Ural Mathematical Journal (UMJ)
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A NEW ROOT–FINDING ALGORITHM USING EXPONENTIAL SERIES
Get PDFIn this paper, we present a new root-finding algorithm to compute a non-zero real root of the transcendental equations using exponential series. Indeed, the new proposed algorithm is based on the exponential series and in which Secant method is special case. The proposed algorithm produces better approximate root than bisection method, regula-falsi method, Newton-Raphson method and secant method. The implementation of the proposed algorithm in Matlab and Maple also presented. Certain numerical examples are presented to validate the efficiency of the proposed algorithm. This algorithm will help to implement in the commercial package for finding a real root of a given transcendental equation
RESTRAINED DOUBLE MONOPHONIC NUMBER OF A GRAPH
Get PDFFor a connected graph of order at least two, a double monophonic set of a graph is a restrained double monophonic set if either or the subgraph induced by has no isolated vertices. The minimum cardinality of a restrained double monophonic set of is the restrained double monophonic number of and is denoted by . The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers with , there is a connected graph with , and , where is the monophonic number and is the restrained monophonic number of a graph
INTERPOLATING WAVELETS ON THE SPHERE
Get PDFThere are several works where bases of wavelets on the sphere (mainly orthogonal and wavelet-like bases) were constructed. In all such constructions, the authors seek to preserve the most important properties of classical wavelets including constructions on the basis of the lifting-scheme. In the present paper, we propose one more construction of wavelets on the sphere. Although two of three systems of wavelets constructed in this paper are orthogonal, we are more interested in their interpolation properties. Our main idea consists in a special double expansion of the unit sphere in such that any continuous function on this sphere defined in spherical coordinates is easily mapped into a -periodic function on the plane. After that everything becomes simple, since the classical scheme of the tensor product of one-dimensional bases of functional spaces works to construct bases of spaces of functions of several variables.There are several works where bases of wavelets on the sphere (mainly orthogonal and wavelet-like bases) were constructed. In all such constructions, the authors seek to preserve the most important properties of classical wavelets including constructions on the basis of the lifting-scheme. In the present paper, we propose one more construction of wavelets on the sphere. Although two of three systems of wavelets constructed in this paper are orthogonal, we are more interested in their interpolation properties. Our main idea consists in a special double expansion of the unit sphere in such that any continuous function on this sphere defined in spherical coordinates is easily mapped into a -periodic function on the plane. After that everything becomes simple, since the classical scheme of the tensor product of one-dimensional bases of functional spaces works to construct bases of spaces of functions of several variables
COMMUTATIVE WEAKLY INVO–CLEAN GROUP RINGS
Get PDFA ring is called weakly invo-clean if any its element is the sum or the difference of an involution and an idempotent. For each commutative unital ring and each abelian group , we find only in terms of , and their sections a necessary and sufficient condition when the group ring is weakly invo-clean. Our established result parallels to that due to Danchev-McGovern published in J. Algebra (2015) and proved for weakly nil-clean rings
HARMONIC INTERPOLATING WAVELETS IN NEUMANN BOUNDARY VALUE PROBLEM IN A CIRCLE
Get PDFThe Neumann boundary value problem (BVP) in a unit circle is discussed. For the solution of the Neumann BVP, we built a method employing series representation of given -periodic continuous boundary function by interpolating wavelets consisting of trigonometric polynomials. It is convenient to use the method due to the fact that such series is easy to extend to harmonic polynomials inside a circle. Moreover, coefficients of the series have an easy-to-calculate form. The representation by the interpolating wavelets is constructed by using an interpolation projection to subspaces of a multiresolution analysis with basis -periodic scaling functions (more exactly, their binary rational compressions and shifts). That functions were developed by Subbotin and Chernykh on the basis of Meyer-type wavelets. We will use three kinds of such functions, where two out of the three generates systems, which are orthogonal and simultaneous interpolating on uniform grids of the corresponding scale and the last one generates only interpolating on the same uniform grids system. As a result, using the interpolation property of wavelets mentioned above, we obtain the exact representation of the solution for the Neumann BVP by series of that wavelets and numerical bound of the approximation of solution by partial sum of such series
A NUMERICAL TECHNIQUE FOR THE SOLUTION OF GENERAL EIGHTH ORDER BOUNDARY VALUE PROBLEMS: A FINITE DIFFERENCE METHOD
Get PDFIn this article, we present a novel finite difference method for the numerical solution of the eighth order boundary value problems in ordinary differential equations. We have discretized the problem by using the boundary conditions in a natural way to obtain a system of equations. Then we have solved system of equations to obtain a numerical solution of the problem. Also we obtained numerical values of derivatives of solution as a byproduct of the method. The numerical experiments show that proposed method is efficient and fourth order accurate
EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART II
Get PDFThe non-elementary integrals \mbox{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, and \mbox{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, where are evaluated in terms of the hypergeometric function . On the other hand, the exponential integral \mbox{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx, is expressed in terms of . The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term
AUTOMORPHISMS OF DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {39; 36; 4; 1; 1; 36}
Get PDFMakhnev and Nirova have found intersection arrays of distance-regular graphs with no more than vertices, in which and . They proposed the program of investigation of distance-regular graphs with and . In this paper the automorphisms of a distance-regular graph with intersection array are studied
ON THE 75TH BIRTHDAY OF PROFESSOR VITALII VLADIMIROVICH ARESTOV
Get PDFJuly 16, 2018, was the 75th birthday of famous Russian scientist, prominent mathematician, Doctor of Physics and Mathematics, Professor Vitalii Vladimirovich Arestov
ASYMPTOTIC EXPANSION OF A SOLUTION FOR THE SINGULARLY PERTURBED OPTIMAL CONTROL PROBLEM WITH A CONVEX INTEGRAL QUALITY INDEX AND SMOOTH CONTROL CONSTRAINTS
Get PDFThe paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with smooth control constraints. In a general case, to solve such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. The main difference from the preceding article [10] is that the terminal part of the convex integral quality index depends not only on slow, but also on fast variables. In a particular case, we derive an equation that is satisfied by an initial vector of the conjugate system. Then this equation is extended to the optimal control problem with the convex integral quality index for a linear system with the fast and slow variables. It is shown that the solution of the corresponding equation as tends to the solution of an equation corresponding to the limit problem. The results obtained are applied to study a problem which describes the motion of a material point in for a fixed interval of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion