Galoitica: Journal of Mathematical Structures and Applications

In this work, we study the split-complex integer solutions for the split-complex linear Diophantine equation in two variables  where  are split-complex integers. An algorithm for generating all solutions will be obtained by transforming the split-complex equation to a classical equivalent system of linear Diophantine equations in four variables.

This paper is dedicated to study the analytical relations between Abel's double summability method and Natarajan's double summability method, where many theorems that draw a bridge between the mentioned methods will be obtained. The main result of our work is to prove that that summability by Natarajan's method implies summability by Abel's method in one or two variables. On the other hand, we illustrate some related examples to clarify the validity of our approach.

In this paper, a numerical method is suggested for solving general a nonlinear third order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.

This paper Deals with the complete bipartite graph K(r, n-r) and the cycle . The matrix of concern is the matrix B which is the (n, n) matrix and whose non zero entries are the reciprocals of the non zero entries of the distance matrix D. A complete characterization of the spectrum of B and a set of n independent eigenvectors of B will be presented. Two special cases will be mentioned, namely the star K(1, n-1) and the graph K(2, n-2). We will also look at the case of infinite graph, i. e if the size n grows big while r stays finite. Finally, some numerical data will be presented. As for the cycle, we present the complete set of eigenvalues of the matrix B.