### A new Lanczos-type algorithm for systems of linear equations

JPRM-Vol. 1 (2014), Issue 1, pp. 104 – 119 Open Access Full-Text PDF
Muhammad Farooq, Abdellah Salhi
Abstract: Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal polynomials of low degree. In this paper, we consider a recurrence relation that has not been studied before and which involves a relatively higher degree polynomial. Interestingly, it leads to an algorithm that shows superior stability when compared to existing Lanczos-type algorithms. This new algorithm is derived and described. It is then compared to the best known algorithms of this type, namely $$A_5/B_{10}$$, $$A_8/B_{10}$$, as well as Arnoldi’s algorithm, on a set of standard test problems. Numerical results are included.

### Topological structure of 2-normed space and some results in linear 2-normed spaces analogous to baire’s theorem and banach Steinhaus theorem

JPRM-Vol. 1 (2014), Issue 1, pp. 92 – 103 Open Access Full-Text PDF
P. Riyas, K. T. Ravindran
Abstract: In this paper we construct the topological structure of linear 2-normed space. This enable us to define the concept of open sets in linear 2-normed space and derive an analogue of Baire’s theorem and Banach Steinhaus theorem in linear 2-normed spaces.

### Zagreb indices and coindices of product graphs

JPRM-Vol. 1 (2014), Issue 1, pp. 80 – 91 Open Access Full-Text PDF
K. Pattabiraman, S. Nagarajan, M. Chendrasekharan
Abstract: For a (molecular) graph, the first Zagreb index $$M_1$$ is equal to the sum of squares of the degrees of vertices, and the second Zagreb index $$M_2$$ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Similarly, the first and second Zagreb coindices are defined as $$\overline{M}_1(G) = \sum_{ uv \notin E(G)} (d_G(u) + d_G(v))$$ and $$\overline{M}_2(G)=\sum_{ uv \notin E(G)} d_G(u)d_G(v)$$. In this paper, we compute the Zagreb indices and coindices of strong, tensor and edge corona product of two connected graphs. We apply some of our results to compute the Zagreb indices and coindices of open and closed fence graphs.

### A comparison of perturbation techniques for nonlinear problems

JPRM-Vol. 1 (2014), Issue 1, pp. 59 – 79 Open Access Full-Text PDF
H. A. Wahab, Saira Bhatti, Muhammad Naeem
Abstract: The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy perturbation method gives the same series solution as in Adomian Decomposition Method and we get the exact solution using two iterations only.

### Magnetohydrodynamics of rotating fractional second grade fluid in porous medium

JPRM-Vol. 1 (2014), Issue 1, pp. 45 – 58 Open Access Full-Text PDF
Azhar Ali Zafar, Dumitru Vieru, Shahraz Akhtar
Abstract: Exact solution for the unsteady flow of a fractional second grade fluid through the porous medium under the influence of magnetic field in the direction normal to the flow has been investigated using the integral transforms. Expressions for dimensionless velocity have been obtained and are presented in terms of Fox’s H–function. The influence of the fractional parameter on the fluid motion is studied and a comparison between velocity of the fractional and classical fluid is made.

### A numerical approach for solving hammerstein integral equations in banach spaces

JPRM-Vol. 1 (2014), Issue 1, pp. 37 – 44 Open Access Full-Text PDF
Abstract: In this work, we give a weaker conditions guarantee the boundedness of the Hammerstein integral equation in $$L^p$$ spaces, also we study conditions of the convergence of the approximate solution to the exact one of the integral equation using the successive approximations method. Finally, we treat numerical examples compared with other papers in order to confirm the efficiency of our results.

### Some characterizations of semigroups in terms of intuitionistic fuzzy interior ideals

JPRM-Vol. 1 (2014), Issue 1, pp. 19 – 36 Open Access Full-Text PDF
Hidayat Ullah Khan, Nor Haniza Sarmin, Asghar Khan,Faiz Muhammad Khan
Abstract: The importance of semigroups and their fuzzy subsystems is evident from their applications and significant role in several applied disciplines like computer sciences, control engineering, error-correcting codes and fuzzy automata theory. In this paper, we give generalizations of intuitionistic fuzzy interior ideals of semigroups and introduced the notions of intuitionistic fuzzy interior ideals of type $$(\overline{∈}, \overline{∈} ∨ \overline{q}_{k})$$ and $$(\overline{∈}, \overline{∈})$$ of semigroups. The important mile stone of the present paper is to link ordinary intuitionistic fuzzy interior ideals, $$(\overline{∈}, \overline{∈})$$-intuitionistic fuzzy interior ideals and $$(\overline{∈}, \overline{∈} ∨ \overline{q}_{k})$$-intuitionistic fuzzy interior ideals. Moreover semigroups are characterized by the properties of these notions.
Abstract: The Wiener index, denoted by $$W(G)$$, of a connected graph $$G$$ is the sum of all pairwise distances of vertices of the graph, that is, $$W(G)=\frac{1}{2}\sum_{u,v\in V(G)}d(u,v)$$. In this paper, we obtain the Wiener index of the tensor product of two cycles.