MATHEMATICAL SCIENCES
Year:2012 | Volume:6 | Issue:-|Papers Count:22

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation:11f (x+2y)+11f (x−2y)=44f(x+y)+44f (x−y)+12f (3y)−48f (2y)+60f (y)−66f(x)in fuzzy Banach spaces.

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In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A0⊂A1⊂…⊂At−1⊂At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0⊂K1⊂…⊂Kt−1⊂Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A∗i and K∗i are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A∗i and K∗i for each i, where 0≤i≤t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

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This article deals with the general oblique derivative boundary value problem for nonlinear elliptic equations of second order in an unbounded multiply connected domain. The problem includes the Dirichlet problem, the Neumann problem and the third boundary value problem as its spacial cases. We first provide the formulation of the above boundary value problem and obtain the representation theorem for the problem. Then, we give a priori estimates of solutions for the boundary value problem by using the reduction to absurdity and the uniqueness of solutions. Finally, by the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the above problem for the nonlinear elliptic equations of second order can be proved.

Purpose: Our aim in this study is to generate some partial differential equations (PDEs) with variable coefficients by using the PDEs with non-constant coefficients.Methods: Then by applying the single and double convolution products, we produce some new equations having polynomials coefficients. We then classify the new equations on using the classification method for the second order linear partial differential equations.Results: Classification is invariant under single and double convolutions by applying some conditions, that is, we identify some conditions where a hyperbolic equation will be hyperbolic again after single and double convolutions.Conclusions: It is shown that the classifications of the new PDEs are related to the coefficients of polynomials which are considered during the process of convolution product.

A labeling of the edges of a graph is called vertex-coloring if the labeled degrees of the vertices yield a proper coloring of the graph. In this paper, we show that such a labeling is possible from the label set 1, 2, 3 for the complete graph Kn, n≥3.

In this paper, we report the application of two parameter coupled alternating group explicit (CAGE) iteration and Newton-CAGE iteration methods for the cubic spline solution of non-linear differential equation u" = f (r, u, u') subject to given natural boundary conditions. The error analysis for CAGE iteration method is discussed in details. We compared the results of proposed CAGE iteration method with the results of corresponding two parameter alternating group explicit (TAGE) iteration method to demonstrate computationally the efficiency of the proposed method.

In this paper, we describe the geometry of distributions by their symmetries and present a simplified proof of the Frobenius theorem and some related corollaries. Then, we study the geometry of solutions of the F-Gordon equation, a PDE which appears in differential geometry and relativistic field theory.

In this article we introduce some I-convergent sequence spaces of fuzzy real numbers defined by Orlicz function and study their different properties such as solidity and symmetricity. The notion of I-convergence generalizes the notion of some particular type of convergence of sequences.

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Purpose: In this paper, we construct the operational matrix of fractional integration of arbitrary order for Laguerre polynomials.Methods: We introduce some necessary definitions and give some relevant properties of Laguerre polynomials. The fractional integration is described in the Riemann-Liouville sense. We develop a direct solution technique for solving the integrated forms of fractional differential equations (FDEs) on the half line using the Laguerre tau method based on operational matrix of fractional integration in the Riemann-Liouville sense.Results: In order to show the fundamental importance of the Laguerre operational matrix, we apply it together with the spectral Laguerre tau method for the numerical solution of general linear multi-term FDEs on a semi-infinite interval.Conclusions: The results obtained by the present methods reveal that the present method is very effective and convenient for linear FDEs. Illustrative examples are included to demonstrate the validity and applicability of the new technique for linear muti-term FDEs on a semi-infinite interval.

Brianciari (‘A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces,’ Publ. Math. Debrecen 57 (2000) 31-37) initiated the notion of the generalized metric space as a generalization of a metric space in such a way that the triangle inequality is replaced by the ‘quadrilateral inequality,’ d (x, y) ≤d (x, a)+d (a, b)+d (b, y) for all pairwise distinct points x, y, a, and b of X. In this paper, we establish a fixed point result for weak contractive mappings T: X®X in complete Hausdorff generalized metric spaces. The obtained result is an extension and a generalization of many existing results in the literature.

It is more appropriate that many industrial products be evaluated and qualified by an imprecise (fuzzy) quality. By this idea the products could be evaluated using two membership functions for specification limits rather than two real numbers used in classical quality control. This idea leads the researchers to be able to deal with the vague process capability indices modeled as triangular fuzzy numbers. In this paper, we discuss on such fuzzy qualities and review some fuzzy process capability indices. Then we will bring them up to analyze several educational systems, such as comparing capability indices of two or more teachers, schools, and so on. The idea of this paper could be applied in other similar evaluation schemes as well.

In this paper, we introduce the idea of constructing sequence spaces with elements in an n-norm space in comparison with the spaces c0, c, L¥ and the Orlicz space LM and extend the notion of n-norm to such spaces. Further we state and define some statements about the n-best approximation in n-normed spaces.

Purpose: In the present paper, we describe a survey of B-spline techniques which have been used for numerical solutions of mathematical problems recently.Methods: Here, we discussed the definition of B-splines of various degrees by two different approaches to generate the recurrence relation to drive the formulation of B-splines.Results: Cubic B-spline applied on two test equations and absolute errors in interpolation are compared with cubic and quintic splines. Some remarks have been included.Conclusions: Numerical results are tabulated in tables; these tables show that the results obtained by cubic B-spline are considerable and accurate with respect to the cubic spline and more or less similar to the quintic spline.

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For a sequence of Hilbert spaces and continuous linear operators, the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra.Purpose: We wish to study those sequences for which the curvature is ‘small’ at each step, e.g., belongs to a fixed operator ideal.Methods: Our methods are based on combining homological algebra with the theory of Fredholm operators in Hilbert spaces.Results: We elaborate the theory of Fredholm sequences and show that any Fredholm sequence of trace class curvature can be reduced to a Fredholm complex. This allows one to introduce the Lefschetz number for cochain self-mappings of Fredholm sequences of ‘small’ curvature.Conclusion: Our results raise fixed point theory for Fredholm complexes of trace class curvature.

Purpose: This paper studies the F-four equation that appears in relativistic quantum mechanics. The primary purpose of this paper is to address the numerical simulations of this model.Methods: The singular soliton solution is also obtained by the ansatz method.Results: The constraint conditions are indicated for the existence of the soliton. Numerical solutions are obtained by using the spectral method where rational Chebyshev functions are used as basis functions.Conclusions: Results are validated by finding error estimates.

Purpose: In this paper, we will consider the interpolation of fuzzy data by using the fuzzy-valued piecewise quartic polynomials Qy0, y1, ..., yn (x) induced from E (3) cubic spline functions.Method: It has been many years since researchers have attended to the problem of interpolation of fuzzy data. Here, for Lagrange interpolation of fuzzy data, we will use the piecewise quartic polynomial induced from E (3) cubic spline functions to interpolate the fuzzy data. To do this, we will apply the extension principle to construct the membership function of Qy0, y1, ..., yn (x). Results: By using piecewise quartic polynomials, a new set of fuzzy spline functions was defined to interpolate given fuzzy data.Conclusions: In our previous study, we used E (3) cubic spline to construct E (3) fuzzy cubic spline. In this article, we added one extra term to this spline to compute the piecewise quartic polynomials and hence the fuzzy-valued piecewise quartic polynomials.

When we discretize nonlinear Volterra integral equations using some numerical, such as collocation methods, the arising algebraic systems are nonlinear. Applying quasilinear technique to the nonlinear Volterra integral equations gives raise to linear Volterra integral equations. The solutions of these equations yield a functional sequence quadratically convergent to the solution. Then, we use collocation method based on Chebyshev polynomials and a modified Clenshaw-Curtis quadrature and obtain a numerical solution. Error analysis has been performed, and the method has been applied on three numerical examples.

In this paper some results about the Hyers-Ulam-Rassias stability for the linear functional equations in general form and its Pexiderized can be proved for given functions on general domain to a complex Banach spaces under some suitable conditions. In connection with the problem of G. L. Forti in the 13st ICFEI we consider the common stability for the systems of functional equations and our aim is to establish some common Hyers-Ulam-Rassias stability for systems of homogeneous linear functional equations. The results is applied to the study of some superstability results for the exponential functional equation.