GANIT: Journal of Bangladesh Mathematical Society <p>The official journal of the Bangladesh Mathematical Society. Full text articles available.</p> en-US <p>The copyright of GANIT: Journal of Bangladesh Mathematical Society is reserved by Bangladesh Mathematical Society (web: <a href=""></a>)</p> (Associate Editor) (Md Fahmid Uddin Khondoker) Mon, 14 Jan 2019 11:43:33 +0000 OJS 60 Numerical Bifurcation Analysis to Study Periodic Traveling Wave Solutions in a Model of Young Mussel Beds <p>Self-bottomed mussel beds are dominant feature of ecosystem-scale self-organization. Regular spatial patterns of mussel beds in inter-tidal zone are typical, aligned perpendicular to the average incoming tidal flow. In this paper, we consider a two-variable partial differential equations model of young mussel beds. Our aim is to study the existence and stability of periodic traveling waves in a one-parameter family of solutions. We consider a parameter regime to show pattern existence in the model of young mussel beds. In addition, it is found that the periodic traveling waves changes their stability by two ways: Hopf type and Eckhaus type. We explain this stability by the calculation of essential spectra at different grid points in the two-dimensional parameter plane.</p> <p>GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 1-10</p> Md Ariful Islam Arif, M Osman Gani ##submission.copyrightStatement## Mon, 14 Jan 2019 00:00:00 +0000 Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods <p>In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2<sup>nd</sup> kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1<sup>st</sup> kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 11-25</p> Hasib Uddin Molla, Goutam Saha ##submission.copyrightStatement## Mon, 14 Jan 2019 00:00:00 +0000 Existence of Periodic Traveling Waves in the Klausmeier Model of Desertification and its Modification: A Comparative Study <p>Self-organized and spatially periodic banded vegetation patterns have been observed in many semi-arid ecosystems. In order to understand the mechanism of these patterns, we consider a system of reaction-advection-diffusion equations in a two-variable model of desertification. This work deals with the investigation of the existence of periodic traveling waves in a one-parameter family of solutions. In addition, we investigate the existence of periodic traveling waves as a function of water transport parameter in the model.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 27-46</p> Md AS Howlader, Md Ariful Islam Arif, LS Andallah, M Osman Gani ##submission.copyrightStatement## Mon, 14 Jan 2019 00:00:00 +0000 Sequentially Updated Weighted Cost Opportunity Based Algorithm in Transportation Problem <p>Transportation models are of multidisciplinary fields of interest. In classical transportation approaches, the flow of allocation is controlled by the cost entries and/or manipulation of cost entries – so called Distribution Indicator (DI) or Total Opportunity Cost (TOC). But these DI or TOC tables are formulated by the manipulation of cost entries only. None of them considers demand and/or supply entry to formulate the DI/ TOC table. Recently authors have developed weighted opportunity cost (WOC) matrix where this weighted opportunity cost matrix is formulated by the manipulation of supply and demand entries along with cost entries as well. In this WOC matrix, the supply and demand entries act as weight factors. Moreover by incorporating this WOC matrix in Least Cost Matrix, authors have developed a new approach to find out Initial Basic Feasible Solution of Transportation Problems. But in that approach, WOC matrix was invariant in every step of allocation procedures. That is, after the first time formulation of the weighted opportunity cost matrix, the WOC matrix was invariant throughout all allocation procedures. On the other hand in VAM method, the flow of allocation is controlled by the DI table and this table is updated after each allocation step. Motivated by this idea, we have reformed the WOC matrix as Sequentially Updated Weighted Opportunity Cost (SUWOC) matrix. The significance difference of these two matrices is that, WOC matrix is invariant through all over the allocation procedures whereas SUWOC &nbsp;&nbsp;matrix is updated in each step of allocation procedures. Note that here update (/invariant) means changed (/unchanged) the weighted opportunity cost of the cells. Finally by incorporating this SUWOC matrix in Least Cost Matrix, we have developed a new approach to find out Initial Basic Feasible Solution of Transportation Problems.&nbsp; Some experiments have been carried out to justify the validity and the effectiveness of the proposed SUWOC-LCM approach. Experimental results reveal that the SUWOC-LCM approach outperforms to find out IBFS. Moreover sometime this approach is able to find out optimal solution too.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 47-55</p> ARM Jalal Uddin Jamali, Pushpa Akhtar ##submission.copyrightStatement## Mon, 14 Jan 2019 00:00:00 +0000 On Codes over the Rings fq + ufq + vfq + uvfq <div> <p>In this paper, we study the structure of linear and self dual codes of an arbitrary length <em>n </em>overhearing <em>F<sub>q </sub>+ uF<sub>q </sub>+ vF<sub>q </sub>+ uvF<sub>q</sub></em>, where <em>q </em>is a power of the prime <em>p</em> and <em>u</em><sup>2 </sup>= <em>v</em><sup>2 </sup>= 0, <em>uv </em>= <em>vu</em>, Also we obtain the structure of consta-cyclic codes of length <em>n </em>= <em>q </em>− 1 over the ring <em>F<sub>q </sub>+ uF<sub>q </sub>+ vF<sub>q </sub>+ uvF<sub>q </sub></em>in the light of studying cyclic codes over <em>F<sub>q </sub>+ uF<sub>q </sub>+ vF<sub>q </sub>+ uvF<sub>q </sub></em>in [6]. This study is a generalization and extension of the works in [7], [8], and [10].</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 57-71</p> </div> <p>&nbsp;</p> Ibrahim M Yaghi, Mohammed M Ali Ashker ##submission.copyrightStatement## Mon, 14 Jan 2019 00:00:00 +0000 Heat-Mass Transfer of Nanofluid in Lid-Driven Enclosure under three Convective Modes <p>Heat is a form of energy which transfers between bodies which are kept under thermal interactions. When a temperature difference occurs between two bodies or a body with its surroundings, heat transfer occurs. Heat transfer occurs in three modes. Three modes of heat transfer are c<a href="">onduction</a>, <a href="">convection</a>&nbsp;and <a href="">radiation</a>. Convection is a very important phenomenon in heat transfer applications and it occurs due to two different gradients, such as, temperature and concentration. This paper reports a numerical study on forced-mixed-natural convections within a lid-driven square enclosure, filled with a mixture of water and 2% concentrated Cu nanoparticles. It is assumed that the temperature difference driving the convection comes from the side moving walls, when both horizontal walls are kept insulated. In order to solve general coupled equations, a code based on the Galerkin's finite element method is used. To make clear the effect of using nanofluid on heat and mass transfers inside the enclosure, a wide range of the Richardson number, taken from 0.1 to 10 is studied. A fair degree of precision can be found between the present and previously published works. The phenomenon is analyzed through streamlines, isotherm and iso-concentration plots, with special attention to the Nusselt number and Sherwood number. The larger heat and mass transfer rates can be achieved with nanofluid than the base fluid for all conditions at Richardson number, <em>Ri</em> = 0.1 to 10. It has been found that the heat and mass transfer rate increase approximately 6% for water with the increase of <em>Ri </em>= 0.1 to 10, whereas these increase about 34% for nanofluid.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 73-83</p> MS Rahman, R Nasrin, MI Hoque ##submission.copyrightStatement## Mon, 14 Jan 2019 11:20:07 +0000 An Inductive Proof of Bertrand's Postulate <p>In this paper, we are going to prove a famous problem concerning the prime numbers called Bertrand's postulate. It states that there is always at least one prime, <em>p</em> between <em>n</em> and <em>2n</em>, means, there exists <em>n &lt; p &lt; 2n </em>where<em> n &gt; 1</em>. It is not a newer theorem to be proven. It was first conjectured by Joseph Bertrand in 1845. He did not find a proof of this problem but made important numerical evidence for the large values of <em>n</em>. Eventually, it was successfully proven by Pafnuty Chebyshev in 1852. That is why it is also called Bertrand-Chebyshev theorem. Though it does not give very strong idea about the prime distribution like Prime Number Theorem (PNT) does, the beauty of Bertrand's postulate lies on its simple yet elegant definition. Historically, Bertrand's postulate is also very important. After Euclid's proof that there are infinite prime numbers, there was no significant development in the prime number distribution. Peter Dirichlet stated the standard form of Prime Number Theorem (PNT) in 1838 but it was merely a conjecture that time and beyond the scope of proof to the then mathematicians. Bertrand's postulate was a simply stated problem but powerful enough, easy to prove and could lead many more strong assumptions about the prime number distribution. Illustrious Indian mathematician, Srinivasa Ramanujan gave a shorter but elegant proof using the concept of Chebyshev functions of prime, <em>υ</em>(<em>x</em>), <em>Ψ</em>(<em>x</em>)and Gamma function, Γ(<em>x</em>) in 1919 which led to the concept of Ramanujan Prime. Later Paul Erdős published another proof using the concept of Primorial function, <em>p# </em>in 1932. The elegance of our proof lies on not using Gamma function yet finding the better approximations of Chebyshev functions of prime. The proof technique is very similar the way Ramanujan proved it but instead of using the Stirling's approximation to the binomial coefficients, we are proving similar results using well-known proving technique the mathematical induction and they lead to somewhat stronger than Ramanujan's approximation of Chebyshev functions of prime.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 85-87</p> Bijoy Rahman Arif ##submission.copyrightStatement## Mon, 14 Jan 2019 11:20:26 +0000 A Bypassing Technique for the Remedy of Portal Hypertension through Extra Hepatic Portal Vein Obstruction by CFD Analysis <p>Extra-hepatic portal vein obstruction (EHPVO) is the blockage to the flow of blood in the portal vein before reaches to the liver. EHPVO is the common cause of portal hypertension in children in the most Asian countries. Examination reveals that the presence of block in the main portal vein may be responsible for the shrinkage of vein with manifold pernicious complication. The “shunt” policy is a fruitful source of restoration of the hepatic portal flow. This study shows that a new approach of bypassing (or shunting) to the blocked (thrombosed) region of the portal vein is a significant way of reducing portal hypertension and restoration of blood circulation. We studied EHPVO case through computational fluid dynamics (CFD) analysis by considering partial block formation and side to side shunt scheme inside the main portal vein. The constitutive equation for non-Newtonian fluidand energy equation are solved by control volume technique. Our study reveals that the shunting technique is strongly effective for the reconstitution of portal venous flow to the liver with lower tissue stress and rapid regression of clinical signs of portal hypertension. This new technique may potentially applicable for medication of EHPVO when shunting procedures are indicated.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 89-104</p> Mst Khorseda Atkar, Md Tajul Islam ##submission.copyrightStatement## Mon, 14 Jan 2019 11:20:38 +0000 Approximation of a Complex Geometric Domain in Polar Coordinates <p>In this study, a complex geometric domain having a colour picture is approximated through a stair- step representation of the coastal and island boundaries to make it suitable for implementing finite difference method in solving shallow water equations (SWEs) in polar coordinates. As a complex domain, we choose the coastal region of Bangladesh situated at the northern tip of the Bay of Bengal (BOB). To cover the whole coastal region, the pole is selected at the point in the &nbsp;plane assuming it on the mean sea level (MSL). Along the tangential direction, 265 uniformly distributive straight lines are considered through the pole and 959 circular grid lines centered at &nbsp;are drawn towards the radial direction covering up to &nbsp;latitude in the BOB. Firstly, a matrix with 960´265 computational grids is constructed from the colour information of the picture. By representing the grids with suitable notations, a proper stair-step algorithm is employed to the matrix obtained with the 960´265 grids to approximate the coastal and island boundaries to the nearest finite difference grid lines using an Arakawa C-grid system. The whole procedure is done with our developed MATLAB program. The grids representing the coastal stations are also identified closely in the obtained approximated domain. Such a type of presentation of the coastal geometry of the region of interest is found to incorporate its complexities properly with minimum computational grid points.</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 105-118</p> Gour Chandra Paul, Md Masum Murshed, Md Mamunur Rasid, Md Morshed Bin Shiraj ##submission.copyrightStatement## Mon, 14 Jan 2019 11:20:49 +0000 Real Analyticity of Hausdorff Dimension Function of Disconnected Julia Sets of Parabolic Polynomials fa,b(z) = z(1 – az – bz<sup>2</sup>) <p>Abstract not available</p> <p>GANIT <em>J. Bangladesh Math. Soc.</em>Vol. 38 (2018) 119-125</p> Hasina Akter ##submission.copyrightStatement## Mon, 14 Jan 2019 11:20:58 +0000