Marko D. Petković

Modified simple equation method for nonlinear evolution equations

Abstract This paper reflects the implementation of a reliable technique which is called modified simple equation method (MSEM) for solving evolution equations. The proposed algorithm has been successfully tested on two very important evolution equations namely Fitzhugh–Nagumo equation and Sharma–Tasso–Olver equation. Numerical results are very encouraging.

Perfect state transfer in integral circulant graphs

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. We give the simple condition for characterizing integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. Using that, we complete the proof of results stated by Saxena, Severini and Shparlinski. Moreover, it is shown that in the class of unitary Cayley graphs there are only two of them allowing perfect state transfer.

Iterative method for computing the Moore-Penrose inverse based on Penrose equations

An iterative algorithm for estimating the Moore-Penrose generalized inverse is developed. The main motive for the construction of the algorithm is simultaneous usage of Penrose equations (2) and (4). Convergence properties of the introduced method as well as their first-order and second-order error terms are considered. Numerical experiment is also presented.

Soliton solutions of Burgers equations and perturbed Burgers equation

Abstract This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV–Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method.

Some classes of integral circulant graphs either allowing or not allowing perfect state transfer

Abstract The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. Motivated by the aforementioned work, Basic, Petkovic and Stevanovic give the simple condition for the characterization of integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. They stated that the integral circulant graphs with minimal vertices allowing perfect state transfer, other than unitary Cayley graphs, are ICG 8 ( { 1 , 2 } ) and ICG 8 ( { 1 , 4 } ) . Moreover, it is also conjectured that two classes of integral circulant graphs ICG n ( { 1 , n / 4 } ) and ICG n ( { 1 , n / 2 } ) allow PST where n ∈ 8 N . These conjectures are confirmed in this work. Moreover, it is shown that there are no integral circulant graphs allowing perfect state transfer in the class of graphs where the number of vertices is a square-free integer.

Symbolic computation of weighted Moore-Penrose inverse using partitioning method

Abstract We propose a method and algorithm for computing the weighted Moore–Penrose inverse of one-variable rational matrices. Continuing this idea, we develop an algorithm for computing the weighted Moore–Penrose inverse of one-variable polynomial matrix. These methods and algorithms are generalizations of the method for computing the weighted Moore–Penrose inverse for constant matrices, originated in Wang and Chen [G.R. Wang, Y.L. Chen, A recursive algorithm for computing the weighted Moore–Penrose inverse A MN † , J. Comput. Math. 4 (1986) 74–85], and the partitioning method for computing the Moore–Penrose inverse of rational and polynomial matrices introduced in Stanimirovic and Tasic [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137–163]. Algorithms are implemented in the symbolic computational package MATHEMATICA .

Gauss-Jordan elimination method for computing outer inverses

This paper deals with the algorithm for computing outer inverse with prescribed range and null space, based on the choice of an appropriate matrix G and Gauss-Jordan elimination of the augmented matrix [G|I]. The advantage of such algorithms is the fact that one can compute various generalized inverses using the same procedure, for different input matrices. In particular, we derive representations of the Moore-Penrose inverse, the weighted Moore-Penrose inverse, the Drazin inverse as well as {2,4} and {2,3}-inverses. Numerical examples on different test matrices are presented, as well as the comparison with well-known methods for generalized inverses computation.

The Hankel transform of the sum of consecutive generalized Catalan numbers

We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.

Soliton solutions for nonlinear Calaogero-Degasperis and potential Kadomtsev-Petviashvili equations

This paper obtains the soliton solution for the Calogero-Degasperis and the potential Kadomtsev-Petviashvili equations. The tanh-coth and the tan-cot methods are used to retrieve the solutions. Finally, the ansatz method is also used to integrate these equations with any arbitrary constant coefficients. Finally, a few numerical simulations are also given.

Computing generalized inverse of polynomial matrices by interpolation

Abstract We investigated an interpolation algorithm for computing the Moore–Penrose inverse of a given polynomial matrix, based on the Leverrier–Faddeev method. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier–Faddeev algorithm is given as the improvement of the interpolation algorithm. Algorithms are implemented in the symbolic programming language MATHEMATICA , and tested on several different classes of test examples.