown absorption coefficient depending both on the excitation field and the emission one with a penalty term. The existence of minimizer of the cost functional is proved rigorously, with explicit expression for the Fréchet derivative of the cost functional. For seeking the local minimizer considered as the approximate solution to the inverse problem efficiently, we develop a dynamical process which makes the non-convex cost functional quadratic at each iteration step, by freezing the unknown coefficient for the excitation field. This new dynamical quadratic optimization process is proven convergent and decreases the amount of computations for the fluorescence imaging, while the reconstruction accuracy keeps almost unchanged. Such advantages are verified by several numerical examples for different configurations of the absorption coefficient to be identified, comparing our proposed scheme with the algorithm for the original non-convex cost functional.