Undetermined Coefficients Research Articles

Computational Fuzzy Modelling Approach to Analyze Neuronal Calcium Dynamics With Intracellular Fluxes.

Mathematical neuroscience investigates how calcium distribution in nerve cells affects the neurological system. The interaction of numerous systems is necessary for the operation of several cellular processes in neuron cells, such as calcium, buffer, ER etc. The dynamics of interacting parameters give useful information on neural cell function. This work uses a mathematical model to analyze the dynamic interactions of buffer and ER inside neurons, considering their spatial properties. While buffers bind to calcium ions and lower their concentration, the endoplasmic reticulum (ER) serves as a reservoir, holding a significant number of free calcium ions. The uncertainty of initial values of calcium concentration poses challenges for researchers to develop calcium signaling models. In this article, we examined the exact solution and approximate solution of the mathematical model that was analyzed using the fuzzy undetermined coefficient approach. MATLAB is being used to perform the simulation. Endoplasmic reticulum and buffer have been found to have a substantial impact on calcium signaling. Fuzzy differential equation Provides a useful tool for evaluating complicated processes with imprecise values when ordinary differential equations perform not precisely. They allow for the examination of dynamic processes under fuzzy settings, which contributes to advances research.

Higher order fractal differential equations

This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with [Formula: see text]-order. Solutions for these equations with constant coefficients are obtained through the method of variation of parameters and the method of undetermined coefficients. The solution space for higher [Formula: see text]-order linear fractal differential equations is defined, showcasing its non-integer dimensionality. The solutions to [Formula: see text]-order linear fractal differential equations are graphically depicted to illustrate their non-differentiability. Additionally, equations of motion governing the behavior of two masses in fractal time are proposed and solved.

Polynomial solutions to the first order difference equations in the bivariate difference field

The bivariate difference field provides an algebraic framework to study sequences satisfying a recurrence of order two, and it can be used to transform summations involving such sequences into first order difference equations over the bivariate difference field. Based on this, we present an algorithm for finding a series of polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree of any possible polynomial solutions, which in turn is sufficient to compute all polynomial solutions by using the method of undetermined coefficients.

Symplectic analytical solutions for free vibration of elastically line-hinged orthotropic rectangular plates with rotationally restrained edges

The elastically line-hinged orthotropic rectangular plates with rotationally restrained edges are commonly used in engineering applications such as deployable structures. However, it is intractable to obtain the analytical solutions for the free vibration problems of such structures owing to the challenges in processing the high-order partial differential equations. Here, we make a first attempt to deal with such issues within the Hamilton system-based symplectic space. The problem of a plate is transformed into the symplectic space from the Euclidean space, and the subplates that may be analyzed analytically by the symplectic superposition method are then obtained with the division of the entire plate. The complex boundary and connection forms are achieved by enforcing mechanical quantities with undetermined expansion coefficients on the edges of the subplates. By integrating the solutions of the subplates, the final solution of an elastically line-hinged orthotropic rectangular plate with rotationally restrained edges is accessible. The proposed solution scheme is performed with rational derivations, with no requirement for pre-defined solution forms. Comprehensive results with validations under various boundary and hinge connection cases are presented. Moreover, detailed parametric investigations are conducted, which could facilitate the engineering design of deployable structures.

Observation vector reconstruction-based nonparametric nonlinear restoring force identification for granules-structures coupled vibrating system

The nonlinear restoring force (NRF) generated by the collision and friction between particles and structures is the leading cause of the complex dynamic response of the granules-structures coupled vibrating system (GSCVS). Identification of NRF can provide critical information for post-event damage diagnosis and structural design of immersed structures. However, the spatial distribution and dynamic response of the particles near the structures are diverse and complex, making it difficult to describe the NRF with an accurate mathematical model. This paper proposed a data-based nonparametric method to estimate the NRF in the GSCVS. A nonparametric model of NRF that considered the additional effects of particles on both sides of the structures and consisted of system response and undetermined coefficients was developed. The observation vector of the conventional Extended Kalman Filter (EKF) was reconstructed by the sparse measurement of the strain response. The reconstructed observation vector contains three response components: translational displacement, translational acceleration, and rotational acceleration, in which the rotational acceleration response is difficult to measure in engineering applications. The proposed EKF based on observation vector reconstruction (EKF-OVR) can identify the undetermined coefficients in the nonparametric model, and then the NRF can be calculated. Numerical studies showed that EKF-OVR achieved higher accuracy and noise robustness than the conventional EKF and the data fusion based EKF. A dynamic experimental study on granules-beam coupled vibrating system (GBCVS) was conducted, and the proposed algorithm was employed to identify the NRF of the GBCVS. The effects of excitation amplitude, particle size, and immersed depth on NRF are analyzed, and it is found that higher harmonic components in the NRF led to period doubling and chaos of the beam response.

Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

Finding The Inverse of A Polynomial Modulo in The Ring Z[X] Based on The Method of Undetermined Coefficients

This paper presents the theoretical foundations of finding the inverse of a polynomial modulo in the ring Z[x] based on the method of undetermined coefficients. The use of the latter makes it possible to significantly reduce the time complexity of calculations avoiding the operation of finding the greatest common divisor. An example of calculating the inverse of a polynomial modulo in the ring Z[x] based on the proposed approach is given. Analytical expressions of the time complexities of the developed and classical methods depending on the degrees of polynomials are built. The graphic dependence of the complexity of performing the operation of finding the inverse of a polynomial in the ring Z[x] is presented, which shows the advantages of the method based on undetermined coefficients. It is found that the efficiency of the developed method increases logarithmically with an increase in the degrees of polynomials.

The use of Cauchy-type singular integrals over neighboring intervals to compute induced slip in displaced faults

We present expressions to compute the inverse of a Cauchy-type singular integral equation representing the relation between a double-peaked Coulomb stress in a fault or fracture and the resulting slip gradient in two distinct collinear slip patches. In particular we consider a situation where the patches are close enough to account for the influence of the slip gradient in one patch on the slip-induced shear stress in the other patch and vice versa. This situation can occur during depletion-induced or injection-induced fault slip in subsurface reservoirs for, e.g., natural gas production, hydrogen or CO2 storage, or geothermal operations. The theory for a single slip patch is well-developed but the situation is less clear for a configuration with two patches although the monographs of Muskhelishvili (1953) and Weertman (1996) provide earlier results. We show that the general inverse solution for the coupled two-patch problem requires six auxiliary conditions to ensure six physical requirements: boundedness of the slip gradient at the four end points of the slip patches and vanishing of the integrals of the slip gradient over the patches. Mathematically, the presence of two additional conditions, as compared to earlier formulations, corresponds to two undetermined coefficients in the general solution of the governing integral equation. Numerical simulation confirms that at least one of these is always non-zero in the coupled situation. For a coupled double-patch case with a symmetric pre-slip Coulomb stress pattern, the general inverse solution requires three auxiliary conditions. Moreover the conditions for the asymmetric case may be reduced to a set of four again, but these are different from the sets of four obtained earlier by Muskhelishvili (1953) and Weertman (1996). We illustrate the theory with a numerical example in which the evaluation of the Cauchy integrals is performed with a modified version of augmented Gauss–Chebyshev quadrature that relies on analytical inversion.

A Comprehensive Fuzzy Model for Understanding Neuronal Calcium Distribution in Presence of VGCC, Na+/Ca2+ Exchanger, Buffer, and ER Fluxes.

Free Calcium ions in the cytosol are essential for many physiological and physical functions. The free calcium ions are commonly regarded as a second messenger, are an essential part of brain communication. Numerous physiological activities, such as calcium buffering and calcium ion channel flow, etc. influence the cytosolic calcium concentration. In light of the above, the primary goal of this study is to develop a model of calcium distribution in neuron cells when a Voltage-Gated Calcium Channel and Sodium Calcium Exchanger are present. As we know, decreased buffer levels and increased calcium activity in the Voltage-Gated Calcium Channel and Sodium Calcium Exchanger lead to Alzheimer's disease. Due to these changes, the calcium diffusion in that location becomes disrupted and impacted by Alzheimer's disease. The model has been constructed by considering key factors like buffers and ER fluxes when Voltage-Gated Calcium Channels and Sodium Calcium Exchangers are present. Based on the physiological conditions of the parameters, appropriate boundary conditions have been constructed in the fuzzy environment. This model is considered a fuzzy boundary value problem with the source term and initial boundary conditions are modeled by triangular fuzzy functions. In this, paper we observed the approximate solution of the mathematical model which was investigated by the fuzzy undetermined coefficient method. The solution has been performed through MATLAB and numerical results have been computed using simulation. The observation made that the proper operation of the Voltage-Gated Calcium Channel and Sodium Calcium Exchanger is critical for maintaining the delicate equilibrium of calcium ions, which regulates vital cellular activities. Dysregulation of Voltage-Gated Calcium Channel and Sodium Calcium Exchanger activity has been linked to neurodegenerative illnesses like Alzheimer's disease.

Series expansion of some elementary functions without using mathematical analysis

We present a method to compute the power series expansions of e x , ln ⁡ ( 1 + x ) , sin ⁡ x , and cos ⁡ x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of mathematical induction. The Newton binomial formula is obtained using the expansions of the exponent and the logarithm in their power series. We build upon the approach outlined in a paper by L. P. Mironenko and O. A. Rubtsova (Approximations of Some Functions by Polynomials and the Method of Undefined Coefficients, Scientific Papers of Donetsk National Technical University. Series: Computing and Automation, No. 2 (25), p. 128–135, 2013) and complement their approach. This approach does not depend on the application of Taylor's theorem to the expansion of functions. Consequently, these expansions can be integrated into the educational process before students become acquainted with the concept of derivatives. It simplifies the study of the theory of limits, especially in the computation of standard limits such as lim x → 0 sin ⁡ ( x ) x , lim x → 0 1 − cos ⁡ ( x ) x 2 , and lim x → 0 ( 1 + x ) 1 / x .

A failure criterion for genuinely orthotropic materials and integration of a series of criteria for materials of different degrees of anisotropy.

Existing failure criteria for orthotropic materials are subject to an underlying assumption which cause contradictions when applied to genuinely orthotropic materials that are significantly anisotropic in elasticity as well as in strengths. For such materials, there is lack of consistent failure criteria to support their applications in engineering structures. A general quadratic failure criterion tends to leave undetermined coefficients for interactive terms. A rational approach is adopted in this paper based on mathematical and logical considerations to determine these coefficients as the objective of this paper. Considerations are based on the intrinsic characteristics of the quadric surfaces introduced by the quadratic failure criterion. These coefficients must take the values as obtained, leaving no alternatives if logic prevails. The obtained criterion integrates for the first time a range of criteria separately formulated for materials of different degrees of anisotropy, from genuinely orthotropic, through transversely isotropic, cubically symmetric, to completely isotropic ones with different or identical tensile and compression strengths.

Reliability control of nonlinear stochastic dynamical system based on discrete data

Stochastic optimal control based on reliability plays a crucial role in mitigating structural failure and ensuring safe operation. For general stochastic vibration systems, a data-driven method for reliability-based optimal control is proposed based on random state data. Firstly, the feedback control is split into conservative and dissipative components in coherence with physical intuition, and each component is expanded upon using pre-selected basic functions, resulting in a modified system with undetermined coefficients. Then, from discrete random samples, the expressions for the probability densities of the first-passage time and the stationary response can be identified, explicitly including system and excitation parameters, as well as initial and safety boundary conditions. Finally, a performance index, combining system reliability and control cost, is constructed and can be explicitly expressed in terms of the undetermined coefficients of the control forces. The optimal control problem for determining the optimal control forces is reformulated as an optimization problem for determining the coefficients that minimize the performance index. This unconstrained optimization problem can be easily solved. Three typical nonlinear stochastic systems, a controlled Duffing oscillator, a controlled nonlinear hysteretic system, and a controlled 2-DOF nonlinear dynamical system, are given to illustrate the validity and accuracy of the proposed data-driven method. Additionally, the control performance of the optimal control strategy in improving the system reliability is also discussed.

A Data–Physics-Driven Modeling Approach of Key Equipment for Large-Scale Distribution Network Simulation

Fueled by pressing global climate concerns, the integration of large-scale renewable distributed generation sources, including distributed wind power and photovoltaics, along with electricity substitution loads into the distribution network has been accelerated to diminish carbon emissions. This shift introduces significant challenges and necessitates the advanced operation and control of distribution systems to accommodate these changes effectively. Against this backdrop, there is a growing expectation for an open and scalable central control mode, equipped with compatible interfaces, to offer a visionary development platform for the grid. This platform is anticipated to meet the evolving needs of future distribution system development, ensuring adaptability and forward compatibility. The aforementioned platform requires open, scalable, and interface-compatible models of key distribution network equipment as its foundation. To address the challenges presented, this paper proposes a data–physics-driven modeling approach for automating simulations in distribution systems. This method employs a simplified and standardized system of linear differential equations with undetermined coefficients to capture the common physical characteristics of specific device types. The models designed through this approach are notably open, allowing for real-time data to refine undetermined coefficients and accurately depict the dynamic behavior of equipment over various periods. Their scalability also stands out, rendering them apt for large-scale distribution network simulations. The paper elaborates on models for distributed photovoltaic, wind turbine, energy storage, and electric vehicle, and demonstrates their application within an IEEE-33 node distribution network topology built on Python.

Revised inflow performance relationship for productivity prediction from marine sandy hydrate reservoirs in Nankai Trough

Productivity prediction is crucial for the safe and efficient exploitation of marine methane hydrate. As a feasible method, inflow performance relationship (IPR) can characterize productivity results and provide a theoretical basis for optimization of depressurization schemes. In this study, we propose two modified IPR models based on field data from limited trial production and numerical fitting in the Nankai Trough. Prediction results are obtained by depressurization, together with IPR curves under different recovery ratios. According to the conventional formulas, the reservoir pressure of the deliverability equation is revised, considering the dual contributions of hydrate dissociation and pressure drop. The comparison results show that the modified Vogel formula and Fetkovich formula can suitably describe the IPR for the single vertical well and horizontal well, respectively. Sensitivity analyses of various reservoir physical parameters indicate that changes in porosity and hydrate saturation cause diverse values for the undetermined coefficient A, primarily due to the alterations in the key coefficient pr′ in the revised Vogel formula for the vertical well scenario. However, absolute permeability has only a slight influence on the shape of the IPR curve. In contrast, absolute permeability plays a crucial role in affecting the Fetkovich-modified IPR curves for horizontal wells. Productions under varying porosities exhibit a consistent linear trend on the Fetkovich-modified IPR curve because of the comparable impact of gas production rate and pressure square variance. Besides, changing hydrate saturation can also have some effect on the IPR. Our research provides a feasible productivity prediction methodology for the safe and efficient extraction of similar hydrate reservoirs.

Optical solitons for the dispersive concatenation model with power law of self-phase modulation: undetermined coefficients

AbstractThis paper recovers 1-soliton solutions to the dispersive concatenation model that comes with power law of self-phase modulation. The method of undetermined coefficients has made this retrieval possible. The parameter constraints are listed for the existence of the solitons. While a full spectrum of solitons is enumerated and exhibited, it is proved that dark 1-solitons exist only for Kerr law of nonlinear refractive index change.

Bearing condition assessment of a simply elastic supported beam based on impact vibration testing

The mechanical properties of bridge bearings gradually deteriorate over time, resulting from daily traffic loading and harsh environmental conditions. However, efficient detection of in-service bridge bearings is still challenging, especially in quantifying support stiffness. This study presents a bridge bearing condition assessment framework of a simply elastic supported beam based on impact vibration testing and structural flexibility identification. Firstly, the dynamic response equation of a simply supported beam with elastic boundary constraints is established. The modal orthogonality of Euler-Bernoulli beams and the corresponding overall matrix of undetermined coefficients are derived using the separate variable method. Subsequently, the flexibility matrix of the simply elastic supported beam is derived and verified in the cases of forward and inverse analysis, respectively. Furthermore, the vertical and rotational stiffness of the bearings are established based on the structural flexibility matrix. Finally, experimental verification is performed to identify the flexibility and the vertical support stiffness of the simply elastic supported beam based on impact vibration tests. The results indicate that the proposed the bearing condition assessment framework can straightforwardly quantify support stiffness.

Effects of joule heating and viscous dissipation on electromagneto-hydrodynamic flow in a microchannel with electroosmotic effect: Enhancement of MEMS cooling

This paper inspects the effect of Joule heating and viscous dissipation due to electric double layer (EDL) and electroosmotic effect on steady fully developed electromagnetohydrodynamic flow in a microchannel. Dimensionless formulations of the Poisson-Boltzmann, momentum, and energy equations are derived for the electric potential, velocity profile, and temperature distribution in the microchannel. Exact solutions for the temperature distributions and velocity profile were obtained using the method of undetermined coefficients. The Debye-Hückel linearization is used to get exact solution for the electric potential. The results showed that Brinkmann number [Formula: see text], Joule heating parameter [Formula: see text], Debye-Hückel parameter [Formula: see text], Hartmann number [Formula: see text], and electric field [Formula: see text] have a substantial impact on flow formation and heat transfer. The complex interaction between joule heating, viscous dissipation, and the EOF effect were accurately captured. The range values for the governing parameter for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are [Formula: see text], and[Formula: see text] respectively.

Random wavenumber and nonlinear parametric effect of speckle field

Speckle field is a relatively common phenomenon. But the speckle has special application value in nonlinear optical domain because it can be used to suppress different nonlinear processes that are caused by high power laser. To enhance the suppression capability, it is necessary to reveal the basic mechanism of the speckle parameter nonlinear optical interaction process. In this work, the coupling wave equation under the wave number mismatch condition is used to analyze the parameter process of speckles field. The solving process of the coupling wave equation is introduced in detail. And the wave number or phase matching condition is fully discussed. Furthermore, the threshold of the nonlinear gain is analyzed when the wave number does not fully meet the matching condition. To describe the solution of the coupling wave equation more clearly, the undetermined coefficient of the exact analytical solution is discussed. Since the boundary field will affect the confirmation of the undetermined coefficient, the characteristic of boundary field should be analyzed first. The nonlinear process of the speckle field is a three-wave interaction process. The different boundary conditions will affect the three-wave interaction process. And it is found that if the complex amplitudes of the three waves at the boundary are not zero, the undetermined coefficient will be changed with the phrase parameters of the three waves. To achieve the maximum value, the boundary waves must meet the phase matching condition. The wave number of the speckle filed is not an invariant, because of its random distribution characteristic. Therefore, during the analysis of the three-wave interaction process, the segment handling method is used to ensure the effective solving of the first order coupling wave equation. On the other hand, the randomly fluctuation of the wave number destroys the phase matching condition of the boundary. It is just through the basic mechanism that the speckle field can be used to suppress the nonlinear gain of high-power optical field. Both the theoretical analyses and the numerical calculation results show that the speckle field has good suppression effect for some typical nonlinear parameter process, such as stimulated Brillouin scattering.

Symmetric Encryption Algorithms in a Polynomial Residue Number System

In this paper, we develop the theoretical provisions of symmetric cryptographic algorithms based on the polynomial residue number system for the first time. The main feature of the proposed approach is that when reconstructing the polynomial based on the method of undetermined coefficients, multiplication is performed not on the found base numbers but on arbitrarily selected polynomials. The latter, together with pairwise coprime residues of the residue class system, serve as the keys of the cryptographic algorithm. Schemes and examples of the implementation of the developed polynomial symmetric encryption algorithm are presented. The analytical expressions of the cryptographic strength estimation are constructed, and their graphical dependence on the number of modules and polynomial powers is presented. Our studies show that the cryptanalysis of the proposed algorithm requires combinatorial complexity, which leads to an NP‐complete problem.

Experimental verification and electromagnetic analysis of axial flux permanent magnet motor using the subdomain method

This study investigates a subdomain method for performing the characteristic analysis of an axial flux permanent magnet motor (AFPM). Initially, the AFPM was simplified for the analytical model and analysis regions were defined to apply the subdomain method. Subsequently, a general solution was derived for the governing equations of each analysis region using the variable separation method. Subsequently, the undetermined coefficients of the general solution were obtained from the matrix using appropriate boundary conditions, and the magnetic flux density in the air-gap region was derived based on the defined magnetic vector potential. Finally, the induced voltage and electromagnetic torque of the analytical model were calculated. The analytical results were verified using the finite element method and the experimental results.