Method Of Functional Equations Research Articles

The viscoelastic behavior of lignin: Quantification through nanoindentation relaxation testing on hot-pressed technical lignin samples from various origins

Lignin, the second most abundant organic polymer on earth, is one of the primary causes of the viscoelastic behavior of plants. An accurate characterization of its viscoelastic properties is essential for predicting the time-dependent response of natural materials, including wood and plant fibers, and for advancing lignin-based materials and their production methods, such as 3D printing of biocomposites. To enrich the still rather sparse knowledge on the viscoelasticity of lignin, we re-evaluate nanoindentation relaxation tests performed on five hot-pressed technical lignins extracted from different feedstocks, using three different extraction methods. The viscoelastic indentation problem is addressed using the method of functional equations combined with the homogenization theory to account for the production-induced porosity. This evaluation procedure allows for quantitatively assessing the viscoelastic properties of lignin, which can be very accurately described by an isochoric four-parameter Burgers model. Remarkably, the viscoelastic properties of all tested lignins are practically identical and independent of the feedstock and the extraction processes.

NEW ANALYSIS METHODS FOR THE COUPLED FRACTIONAL NONLINEAR HIROTA EQUATION

In this work, the coupled fractional nonlinear Hirota equation is defined by using a powerful fractional derivative sense, which is M-truncate derivative. We explore the fractional functional method and fractional simple equation method to investigate the structure of the solutions of the coupled fractional nonlinear Hirota equations, and some new periodic solutions and solitary wave solutions are successfully acquired. The two proposed approaches are simple, effective and direct. Moreover, some 3D and 2D graphs are sketched to elaborate the behavior of these solutions. These obtained solitary wave and periodic solutions are helpful to improve the understanding of the physical behavior of the corresponding mathematical model.

On parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus

<abstract><p>This paper aims to give generating functions for the new family of polynomials, which are called parametric types of the Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials by using Golden calculus. Numerous properties of these polynomials with their generating functions are investigated. These generating functions give us a generalization of some well-known generating functions for special polynomials such as Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials. Using the Golden differential operator technique, the functional equation method for generating function, we present some properties of these newly established polynomials.</p></abstract>

Stability analysis of implicit–explicit [formula omitted]-methods for composite stiff neutral functional differential equations in Banach space

This paper is concerned with the stability and asymptotic stability of implicit–explicit (IMEX) θ-methods for composite stiff neutral functional differential equations in Banach space. The stability and asymptotic stability conditions of the methods are obtained. The given numerical experiments confirmed the accuracy of the obtained theoretical results and showed that the IMEX θ-methods have a higher computational efficiency compared to the θ-methods in the end.

Stage-based interpolation Runge–Kutta methods for nonlinear Volterra functional differential equations

Stage-based interpolation Runge–Kutta methods for nonlinear Volterra functional differential equations

Application of numerical method of functional differential equations in fair value of financial accounting

Abstract In order to improve the information quality of financial accounting reports, this paper puts forward a functional differential equation, establishes the determination model of fair value by mathematical analysis method and studies the correlation of financial accounting information of fair value through this model. The results show that according to the fair value determination model, in the free accounting environment, fair value financial accounting information has an impact.

Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting

Abstract Financial accounting, the use of historical cost of assets, is an important basic principle of historical cost, which is to become the dominant mode of accounting measurement. Background analyses, as well as the historical cost basis and fair value, result from the development of the theory of historical cost and fair value. Historical cost and fair value measurement model has its own advantages and problems. Based on this background, the paper applies B-theoretical numerical methods to differential equations pan function analysis for calculation of fair value accounting and conducts theoretical analysis of their stability and convergence. Finally, numerical examples with different methods of calculating an approximate solution are provided and a comparison of the various methods is done based on the results obtained. The results show fair value accounting better meets the needs of the target –decision-making availability, compared to historical cost or fair value, more in line with the requirements of Accounting Information Quality.

Stability analysis of theθ-method for hybrid neutral stochastic functional differential equations with jumps

• Few results on the stability of numerical methods for the neutral stochastic functional differential equations with Markovian switching and jumps, results obtained in this manuscript can enrich this area. • Illustrate the almost sure exponential stability and the mean-square exponential stability of the trivial solution. • Show the θ -method can preserve the almost sure exponential stability and the mean-square exponential stability of the trivial solution under some appropriate condition. • The almost sure exponential stability of the θ -method is obtained by the discrete semi-martingale convergence theorem without resorting to using the Borel-Cantelli lemma and the Chebyshev inequality. Few results seems to be known about the stability of numerical methods for the hybrid neutral stochastic functional differential equations with jumps (also known as the neutral stochastic functional differential equations with Markovian switching and jumps (NSFDEwMJs)). This paper mainly investigates the exponential stability (both the almost sure and the mean-square exponential stability) of the θ -method for NSFDEwMJs. Precisely, it is first illustrated that the trivial solution of the NSFDEwMJ is almost surely and mean-square exponentially stable. It is then shown that the θ -method can preserve the same conclusions of the trivial solution. Numerical examples are demonstrated to illustrate the obtained results.

Time-delay effect on a diffusive predator\u2013prey model with habitat complexity

Based on the predator–prey system with a Holling type functional response function, a diffusive predator–prey system with digest delay and habitat complexity is proposed. Firstly, the stability of the equilibrium of diffusion system without delay is studied. Secondly, under the Neumann boundary conditions, taking time delay as the bifurcation parameter, by analyzing the eigenvalues of linearized operator of the system and using the normal form theory and center manifold method of partial functional differential equations, the effect of time delay on the stability of the system is studied and the conditions under which Hopf bifurcation occurs are given. In addition, the calculation formulas of the bifurcation direction and the stability of bifurcating periodic solutions are derived. Finally, the accuracy of theoretical analysis results is verified by numerical simulations and the biological explanation is given for the analysis results.

Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

Dissipativity of variable-stepsize Runge-Kutta methods for nonlinear functional differential equations with application to Nicholson’s blowflies models

This paper deals with dissipativity of the variable-stepsize Runge-Kutta methods applied to nonlinear Volterra functional differential equations in Hilbert spaces. The conditional dissipativity of variable-stepsize Runge-Kutta methods is analyzed and hence some new numerical dissipative criteria are derived. The resulting dissipative criteria extend and improve the existing results. Especially, for the algebraically stable variable-stepsize Runge-Kutta methods, we obtain a sharper dissipative result. In the end, we apply some concrete variable-stepsize Runge-Kutta methods to the three classes of nonlinear Nicholson’s blowflies models. The presented numerical examples further illustrate the theoretical results.

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

The aim of this paper is to construct generating functions for a new family of polynomials, which are called parametric Hermite‐based Milne–Thomson type polynomials. Many properties of these polynomials with their generating functions are investigated. These generating functions give us generalization of some well‐known generating functions for special polynomials such as Hermite type polynomials, Milne–Thomson type polynomials, and Apostol type polynomials. Using the Euler formula, functional equation method for generating function, and differential operator technique, we give relations among parametric Hermite‐based Milne–Thomson type polynomials, the Bernoulli numbers, the Euler numbers, the Chebyshev polynomials, the Bernstein basis functions, homogeneous harmonic polynomials, and parametric kinds of Apostol type polynomials. Moreover, some computational formulas for these polynomials are derived. Finally, using Wolfram Mathematica version 12.0, some of these polynomials and their generating functions are illustrated by their plots under the special conditions. Potential relationships and connections of this paper's results with the results of previous and future research are pointed out.

Families of Commuting Formal Power Series and Formal Functional Equations

Abstract In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F (x) = σx + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation.

Application of B-theory for Numerical Method of Functional Differential Equations in the Analysis of Fair Value in Financial Accounting

Application of B-theory for Numerical Method of Functional Differential Equations in the Analysis of Fair Value in Financial Accounting

A survey on new methods for partial functional differential equations and applications

This work is a survey of many papers dealing with new methods to study partial functional differential equations. We propose a new reduction method of the complexity of partial functional differential equations and its applications. Since, any partial functional differential equation is well-posed in a infinite dimensional space, this presents many difficulties to study the qualitative analysis of the solutions. Here, we propose to reduce the dimension from infinite to finite. We suppose that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition. The delayed part is continuous. We prove the dynamic of solutions are obtained through an ordinary differential equations that is well-posed in a finite dimensional space. The powerty of this results is used to show the existence of almost automorphic solutions for partial functional differential equations. For illustration, we provide an application to the Lotka–Volterra model with diffusion and delay.

Effective conductivity of a random suspension of highly conducting spherical particles

• A constructive approach for random 3D composites is developed. • The question of conditionally divergent series for random composites is resolved. • The famous Jeffreys formula for the effective conductivity of suspensions is revised and extended. • A theory of 3D Eisenstein functions is developed. Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions f . Jeffrey (1973) suggested an analytical formula valid up to O ( f 3 ) for macroscopically isotropic random composites. A conditionally convergent sum arose in the spatial averaging . In the present paper, we apply a method of functional equations to random composites and correct Jeffrey’s formula. The main revision concerns the proper investigation of the conditionally convergent sum and correction the f 2 -term. An algorithm for symbolic computations of the effective conductivity tensor is developed and performed up to O ( f 10 3 ) . The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the f 2 -term, and in the f 3 -term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.

Bifurcation analysis in a nonlinear electro-optical oscillator with delayed bandpass feedback

The dynamics of a nonlinear electro-optical oscillator with delayed bandpass feedback is investigated. By analyzing the distribution of the eigenvalues, the existence and stability of Hopf bifurcations are obtained. Particularly, the stability switches are found as the delay varies, where the time delay is regarded as bifurcation parameter. And then, by applying the normal form method and center manifold theory of functional differential equations, an algorithm for determining the sense of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived. For illustrating the theoretical results, some numerical simulations are performed.

Semianalytical solution of the random-product problem of matrices and discrete-time random evolution

We consider the product of a large number of matrices chosen randomly (with some correlation) as $A$ or $B$. We introduce an exact functional equation, whose solution gives the limiting distribution and the Lyapunov exponent for the product of large number of matrices. Thus we suggest new semianalytical method to solve the random product problem of correlated matrices. The method can be easily generalized to high order matrices, as we deal with a larger systems of functional equations. We have provided a known numerical algorithm for the solution of the matrix product problem. We have also applied our method for a solution of discrete time random evolution. Our functional equation method provides exact transition point.

Unsteady two-dimensional flows of a nanofluid over a vertical plate with time-exponential temperature

A two dimensional natural convective flow of nanofluid over a vertical plate in the presence of time-exponential temperature has been studied analytically. We have incorporated the effects of thermophoresis and Brownian motion in the considered nanofluid model. Moreover on the boundary of plate, it is assumed that the volumetric concentration of nanoparticles is passively controlled. The non-linear coupled governing equations have been solved with the help of the modified simple function equation method. Exact expressions for the concentration, temperature and velocity distribution profile have been established. Numerical computations are carried out in order to see the influence of different physical parameters like Lewis number, Pradtl number, buoyancy ratio parameter, thermophoresis parameter and Brownian motion parameter on the volumetric concentration of nanoparticles, temperature and velocity distribution profile and results are illustrated graphically and with tables. It is found that for decreasing values of the Brownian diffusion coefficient, therefore for increasing values of the Lewis number, the values of temperature and of particles concentration decrease. The temperature decreases with Prandtl number; while the nanoparticles concentration increases with Prandtl number (the thermal diffusivity has different influence on temperature and concentration).

Green’s functions for problems that simulate potential fields in thin-walled structures of irregular configuration

Matrices of Green’s type are targeted, for the first time, for a specific class of boundary–contact value problems which arise in studying potential fields induced in thin plate and shell assemblies of irregular configuration. A specific semi-analytic approach is proposed for evaluation of such matrices. It is based on our modification of the classical Kupradze’s method of functional equations. The approach appears workable for a broad range of the indicated problem settings. Representing the key component in the approach, matrices of Green’s type are analytically constructed to a number of boundary–contact value problems stated for relevant compound regularly configurated assemblies. This is accomplished prior to the actual computer work maintaining a solid background for fast and accurate solution of targeted problems.