An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Linear Functional Equations and Twisted Polynomials

A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.

Solution for Output Coordination Equations of Several Typical Parallel Six-Dimensional Acceleration Sensing Mechanisms

Aiming at the problem that it is difficult to generate the dynamic decoupling equation of the parallel six-dimensional acceleration sensing mechanism,two typical parallel six-dimensional acceleration sensing mechanisms are taken as examples.By analyzing the scale constraint relationship between the hinge points on the mass block and the hinge points on the base of the sensing mechanism,a new method for establishing the dynamic equation of the sensing mechanism is proposed.Firstly,based on the scale constraint relationship between the hinge points on the mass block and the hinge points on the base of the sensing mechanism,the expression of the branch rod length is obtained.The inherent constraint relationship between the branches is excavated and the branch coordination closed chain of the“12-6”configuration is constructed.The output coordination equation of the sensing mechanism is successfully derived.Secondly,the dynamic equations of“12-4”and“12-6”configurations are constructed by the Newton-Euler method,and the forward decoupling equations of the two configurations are solved by combining the dynamic equations and the output coordination equations.Finally,the virtual prototype experiment is carried out,and the maximum reference errors of the forward decoupling equations of the two configuration sensing mechanisms are 4.23%and 6.53%,respectively.The results show that the proposed method is effective and feasible,and meets the real-time requirements.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.

Chebyshev Polynomials for Solving a Class of Singular Integral Equations

This paper is devoted to studying the approximate solution of singular integral equations by means of Chebyshev polynomials. Some examples are presented to illustrate the method.

Numerical Solutions of Volterra Equations Using Galerkin Method with Certain Orthogonal Polynomials

This work is aim at providing a numerical technique for the Volterra integral equations using Galerkin method. For this purpose, an effective matrix formulation is proposed to solve linear Volterra integral equations of the first and second kind respectively using orthogonal polynomials as trial functions which are constructed in the interval [-1,1] with respect to the weight function w(x)=1+x2. The efficiency of the proposed method is tested on several numerical examples and compared with the analytic solutions available in the literature.

Reconstructed Elzaki Transform Method for Delay Differential Equations with Mamadu-Njoseh Polynomials

One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.

MODELLING OF MECHANICAL MECHANISM OF CHROMATOGRAPHIC SYSTEM AND THEORETICAL EQUATIONS SHOWING DYNAMICAL CHARACTERISTICS OF CHROMATOGRAPHY

A simple model of chromatographic mechanical mechanism is present, and then a scrics of theoretical chromatographic equations and fundamental Formulae are derived. These theoretical equations and formulae not only reserve thermodynamic characteristics in the current fundamental chromatographic formulae, but also introduce one or more kinetic parameter, so it is possible to make the macroscopic-control on the effect of kinetic characteristics on chromatographic system.

Applications of Non-Classical Equations and Their Approaches to the Solution of Some of Classes Equations Arise in the Kelvin-Helmholtz Mechanism and Instability

In the presented work, we consider applications of non-classical equations and their approaches to the solution of some classes of equations that arise in the Kelvin-Helmholtz Mechanism (KHM) and instability. In all areas where the Kelvin-Helmholtz instability (KHI) problem is investigated with the corresponding data unchanged, the solution can be taken directly in a specific form (for example, to determine the horizontal structure of a perturbation in a barotropic rotational flow, which is a boundary condition taken, as well as other types of Kelvin-Helmholtz instability problems). In another example, the shear flow along the magnetic field in the Z direction, which is the width of the contact layer between fast and slow flows, has a velocity gradient along the X axis with wind shear. The most difficult problems arise when the above unmentioned equation has singularities simultaneously at points and in this case, our results also remain valid. In the case of linear wave analysis of Kelvin-Helmholtz instability (KHI) at a tangential discontinuity (TD) of ideal magneto-hydro-dynamic (MHD) plasma, it can be attributed to the presented class, and in this case, as far as we know, solutions for eigen modes of instability KH in MHD plasma that satisfy suitable homogeneous boundary conditions. Based on the above mentioned area of application for degenerating ordinary differential equations in this work, the method of functional analysis in order to prove the generalized solution is used. The investigated equation covers a class of a number of difficult-to-solve problems, namely, generalized solutions are found for classes of problems that have analytical and mathematical descriptions. With the aid of lemmas and theorems, the existence and uniqueness of generalized solutions in the weight space are proved, and then general and particular exact solutions are found for the considered problems that are expressed analytically explicitly. Obtained our results may be used for all the difficult-to-solve processes of KHM and instabilities and instabilities, which cover widely studied areas like galaxies, Kelvin-Helmholtz instability in the atmospheres of planets, oceans, clouds and moons, for example, during the formation of the Earth or the Red Spot on Jupiter, as well as in the atmospheres of the Sun and other stars. In this paper, also, a fairly common class of equations and examples are indicated that can be used directly to enter data for the use of the studied suitable tasks.

Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode Reactions for CE Mechanism at a Spherical Electrode

A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential equation that describes the homogeneous processes coupled to electrode reaction. In this paper the approximate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These approximate results are compared with the available analytical results and are found to be in good agreement.

THE GROWTH OF SOLUTIONS TO HIGHER ORDER DIFFERENTIAL EQUATIONS WITH EXPONENTIAL POLYNOMIALS AS ITS COEFFICIENTS

By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

Application of Bernstein polynomials for solving Fredholm integro-differential-difference equations

In this paper,the Bernstein polynomials method is proposed for the numerical solution of Fredholm integro-differential-difference equation with variable coefficients and mixed conditions.This method is using a simple computational manner to obtain a quite acceptable approximate solution.The main characteristic behind this method lies in the fact that,on the one hand,the problem will be reduced to a system of algebraic equations.On the other hand,the efficiency and accuracy of the Bernstein polynomials method for solving these equations are high.The existence and uniqueness of the solution have been proved.Moreover,an estimation of the error bound for this method will be shown by preparing some theorems.Finally,some numerical experiments are presented to show the excellent behavior and high accuracy of this algorithm in comparison with some other well-known methods.

An Analysis of the Impact of Economic-Ecological Balance Mechanism Based on Non-Linear Partial Differential Equations on Land Financial Teaching Methods

This paper clarifies the relationship between the flow paths of the corresponding ecological flows because of the ecological impact for land consolidation, using external energy methods to measure the external input of the project area or the output of ecological products. The application for nonlinear estimation of partial differential equations to land consolidation, the project ecological flow and system efficiency were quantitatively calculated. It shows that the conflict between fairness and efficiency is caused under conditions and levels of value and ecological compensation mechanism is built as a criterion for this ecological economics. Based on the years of use of the land improvement project, the time evolution of regional net ecological value, natural resource dependence, renewable resource dependence, ecological output ratio, ecological carrying capacity and ecological sustainability after the implementation of the project was assessed.

ORTHOGONAL POLYNOMIALS AND DETERMINANT FORMULAS OF FUNCTION-VALUED PADE-TYPE APPROXIMATION USING FOR SOLUTION OF INTEGRAL EQUATIONS

To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.

On Polynomials Solutions of Quadratic Diophantine Equations

Let P:=P(t) be a polynomial in Z[X]\{0,1} In this paper, we consider the number of polynomial solutions of Diophantine equation E:X2–(P2–P)Y2–(4P2–2)X+(4P2–4P)Y=0. We also obtain some formulas and recurrence relations on the polynomial solution (Xn,Yn) of

Numerical solution of stochastic Ito^(^)-Volterra integral equations based on Bernstein multi-scaling polynomials

In this paper,first,Bernstein multi-scaling polynomials(BMSPs)and their properties are introduced.These polynomials are obtained by compressing Bernstein polynomials(BPs)under sub-intervals.Then,by using these polynomials,stochastic operational matrices of integration are generated.Moreover,by transforming the stochastic integral equation to a system of algebraic equations and solving this system using Newton’s method,the approximate solution of the stochastic Ito^(^)-Volterra integral equation is obtained.To illustrate the efficiency and accuracy of the proposed method,some examples are presented and the results are compared with other methods.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.