Method Of Functional Equations Research Articles

A heat conduction problem of 2D unbounded composites with imperfect contact conditions

We consider a steady-state heat conduction problem in 2D unbounded doubly periodic composite materials with temperature independent conductivities of their components. Imperfect contact conditions are assumed on the boundaries between the matrix and inclusions. By introducing complex potentials, the corresponding boundary value problem for the Laplace equation is transformed into a special R-linear boundary value problem for doubly periodic analytic functions. The method of functional equations is used for obtaining a solution. Thus, the R-linear boundary value problem is transformed into a system of functional equations which is analysed afterwards. A new improved algorithm for solving this system is proposed. It allows to compute the average property and reconstruct the temperature and the flux at an arbitrary point of the composite. Computational examples are presented.

Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials

We consider a boundary value problem (BVP) in unbounded 2D doubly periodic composite with circular inclusions having arbitrary constant conductivities. By introducing complex potentials, the BVP for the Laplace equation is transformed to a special -linear BVP for doubly periodic analytic functions. This problem is solved with use of the method of functional equations. The -linear BVP is transformed to a system of functional equations. A new improved algorithm for solution of the system is proposed. It allows one not only to compute the average property but to reconstruct the solution components (temperature and flux) at an arbitrary point of the composite. Several computational examples are discussed in details demonstrating high efficiency of the method. Indirect estimate of the algorithm accuracy has been also provided.

Hopf bifurcation in a partial dependent predator–prey system with multiple delays

In this paper, a partial dependent predator–prey system with multiple delays is investigated. By choosing τ1, τ2 and τ3 as bifurcating parameters, we show that Hopf bifurcations occur. In addition, by using theory of functional differential equation and Hassard's method, explicit algorithms for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are performed to support the analytical results, and the chaotic behaviors are observed.

To the Efficiency of a Green's Function Modification of the Method of Functional Equations

Analysis of the computational potential is provided for a modification of one of the numerical approaches to the classical boundary integral equations method. The originally proposed name of the approach is the method of functional equations, but in nowadays it is also referred to as the fundamental solutions method. Undesired after effects of this name flip are pointed out. The modification, implemented in this study, requires computer-friendly representations for some relevant Green's function and significantly enhances the resolving potential of the method. This work focuses on the exploration of the computational applicability of this modification. Chosen for that boundaryvalue problems and stated on regions of irregular configuration for second order elliptic equations with discontinuous coefficients.

Stability Analysis of Runge-Kutta Methods for Nonlinear Functional Differential and Functional Equations

This paper is concerned with the numerical stability of Runge-Kutta methods for a class of nonlinear functional differential and functional equations. The sufficient conditions for the stability and asymptotic stability of(k,l)-algebraically stable Runge-Kutta methods are derived. A numerical test is given to confirm the theoretical results.

Explicit and Implicit Difference Methods for Quasilinear First Order Partial Functional Differential Equations

Abstract. Initial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes of the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. Error estimates for both methods are presented. Interpolating operators corresponding to functional variables are constructed.

Implicit Difference Methods for Differential Functional Parabolic Equations with Dirichlet's Condition

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition are approximated in the paper by solutions of associated implicit difference functional equations. The functional dependence is of the Volterra type. Nonlinear estimates of the generalized Perron type for given functions are assumed. The convergence and stability results are proved with the use of discrete functional inequalities and the comparison technique. In particular, these theorems cover quasi-linear equations. However, such equations are also treated separately. The known results on similar difference methods can be obtained as particular cases of our simple result.

Stability and convergence of Runge-Kutta methods for nonlinear neutral functional differential equations

This paper concerns the stability and convergence of Runge-Kutta methods for nonlinear neutral functional differential equations (NFDEs). Based on the B-theory of Runge-Kutta methods for Volterra functional differential equations (VFDEs), we introduce the concepts of EB-stability and EB-convergence of Runge-Kutta methods for NFDEs and obtain EB-stability and EB-convergence results on variable stepsize Runge-Kutta methods for this class of equations. These results are new even for neutral delay differential equations and neutral delay integro-differential equations.

Waveform relaxation methods for fractional functional differential equations

Abstract In this paper, we use waveform relaxation method to solve fractional functional differential equations. Under suitable conditions imposed on the so-called splitting functions the convergence results of the waveform relaxation method are given. Delay dependent error estimates for the method are derived. Error bounds for some special cases are considered. Numerical examples illustrate the feasibility and efficiency of the method. It is the first time for applying the method in the fractional functional differential equations.

Conical Indentation of a Viscoelastic Sphere

Instrumented indentation is commonly used for determining mechanical properties of a range of materials, including viscoelastic materials. However, most—if not all—studies are limited to a flat substrate being indented by various shaped indenters (e.g., conical or spherical). This work investigates the possibility of extending instrumented indentation to nonflat viscoelastic substrates. In particular, conical indentation of a sphere is investigated where a semi-analytical approach based on “the method of functional equations” has been developed to obtain the force–displacement relationship. To verify the accuracy of the proposed methodology selected numerical experiments have been performed and good agreement was obtained. Since it takes significantly less time to obtain force–displacement relationships using the proposed method compared to conducting full finite element simulations, the proposed method is an efficient substitute of the finite element method in determining material properties of viscoelatic spherical particles using indentation testing.

Iterative methods for parabolic functional differential equations

This paper is concerned with iterative methods for parabolic functional differential equations with initial boundary conditions. Monotone iterative methods are discussed. We prove a theorem on the existence of solutions for a parabolic problem whose right

Generalized quadratic revenue functions

In this paper we focus on the specification of revenue functions in their dual price space. We consider two distance functions–the Shephard output distance function and the directional output distance function–and define both in price space. The former is multiplicative in nature and satisfies homogeneity, whereas the latter is additive and satisfies the translation property. Functional equation methods yield the translog specification in the case of the Shephard distance function and a quadratic specification in the case of the directional distance function. Monte Carlo evidence suggests that the quadratic specification outperforms the translog in large samples and in true models with plenty of curvature.

On a Notion of Similarity with Endowments in Public Economics

We establish that a novel notion of similarity with endowments by Shitovitz and Spiegel (1998) amounts to severe functional form restrictions, and these restrictions on utility functions can be determined by functional equation methods. The resulting functional forms are related to the familiar Cobb-Douglas and quasi-linear forms, as well as the classical result of Bergstrom and Cornes (1983) on the independence of allocative efficiency from distribution in the theory of public goods.

Implicit difference schemes for quasilinear parabolic functional equations

AbstractWe present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

Numerical Method of Hybrid Stochastic Functional Differential Equations with the Local Lipschitz Coefficients

Recently, hybrid stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical schemes established for the hybrid stochastic functional differential equations. In this paper, the Euler—Maruyama method is developed, and the main aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition. The result obtained generalizes the earlier results.

Embedded Waveform Relaxation Methods for Parabolic Partial Functional Differential Equations

Waveform relaxation methods are decoupling or splitting methods for large scale ordinary differential equations. In this paper, we apply the meth- ods directly to semi-linear parabolic partial functional differential equations. Taking into consideration of the complicated forms of these parabolic equa- tions, we propose a kind of embedded waveform relaxation methods, which are in fact two-level waveform relaxation methods and which can also be ap- plied to some other systems. We provide explicit iterative expressions of the embedded methods and exhibit the superlinear rate of convergence on finite time intervals. We also apply the two-level idea to the functional differential equations derived from semi-discretization of the original system. The win- dowing technique is employed for the situation of long time intervals. Finally, two numerical experiments are performed to confirm our theory.

HOMOMORPHISMS IN PROPER LIE CQ*-ALGEBRAS

Using the Hyers-Ulam-Rassias stability method of functional equations, we investigate homomorphisms in proper <TEX>$CQ^*$</TEX>-algebras and proper Lie <TEX>$CQ^*$</TEX>-algebras, and derivations on proper <TEX>$CQ^*$</TEX>-algebras and proper Lie <TEX>$CQ^*$</TEX>-algebras associated with the following functional equation <TEX>$$\frac{1}{k}f(kx+ky+kz)=f(x)+f(y)+f(z)$$</TEX> for a fixed positive integer <TEX>$k$</TEX>.

Comparison of explicit and implicit difference methods for quasilinear functional differential equations

Comparison of explicit and implicit difference methods for quasilinear functional differential equations

Associative Binary Operations and the Pythagorean Theorem

I n a recent article [2], L. Berrone presented a new approach to the Pythagorean Theorem (PT). The idea is to derive the geometric theorem from analytic and algebraic properties, by methods of functional equations. (So we are not dealing with a method that was an option for the ancients!) I thought about Berrone’s ideas, within his context of functional equations. Some pleasant surprises fell out. Then a surprising gift of functional equations back to geometry closed the circle for me – and will close this article. Let a b denote the hypotenuse of a right triangle with legs a and b. The operation : ð0;1Þ 7!ð0;1Þ is evidently continuous. It is also (a) Homogeneous (of degree one):

The kernel method and systems of functional equations with several conditions

We generalize the kernel method to equation systems in which the number of unknowns is allowed to exceed the number of equations. With this generalization, we derive the generating functions for several kinds of sequences and generating trees whose recursions are dependent on the parity of the indices.