Method Of Undetermined Coefficients Research Articles

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.

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.

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 .

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.

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.

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.

Gözenekli malzeme ile doldurulmuş yatay mikrokanaldan akışkan akışında kesirli zaman türevi

Fractional time derivative is considered in the description of the unsteady fluid flow through a horizontal microchannel filled with porous material. The resultant governing equations were solved using the Laplace transform technique and the method of undetermined coefficient in the Laplace domain. The Riemann-sum approximation approach was then utilized to obtain the solution in the time domain. The results were then simulated and presented as line graphs utilizing MATLAB (R2015b) to study the effects of the parameters involved in the fluid flow.

Effect of fractional time derivatives to pressure-driven flow through the horizontal microchannel

This research applies fractional time derivatives to fluid flow through a horizontal microchannel. It uses fractional time derivatives with the Laplace transform technique and method of undetermined coefficient to analyze and obtain solutions of the governing equations in the Laplace domain. To this end, the solutions are reversed in the time domain using Riemann-sum approximation methods. In order to obtain the solutions for the pressure-driven flow, the time factional derivative in the Caputo sense is employed. Here, the influence of each governing parameter is explained with a line graph. Results show that with the decreases in fractional order $(\alpha)$, the velocity decreases within the interval $0

Analysis of Joule heating and viscous dissipation on electromagnetohydrodynamic flow with electroosmotic effect in a porous microchannel: A heat transfer miniature enhancement

AbstractThis work investigates 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 porous 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 profiles were obtained using the method of undetermined coefficients. The Debye–Hückel linearization is used to get an exact solution for the electric potential. The results showed that the Brinkmann number , Joule heating parameter , Debye–Hückel parameter , Hartmann number , electric field , and suction/injection parameter have a substantial impact on flow formation and heat transfer. Using MATLAB software, graphical simulations are provided to deliver a greater understanding of the influence of relevant parameters on the results achieved.

Optical Solitons for the Dispersive Concatenation Model: Undetermined Coefficients

This paper recovers optical solitons to the dispersive concatenation model that is studied with Kerr law of self-phase modulation. The method of undetermined coefficients is the adopted integration algorithm, enabling this retrieval possible. A full spectrum of optical solitons is recovered. The parameter constraints for the existence of the solitons, that naturally emerge during the course of their derivation, are also presented. The practical applications of this research include advancements in optical communication, nonlinear optics, and optical signal processing, as well as the potential for optimizing optical soliton-based technologies. In our current work, we have achieved the following novel findings: optical soliton recovery, integration algorithm innovation, and parameter constraints.

A general class of linear unconditionally energy stable schemes for the gradient flows, II

This paper continues to study linear and unconditionally modified-energy stable (abbreviated as SAV-GL) schemes for the gradient flows. The schemes are built on the SAV technique and the general linear time discretizations (GLTD) as well as the extrapolation for the nonlinear term. Different from [44], the GLTDs with three parameters discussed here are not necessarily algebraically stable. Some algebraic identities are derived by using the method of undetermined coefficients and further used to establish the modified-energy inequalities for the unconditional modified-energy stability of the semi-discrete-in-time SAV-GL schemes. It is worth emphasizing that those algebraic identities or energy inequalities are not necessarily unique for some choices of three parameters in the GLTDs. Numerical experiments on the Allen-Cahn, the Cahn-Hilliard and the phase field crystal models with the periodic boundary conditions are conducted to validate the unconditional modified-energy stability of the SAV-GL schemes, where the Fourier pseudo-spectral method is employed in space with the zero-padding to eliminate the aliasing error and the time stepsizes for ensuring the original-energy monotonous decay are estimated by using the stability regions of our SAV-GL schemes for the test equation. The resulting time stepsize constraints for the SAV-GL schemes are almost consistent with the numerical results on the above gradient flow models.

ROLE OF SUCTION/INJECTION ON MIXED CONVECTION FLOW IN A VERTICAL MICROCHANNEL FILLED WITH POROUS MATERIAL

A suction/injection controlled mixed convection flow of an incompressible and viscous fluid in a vertical microchannel filled with porous material with asymmetric plate temperature is presented. The governing equations are derived and solved using the method of undetermined coefficient. The closed form expression for temperature field, velocity field, skin friction and Nusselt number for the steady fully developed flow are obtained analytically. The effects of the governing parameters on the microchannel hydrodynamic and thermal behaviors are determined. It is interesting to point out that growing the Darcy number as well as the suction S accelerate the fluid motion while they reduce the critical values of the ( at which flow reversal sets in ) on both plates. The stability of the flow is affected by variation in the permeability of the porous material.

MODELS OF FIRE EXTINGUISHING WHEN FLAMMABLE LIQUID COMBUSTION

The process of extinguishing a class B fire with sprayed water is described by the differential equation of heat conduction with boundary conditions of the second kind. The solution of this differential equation in a dimensionless form allows to make a transition to the operative form of representation of the mathematical model of the quenching process - to the transfer function. The peculiarity of such a fire transfer function when extinguishing it with sprayed water is its irrationality, which causes difficulties in its use. One of the ways out of this situation is to switch to an equivalent representation of an irrational function of a complex variable in the form of a small-rational function of such a variable. Such a transition is carried out with the help of the minimax approximation using the Remez algorithm. Determination of the maximum order of the characteristic polynomial of the fire transfer function is carried out using the Hurwitz stability criterion. It is shown that for the transfer function of class B fire, the order of the Hurwitz polynomials does not exceed four, the approximation error does not exceed 3.6% using the method of undetermined coefficients, the transfer functions of the fire when they are extinguished with sprayed water are constructed in the form of a superposition of the transfer functions of aperiodic links. Numerical values of the parameters of such transfer functions were obtained. For the case when the intensity of the sprayed water supply is described by the Heaviside function, expressions describing the temperature of the surface of the burning liquid were constructed. It is noted that these expressions correspond to the expressions that describe the dynamic properties of class B fire in the time domain. It is shown that the maximum value of the relative error when using such transfer functions does not exceed 4.0%, if the general transfer function represents the superposition of the transfer functions of two aperiodic links. It is noted that when using the Padé approximation, the transfer function in the form of the transfer functions of eight aperiodic links corresponds to this value of the divergence error. Keywords: flammable liquid, fire extinguishing, sprayed water, transfer function.

Undetermined Coefficients Method for Sequential Fractional Differential Equations

The undetermined coefficients method is presented for nonhomogeneous sequential fractional differential equations involving Caputo fractional derivative of order n\alpha where n-1 n\alpha\le\ n and n\in\mathbb{N}. By employing proposed method, a particular solution of the considered equation is obtained. Some details about estimating the particular solution required to apply this method are explained. This method is shown to be particularly effective for nonhomogeneous fractional differential equations when the fractional differential equations involve some specific right-hand side functions.

Application of Undetermined Coefficients Method in Solving Fractional Integral Problems

Application of Undetermined Coefficients Method in Solving Fractional Integral Problems

Optical Solitons for the Concatenation Model with Differential Group Delay: Undetermined Coefficients

In the current study, the concatenation model of birefringent fibers is explored for the first time, and we present optical soliton solutions to the model. The integration algorithm used to achieve this retrieval is the method of undetermined coefficients, which yields a wide range of soliton solutions. The parameter constraints arise naturally during the derivation of the soliton solutions, which are essential for such solitons to exist.

Singular expansions and collocation methods for generalized Abel integral equations

In this paper, the generalized Abel integral equation with singular solution is studied. First, the finite-term psi-series expansion of the solution about the origin is derived using the method of undetermined coefficients, which gives the complete singular information of the solution. Second, this singular expansion is used to separate the singularity such that the Abel integral equation is converted into a perturbed one with smooth solution on a regular interval. Third, a product trapezoidal method is designed to solve the transformed equation on the regular interval and the convergence analysis is conducted to show that the scheme has second order accuracy combining with a controllable perturbation term. Fourth, a Chebyshev collocation method is further constructed on the regular interval to show that the global orthogonal polynomial interpolation is also suitable for solving the transformed Abel integral equation with high accuracy. Finally, two numerical examples confirm the correctness of the truncated psi-series solution and the effectiveness of the piecewise and global collocation methods with singularity separation for solving this kind of Abel integral equation.