Transactions of A Razmadze Mathematical Institute最新文献

In this paper, we define a generalized $T$-contraction and derive some new coupled fixed point theorems in cone metric spaces with total ordering condition. An illustrative example is provided to support our results. As an application, we utilize the results obtained to study the existence of common solution to a system of integral equations. We also present an application to Markov process.

In this paper, we first introduce strong compact operators in PN-spaces and then we prove some of their properties. After that we prove the Ascoli–Arzela’s theorem in SPN spaces. In addition some of its known consequences such as Schauder’s theorem and strong Banach closed rang theorem in SPN spaces are presented.

In this paper, the aim is to present a high-order compact alternating direction implicit (ADI) scheme for the two-dimensional time fractional diffusion-wave (FDW) equation. The time fractional derivative which has been described in the Caputo’s sense is approximated by a scheme of order $O\left({\tau }^{3-\alpha }\right)$, $1<\alpha <2$ and the space derivatives are discretized with a sixth-order compact procedure. The solvability, stability and $H^1$ norm of the scheme are proved. Numerical results are provided to verify the accuracy and efficiency of the proposed method of solution. The sixth-order accuracy in the space directions has not been achieved in previously studied schemes.

In this paper, a method has been proposed to finding a numerical function for the Riccati differential equations of non integer order (FRDEs), in which trigonometric basic functions are used. First, by defining trigonometric basic functions, we define the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the numerical function is defined as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get a numerical solution function with discrete derivatives of the solution function, we have determined the numerical solution function which satisfies the FRDEs. In the end, the algorithm of the method is elaborated with several examples. Numerical results obtained show that the proposed algorithm gives very good numerical solutions. In one example, we have presented an absolute error comparison of some numerical methods.

This analysis pertains to the non-Newtonian nanofluid impinging the surface of stretching obliquely. The base fluid under discussion obeys the constitutive equation of a UCM fluid. Use of similarity variables and RKF 45 numerical method along with shooting technique enabled us to obtain the solution of the problem. The effects of physical parameters associated with nanofluids on the flow variables are discussed in detail through graphs. The streamlines are skewed towards right of the stagnation point when the stagnation flow parameter is negative and towards left for positive values. Due to Brownian motion, thermophoresis and nonlinear thermal radiation temperature is enhanced. Brownian motion and chemical reaction have an increasing influence on Sherwood number while a reversal effect is noticed with thermophoresis. The results of this study are compared with those available in the existing literature and are found to be in good agreement.

In this article, we will introduce a new concept of gauge spaces induced by a family of vector valued pseudo metrics. After this, we will also prove some results to ensure the existence of fixed points of self mappings. As an application of our result, we will give an existence theorem to ensure the existence of solutions of $n$ different integral equations.

In this paper, an efficient numerical method is used for solving 2D-mixed Volterra–Fredholm integral equations. This method is based on 2D-orthonormal Bernstein polynomials (2D-OBPs) together with collocation method. This approach is applied to convert the problem under study into a system of algebraic equations which can be solved by using a convenient numerical method. Several useful theorems are proved which are concerned with the convergence and error estimate associated to the suggested scheme. Finally, by comparing the values of absolute error achieved from this method with values of absolute error obtained from other previous methods, we show that this method is very accurate and efficient.

In this paper we have discussed an algebraic ordered extension of vector space. This new structure comprises a semigroup structure, a scalar multiplication and a compatible partial order. It is an algebraic axiomatisation of topological hyperspace; also it can be thought of as a generalisation of vector space in the sense that, it always contains a vector space and conversely, every vector space can be embedded maximally into such a structure. Initially the idea of this structure was given by S. Ganguly et al. with the name “quasi-vector space” in “An Associated Structure Of A Topological Vector Space; Bull. Cal. Math. Soc; Vol-96, No.6 (2004), 489-498”. The axioms of this structure evolve a very rapid growth of its elements with respect to the partial order and also evoke some sort of positiveness in each element. Meanwhile, a vector space is evolved within this structure and positivity of each element of the new structure is judged with respect to the elements of the vector space generated. Considering the exponential behaviour of its elements, we have studied this structure in the present paper with a new nomenclature —“exponential vector space” in short ‘evs’. We have developed a quotient structure on an evs by defining ‘congruence’ on it and have shown that the quotient structure also forms an evs with respect to suitably defined operations and partial order. We have obtained an isomorphism theorem and a correspondence theorem in the context of exponential vector space. Further, we have topologised the quotient evs by defining compatibility of the associated congruence with the topology of the base evs. A necessary and sufficient condition has been deduced so that the order-isomorphism stated under the isomorphism theorem becomes topological. Also, we have constructed order-morphisms on a quotient evs corresponding to that on the base evs.

The Dirichlet ordinary and generalized harmonic problems for some 3D finite domains are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. An algorithm of numerical solution by the method of probabilistic solution (MPS) is given, which in its turn is based on a computer simulation of the Wiener process. Since, in the case of 3D generalized problems there are none exact test problems, therefore, for such problems, the way of testing of our method is suggested. For examining and to illustrate the effectiveness and simplicity of the proposed method five numerical examples are considered on finding the electric field. In the role of domains are taken ellipsoidal, spherical and cylindrical domains and both the potential and strength of the field are calculated. Numerical results are presented.