Department of Mathematics

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Adomian decomposition method with Neumann boundary conditions for solution of nonlinear boundary value problem
(Science International, 2015) Sana. A.; N. A. Chaudhry; M.Saeed; M.F.Tabassum
The Adomian decomposition method (ADM) is a creative and effective method for exact solution of functional equations of various kinds. Adomian decomposition method solves wide class of linear and non-linear, ordinary or partial differential equations. This paper presents the Adomian decomposition method for the solution of nonlinear boundary value problem using Neumann boundary conditions. In this approach, the solution is found in the form of a convergent power series with easily computed components. To show the efficiency of the method, numerical results and graphical representation of results are presented and compared with exact solution.
Algebraic properties of the binomial edge ideal of a complete bipartite graph
(An. St. Univ. Ovidius Constanta, 2014) Schenzel, Peter; Sohail Zafar
Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj 􀀀 xjyi; 1 _ i < j _ n; in the polynomial ring S = K[x1; : : : ; xn; y1; : : : ; yn] where fi; jg is an edge of G. We study the arithmetic properties of S=JG for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of decencies, the duals of local co homology modules, and prove the purity of the minimal free resolution of S=JG.
Algebraic properties of the binomial edge ideal of a complete bipartite graph
(Univ. Ovidius Constant a, 2014) Sohail Zafar; Schenzel, Peter
Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj 􀀀 xjyi; 1 i < j n; in the polynomial ring S = K[x1; : : : ; xn; y1; : : : ; yn] where fi; jg is an edge of G. We study the arithmetic properties of S=JG for G, the complete bipartite graph. In particular we compute dimen- sions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of de ciencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S=JG.
Analysis of stability and accuracy for forward time centered space approximation by using modified equation
(Science International, 2015) Tahir Ch; N.A. Shahid; M.F. Tabassum; A. Sana; S.Nazir
In this paper we investigate the quantitative behavior of a wide range of numerical methods for solving linear partial differential equations [PDE’s]. In order to study the properties of the numerical solutions, such as accuracy, consistency, and stability, we use the method of modified equation, which is an effective approach.To determine the necessary and sufficient conditions for computing the stability, we use a truncated version of modified equation which helps us in a better way to look into the nature of dispersive as well as dissipative errors. The heat equation with Drichlet Boundary Conditions can serve as a model for heat conduction, soil consolidation, ground water flow etc.Accuracy and Stability of Forward Time Centered Space (FTCS) scheme is checked by using Modified Differential Equation [MDE].
Analytic solutions of oldyrod-b fluid with fractional derivatives in a circular duct that applies a constant couple
(Alexendria Engineering Journal, Elsevier, 2016) Muhammad Bilal Riaz; Muhammad Imran Asjad
The aim of this article was to analyze the rotational flow of an Oldroyd-B fluid with fractional derivatives, induced by an infinite circular cylinder that applies a constant couple to the fluid. Such kind of problem in the settings of fractional derivatives has not been found in the literature. The solutions are based on an important remark regarding the governing equation for the nontrivial shear stress. The solutions that have been obtained satisfy all imposed initial and boundary conditions and can easily be reduced to the similar solutions corresponding to ordinary Oldroyd-B, fractional/ordinary Maxwell, fractional/ordinary second-grade, and Newtonian fluids performing the same motion. The obtained results are expressed in terms of Newtonian and non-Newtonian contributions. Finally, the influence of fractional parameters on the velocity, shear stress and a comparison between generalized and ordinary fluids is graphically underlined.
Analytical solutions for unsteady flow problems of Maxwell fluid in a porous mediam
(Science International, 2015) Siddique; Abdullah Y. Al Hossain; M. B. Riaz
In this paper, the problems of unsteady unidirectional flow of Maxwell fluid in a porous media are examined. The governing equations of flow are modelled, by employing the modified Darcy's law of a Maxwell fluid. Using Sumudu transform, analytical solutions of modelled equations are established for the following problems: (i) unsteady Couette flow, (ii) unsteady Poiseuille flow and (iii) unsteady generalized Couette flow. Since the Sumudu transform has units preserving properties, therefore aforementioned problems are solved without restoring the frequency domain. This is one of many strength points for this new transform, especially with respect to applications in problems with physical dimensions. Further, the solutions for the velocity fields that have been obtained; have complete agreement with those established by using the Laplace transform. Moreover, the corresponding solutions for Newtonian fluids as well as those for Maxwell fluids can be obtained as limiting cases of our solutions. Finally, the impact of relevant parameters on the velocity of fluids is also analyzed by graphical illustrations.
Approximations for soft fuzzy rough sets.
(Scientific publications of the state University of NOVI Pazar, Scientific, 2016) Tabasam Rashid
In this paper, we introduce a modified soft fuzzy rough set model. The lower and upper approximation operators are presented and their related properties are investigated. It is shown that these new models of approximations are finer than already known in the literature.
Artificial showering algorithm: a new meta-heuristic for unconstrained optimization
(Science International, 2015) Javaid Ali; Muhammad Saeed; Muhammad Luqman; Muhammad Farhan Tabassum
A novel meta-heuristic known as Artificial Showering Algorithm (ASHA) is presented in this paper. The proposed method is based on flow and accumulation phenomena of water units distributed by human controlled equipment in an ideal field representing the search space. The developed method is applied to benchmarking test functions and quality solutions are obtained. Comparisons witness that the method even at its evolvement phase performs better than pioneering algorithms like Genetic Algorithm, Differential Evolution and Simulated Annealing method.
Charged cylindrical polytropes with generalized polytropic equation of state.
(The European Physical Journal C,Springer, 2016) Syed Ali Mardan Azmi; Ifra Noureen; Muhammad Aziz Ur Rehman
We study the general formalism of polytropes in the relativistic regime with generalized polytropic equations of state in the vicinity of cylindrical symmetry. We take a charged anisotropic fluid distribution of matter with a conformally flat condition for the development of a general framework of the polytropes. We discuss the stability of the model by the Whittaker formula and conclude that one of the models developed is physically viable.
Common fixed point theorem for a hybrid pair of mappings in hausdorff fuzzy metric spaces
(University of Management and Technology, 2012) M Abbas; Basit Ali; A Amini-Harandi
In this paper, we prove a coupled fixed point theorem for a multivalued fuzzy contraction mapping in complete Hausdorff fuzzy metric spaces. As an application of the first theorem, a coupled coincidence and coupled common fixed point theorem has been proved for a hybrid pair of multivalued and single-valued mappings. It is worth mentioning that to find coupled coincidence points, we do not employ the condition of continuity of any mapping involved therein. Also, coupled coincidence points are obtained without exploiting any type of commutativity condition. Our results extend, improve, and unify some well-known results in the literature.
Convex hesitant fuzzy sets
(Journal of Intelligent & Fuzzy Systems, IOS Press, 2016) Tabasam Rashid
Convex hesitant fuzzy sets are defined as an extension of convex fuzzy sets. Also level sets are defined for hesitant fuzzy sets and discussed with their convexity. We focus on aggregation functions for hesitant fuzzy elements. These aggregation functions are further extended for hesitant fuzzy sets as well as for the convex structures of these sets.
Cosmological evolution of pilgrim dark energy
(Astrophys Space Sci, 2014) Muhammad Zubair; Muhammad Sharif
We study pilgrim dark energy model by taking IR cut-offs as particle and event horizons as well as conformal age of the universe. We derive evolution equations for fractional energy density and equation of state parameters for pilgrim dark energy. The phantom cosmic evolution is established in these scenarios which is well supported by the cosmological parameters such as deceleration parameter, statefinder parameters and phase space of ωϑ and ω ϑ . We conclude that the consistent value of parameter μ is μ<0 in accordance with the current Planck and WMAP9 results.
Coupled lower and upper solution approach for the existence of solutions of nonlinear coupled system with nonlinear coupled boundary conditions.
(Proyecciones Journal of Mathematics,, 2016) Imran Talib; Naseer Ahmad Asif
The present article investigates the existence of solutions of the following nonlinear second order coupled system with nonlinear coupled boundary conditions (CBCs) ⎧ ⎪⎪⎨ ⎪⎪⎩ −u 00 (t) = f1(t, v(t)), t ∈ [0, 1], −v 00 (t) = f2(t, u(t)), t ∈ [0, 1], µ(u(0), v(0), u0 (0), v0 (0), u0 (1), v0 (1)) = (0, 0), ν(u(0), v(0)) + (u(1), v(1)) = (0, 0), where f1, f2 : [0, 1] × R → R, µ : R6 → R2 and ν : R2 → R2 are continuous functions. The results presented in [7, 11] are extended in our article. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder’s fixed point theorem play an important role in establishing the arguments. Some examples are taken to ensure the validity of the theoretical results.
Cracking of compact objects with electromagnetic field.
(Astrophysics and Space Science, 2015) Syed Ali Mardan; Muhammad Aziz Ur Rehman
In this paper, we investigate the role of electromagnetic field on the stability regions of charged selfgravitating compact objects by using the concept of cracking. For this purpose, we have applied local density perturbation scheme to the hydrostatic equilibrium equation as well as on physical parameters involved in the model. In particular, we have examined the cracking of charged compact objects like PSR J1614-2230, PSR J1903+327, Vela X-1, SMC X-1 and Cen X-3 with different values of charge. We conclude that these objects exhibit cracking and stability regions decreases with the increase of charge.
Cracking of some compact objects with linear regime
(Astrophys Space Sci, 2015) M. Azam; S.A. Mardan; M.A. Rehman
Compact stars serve as a logical regimen for the implementation of theoretical models that are difficult to understand from an experimental setup. In our present work, we discuss the stability of self-gravitating compact objects by using the concept of cracking in the linear regime. We investigate the effect of density perturbation and local anisotropy on the stability regions of the following compact objects, neutron star PSR J1614-2230, the millisecond pulsar PSR J1903+327 and X-ray pulsars Vela X-1, SMC X-1, Cen X-3.We find that SMC X-1 is the stable compact object and all other exhibit cracking.
Dissipative spherical collapse of charged anisotropic fluid in f (r) gravity
(The European Physical Journal C, 2014) H. Rizwana Kausar; Ifra Noureen
This manuscript is devoted to study the combined effect of a viable f(R) = R + Rn model and electromagnetic field on the instability range of gravitational collapse. We assume charged anisotropic fluid that dissipate energy via heat flow and discuss that electromagnetic field, density inhomogeneity, shear and phase transition on astrophysi- cal bodies can be incorporated by locally anisotropic background. Dy- namical equations help to investigate the evolution of self-gravitating objects and leads to the conclusion that adiabatic index depend upon the electromagnetic background, mass and radius of the spherical ob- jects.
Dynamical analysis of charged anisotropic spherical star in f(r) gravity
(The European Physical Journal Plus, 2015) H. Rizwana Kausar; Ifra Noureen; M. Umair Shahzad
We consider a modified gravity theory, f(R) = R + Rn − μ4 Rm , in the metric formulation and analyze the contribution of electromag- netic field on the range of dynamical instability of a star filled with anisotropic matter. The collapse equation is developed by applying conservation on anisotropic matter, Maxwell source and dark source terms arising due to f(R) gravity. Specific perturbation scheme is implemented and it is observed that the inclusion of Maxwell source slows down the collapse and makes system more stable in Newtonian regime. Also, we make comparison of our results with the existing literature.
Dynamical instability and expansion-free condition in f (r, t) gravity
(European. Physics Journal C, 2015) Ifra Noureen; M. Zubair
A dynamical analysis of a spherically symmetric collapsing star surrounded by a locally anisotropic environment under an expansion-free condition is presented in f (R, T ) gravity, where R corresponds to the Ricci scalar and T stands for the trace of the energy momentum tensor. The modified field equations and evolution equations are reconstructed in the framework of f (R, T ) gravity. In order to acquire the collapse equation we implement the perturbation on all matter variables and dark source components comprising the viable f (R, T ) model. The instability range is described in the Newtonian and post-Newtonian approximation. It is observed that the unequal stresses and density profile define the instability range rather than the adiabatic index. However, the physical quantities are constrained to maintain positivity of the energy density and a stable stellar configuration.
Dynamical instability of expansion-free collapse in f (t) gravity
(International Journal of Theoretical Physics, 2014) Muhammad Sharif; Shamaila Rani
In this paper, we analyze the dynamical instability of a spherically symmetric collapsing star in the context of f (T ) gravity. For this purpose, we assume power-law f (T ) model with non-dissipative anisotropic fluid distribution under expansion-free condition. The perturbation scheme is applied to all matter, metric and f (T ) functions. We formulate dynamical equations using contracted Bianchi identities to investigate dynamical instability ranges in Newtonian and post-Newtonian regimes. It is found that the instability ranges of expansion-free fluid are independent of adiabatic index but depend on radial density profile, anisotropic pressure and torsion terms.
Dynamics of axial symmetric system in self-interacting brans–dicke gravity.
(European Physics Journal , Springer link, 2016) Rubab Manzoor
This paper investigates dynamics of axial reflection symmetric model in self-interacting Brans-Dicke gravity for anisotropic fluid. We formulate hydrodynamical equations and discuss oscillations using time-dependent perturbation for both spin as well as spin-independent cases. The expressions of frequency, total energy density and equation of motion of oscillating model are obtained. We study instability of oscillating models in weak approximations. It is found that the oscillations and stability of the model depend upon the dark energy source along with anisotropy and reflection effects. We conclude that the axial reflection system remains stable for stiffness parameter Γ = 1, collapses for Γ > 1 and becomes unstable for 0 < Γ < 1.

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