Authors:Khurshid Ahmad, Serkan Araci, Mirajul Haq, Bilal Khan Abstract: In this paper, we define a new subclass of analytic functions involving the cosine functions. For this function class, we obtain the upper bound of the third Hankel determinant. PubDate: 2022-04-16 Issue No:Vol. 9 (2022)

Authors:Safdar Ali, Fozia Hanif, Muhammad Ilyas, Rehan Shams, Muhammad Rehan, Syed Inayatullah Abstract: The purpose of this study is to develop the third order time fractional partial differential equations (PDEs) in one and higher dimensions, by taking Laplace Adomian decomposition method (LADM) and q-homotopy analysis transform method (q-HATM). To define fractional derivative, the Caputo operator is used for both fractional and integer orders. The solutions are obtained in the form of series. To understand the procedure of the suggested procedure, three numerical examples are taken. The graphs are plotted for the proposed solution at different values of fractional order '''' which is 0< '''' ≤ 1. Both proposed methods are implemented by using (LADM) and (q-HATM) showing that the proposed technique is found to be better and accurate instrument for solving linear and non-linear time fractional PDEs. The Novelty of the proposed study is that the provided solution for fractional order partial differential equations has never been attempted for third order, this means that the provided solution can solve the third order and could be generalized for the higher order also. PubDate: 2022-12-31 Issue No:Vol. 10 (2022)

Authors:Waris Ali, Asif Ali Shaikh, Feroz Shah, Sajjad Hussain Abstract: The energy provision is one of the main concerns of modern technological processes and thermal management systems. Through latent heat energy, the storage of thermal energy using phase-change materials is examined in this paper. Paraffin Rubitherm 50 is filled in the cylinder. The base of the cylinder is heated and the vertical surface is made adiabatic. The melting procedure for two cases namely the plane surface and finned surface of the cylinder are considered. The melt fractions are observed and photographed for fixed intervals of time from solid state to total melt state. Initially, the melting of specified PCM was slow and then it became faster when convection heat transfer is accompanied with the conduction. The melting of PCM is geared with fin presence. PubDate: 2022-12-31 Issue No:Vol. 10 (2022)

Authors:Azam Ali, K. N. Memon, Syed Feroz Shah, Mohsin Amur, A. M. Siddiqui Abstract: This work clarifies the tank drainage problem of unsteady, incompressible and isothermal third-order fluid. The analytical solution is obtained from governing continuity and momentum equations for resulting non-linear PDE with no-slip conditions using the perturbation technique. The V(z) velocity profile, flow rate, depth H(t), time efflux, and time depth of the mass relation have been inspected on various parameters. The consequences of the depth H(t), the pipe having radius R, the density of the fluid ρ pipe of length L, dynamic viscosity µ, and small perturbation parameter ϵ on the velocity profile are beheld graphically. Comparison of Analytical result of the special case of third order fluid with existing literature when ϵ = 0 is given. PubDate: 2022-12-31 Issue No:Vol. 10 (2022)

Authors:Sanaullah Jamali, Zubair Ahmed Kalhoro, Abdul Wasim Shaikh, Muhammad Saleem Chandio, Sanaullah Dehraj Abstract: Abstract In this paper a three-step numerical method, using weight function, has been derived for ﬁnding the root of non-linear equations. The proposed method possesses the accuracy of order eight with four functional evaluations. The eﬃciency index of the derived scheme is 1.682. Numerical examples, application problems are used to demonstrate the performance of the presented schemes and compare them to other available methods in the literature of the same order. Matlab, Mathematica 2021 & Maple 2021 software were used for numerical results. PubDate: 2022-12-31 Issue No:Vol. 10 (2022)

Authors:Sanaullah Jamali, Zubair Ahmed Kalhoro, Abdul Wasim Shaikh, Muhammad Saleem Chandio, Sanaullah Dehraj Abstract: It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in literature to find root but their cost are very high In this regard we introduce a two-step fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods. PubDate: 2022-12-31 Issue No:Vol. 10 (2022)

Authors:Hanan Shiekh, Rao Faisal Rajput, K. B. Amur Abstract: A Long time deflation preconditioner is used to speed up the convergence of the Krylov subspace method. The discretization of Helmholtz equation with Dirichlet boundary condition by finite difference method obtained any linear system. Resolving a large wavenumber requires a larger number of Grid points, i.e. large linear systems. Thus due to the large linear system, many (sparse) direct methods have taken more memory, So we have used the (iterative technique) Krylov subspace method. One of the problems of the Krylov subspace method is the required preconditioner for better convergence. We use (CSLP) as a preconditioner and drive eigenvalues of (CSLP). However, with increasing wavenumber CSLP shows slow convergence behavior. Then we use another projection-type preconditioner as a deflation preconditioner. A rigorous Fourier analysis (RFA) is a separate research idea to examine the con- vergence of the iterative method included in this article. We analyze the deflation preconditioner with a complex shifted Laplace preconditioner (CSLP) which exhibition spectral behavior of the preconditioner, which is favorable to the Krylov method. PubDate: 2022-12-31 Issue No:Vol. 10 (2022)

Authors:Muhammad Bilal, Shakoor Muhammad, Nekmat Ullah, Fazal Hanan, Subahan Ullah Abstract: This paper proposes a new minimization technique for the solutions of partial diﬀerential equation with initial conditions. The proposed procedure is used to minimize the obtained solutions through any numerical technique. For the minimization process, Non-linear Nelder-Mead Simplex algorithm and genetic algorithm are used as optimization techniques. The designed partial diﬀerential equation has been calculated as an error function for the minimization process. Both Non-linear Nelder-Mead Simplex and genetic algorithm guarantees the minimization of nonlinear partial diﬀerential equation with initial conditions. The resultant technique has a global validity for the solutions minimization of partial diﬀerential equations. Non-linear Nelder-Mead simplex showed better performance than genetic algorithm when tested on numerical instances. PubDate: 2022-12-17 Issue No:Vol. 10 (2022)

Authors:Liaquat Ali Zardari, Shakeel Ahmed Kamboh, Abbas Ali Ghoto, Dr. Kirshan Kumar Luhana, Dr. Shah Zaman Nizamani Abstract: In this study the eﬀect of the coecients on the convergence of numerical solution of general second order linear homogeneous partial diﬀerential equation has been investigated. The main objective was to determine the sensitivity of the coecients of the PDE in relation to the domain and mesh size. The nite diﬀerence method was used to discretize the PDE and numerical solution was obtained by implementing the algorithm on MATLAB. The outcomes of the research have provided interesting facts about the stable values of the coecients. From the results it is found that the arbitrary coecients d and e are more sensitive as compared to a, b, c and f. The outcomes of this research study are expected to provide the ways to predict and control the numerical solution convergence behavior obtained by the general second order PDE based on the variable coecients of the PDE. PubDate: 2022-12-09 Issue No:Vol. 10 (2022)

Authors:Tasawar Abbas, Faisal Mumtaz, Zamir Hussain, Rehan Zafar Abstract: In modern social administrative economic activities, we are facing a considerable amount of multi-attribute group decision making problems. The methods and theory related to this method are very useful in the field of particular disciplines as well as in operational research, and a lot of achievements have been described. Obviously the real world is full of uncertainties and classical set theory cannot be used to describe different phenomena such as beauty, intelligence, height (tallness) and age etc. This thing leads mathematicians to develop the notion of fuzzy sets. Later Zadeh introduced the concept of membership and non-membership degree. Definitely human opinion about a phenomenon may be unidirectional or multi-directional, that’s why Atanossov proposed the concept of another advance type of fuzzy sets, which is known as intuitionistic fuzzy sets. His concept is based on a degree of membership and degree of non-membership with a exquisite that their sum must not exceed 1. In our work we introduced cubic linguistic spherical fuzzy sets. Then, we proposed the fundamental operation law for CLSFVs and a series of their average operators (AOs), such as the (cubic linguistic spherical fuzzy power average), (cubic linguistic spherical fuzzy power weighted average), (cubic linguistic spherical fuzzy power hamy mean) and (cubic linguistic spherical fuzzy power weighted hamy mean) operators, was developed by combining the power average and hamy mean operators in cubic linguistic spherical fuzzy environment. Also we described some specific desirable properties of all these operators. In addition, we suggested a new MAGDM method. PubDate: 2022-11-28 Issue No:Vol. 10 (2022)

Authors:Muhammad Bilal, Muhammad Aamir, Saleem Abdullah, Noor Mahmood, Umair Khalil, Nida Khalid, Maqbool Ahmed, Muhammad Naeem, Shakoor Muhammad, Laiba Sultan Dar Abstract: Abstract The COVID-19 virus is a pandemic that, from the outset, alters its appearance and symptoms. It has aggressively spread around the world. The COVID-19-induced fear and uncertainty are disrupting the global economy and exacerbating ﬁnancial market volatility. The most impacted countries were the United States, the United Kingdom, India, and Pakistan. The continuing COVID-19 situation is both a public health and economic concern on a worldwide. This research aims at how the spread of the COVID-19 has affected the cost of gasoline, diesel, and liqueﬁed petroleum gas (LPG). Every week, statistics on COVID-19 instances and pricing are collected. The data was analyzed using the ARDL model and the Bound test to determine the short and long-term association between COVID-19 and prices. The Autoregressive distributive lag model ﬁndings reveal that conﬁrmed and mortality cases impact fuel, diesel, and LPG prices. PubDate: 2022-11-17 Issue No:Vol. 10 (2022)

Authors:Tasawar Abbas, Ehsan Ul Haq, Ambreen Ayub, Asadullah Dawood Abstract: Nowadays, the discovery of the link between entropy and plasma density and temperature opens up new avenues for mathematicians and researchers to examine alternative plasma models in terms of entropy. After including the entropy drift in ETG mode, the linear dispersion relation and KdV equation are produced. In addition, the Variational Iteration Technique (VIM) is used to determine the problem’s analytical solutions. Involving the Langrange multiplier also helps to accelerate computation and lower its cost. Then, it is shown that in ETG mode, entropy impacts both the breadth and amplitude of rarefactive solitons, as well as the effects of the magnetic ﬁeld and inhomogeneity drift on the structure of solitons. In this instance, however, only the rarefactive solitons are present. Since introducing entropy to the system might alter the previous plasma ﬁndings, this study is new. Finally, the current approach will be applied to the entropy-based analysis of solitary waves in magnetically conﬁned plasmas. The graphical ﬁndings are also provided using Maple-18. PubDate: 2022-11-14 Issue No:Vol. 10 (2022)

Authors:Rabnawaz Mallah, Wajid Ahmed Siyal, Saira Aslam, Muhammad Suleman Sial, Inayatullah Soomro Abstract: Abstract Numerous techniques exist for solving and describing the Partial differential equation’s mathematical and computational model. The Laplacian operator is one of the most effective techniques for solving linear and nonlinear partial differential equations. It is quick, and researchers use it frequently because of its modern technique and high accuracy in results. The Crank-Nicolson (CN) scheme in the Cartesian coordinate system has been discussed in this research work. Using this method, a numerical approximation scheme in Cartesian coordinate system has been discretized on a 5 point stencil, extendable to nine points. The Tailor Series was used to discretize this scheme on 5-point stencils, which will be used in FORTRAN code for numerical approximation and can be visualized in OPEDX software. The Nicolson scheme is a ﬁnite difference scheme used to solve partial differential equations such as heat, wave, and diffusion equations in both 1-D and 2-D. Because of his extendable stencil, it will create accuracy and stability in the novel results of the scheme. These extendable stencils will reduce the error of the scheme and will assist researchers in ﬁnding novel results by solving ODES and PDES using the CN method. PubDate: 2022-11-12 Issue No:Vol. 10 (2022)

Authors:Khadija Shaikh, Fozia Shaikh, Rahim Bux Khokhar, K.N. Memon Abstract: This paper investigates the fractional Oldroyd-B fluid flow across two interminable coaxial cylinders, where the fluid’s motion is generated by the oscillatory motion of cylinders and the oscillating pressure gradient. The profile of velocity and shear stress of the flow is derived through a semi-analytical approach with the assistance of the Caputo fractional derivative, Laplace transforms, and the finite Hankel transform. The semi-analytical solution is then displayed as generalized functions of G and R satisfying governing equations as well as all initial and boundary conditions thereupon. For authenticity, certain limits have been imposed on the resultant equations and their results have been contrasted against existing results. Furthermore, the effect of different parameters on the flow of Oldroyd-B fluid is analyzed and illustrated graphically. PubDate: 2022-11-12 Issue No:Vol. 10 (2022)

Authors:Kamran Nazir Memon, Ahsan Mushtaque, Fozia Shaikh, AA Ghoto, A. M. Siddiqui Abstract: This work uses the Delta Perturbation Method (DPM) to theoretically evaluate the steady plane Couette-Poiseuille flowbetween two parallel plates for third-grade fluid.That'sa kind of perturbation approach and was deliveredwith the aid of Bender and his colleagues in the 1980s. Utilizing DPM, analytical solutions have been found from the governing continuity and momentum equations subject to the necessary boundary conditions. In this proposed model, the Newtonian solution is obtained through the substitution. It is possible to measure the velocity field, temperature distribution, volumetric flow rate, and average velocity of the fluid flow. We derived that the third-grade fluid's velocity will change in response to an increasing material constant from the visual and table representations of the impacts of different parameters on the velocity and temperature profiles.The suggested model additionally mentions temperature distribution losses with increases in thermal conductivity and rises as a result of increases of dynamic viscosity , constant parameters and and material constant. Here we have also find out that temperature distribution and velocity profile enhance with higher magnitude of pressure gradient PubDate: 2022-11-12 Issue No:Vol. 10 (2022)

Authors:Muhammad Asif Jamal, M. Faizan, Ahmed Farid, Fozia Shaikh, Fozia hanif Abstract: Abstract The recent article addresses the unsteady ﬂow of MHD incompressible tangent hyperbolic ﬂuid with Nanoﬂuid particles in the direction of a stretching surface. Nano-ﬂuid is related to thermo-phoretic and Brownian movement. With proper help through the transformation procedure, the set of non-linear (PDEs) is re-framed into (ODEs). The initiate expressions of momentum, temperature ﬁeld, and nano-particle concentration are composed into groups of nonlinear equations. That consequential terminology is computed shooting system. The impact of fundamental parameters on the ﬂow ﬁeld, thermal circulation, and meditation is described. Moreover, the ﬂow ﬁeld behavior due to the Wall friction, local Nusselt, and Sherwood numbers are examined. This study is signiﬁcant as this transformation determined the shooting technique’s numerical result and ensured the physical parameters’ behavior graphically. The results show that the velocity ﬁeld diminishes by escalating the Weissenberg (We) ﬁgure and power-law index (n), while thermal and concentration ﬁelds remain to detect elevating at similar parameters. Furthermore, the computed result is compared with existing literature and gets accuracy. PubDate: 2022-06-30 Issue No:Vol. 10 (2022)

Authors:Memoona Pirzada, Khuda Bux Amur, Muzaffar Bashir Arain, Rajab Ali Malookani Abstract: We present the idea of image driven isotropic diffusivity along with complementary regularization for image denoising problem. The method is based on the optimization of a quadratic function in L2 norm. The minimization of the energy functional leads to the Partial Differential equation (PDE)-based problem. We are looking for a steady state solution of equivalent time dependent problem. We discretize the problem with standard ﬁnite differences. The steady-state numerical solution of the time dependent problem leads to the iterative procedure, which allow to compute a regularized version of the solution as a denoised image. We have applied our designed model on synthetic as well as real images. The numerous experiments have been conducted to analyse the performance of the method for the different choices of scaling parameters. From the quality of the obtained results and comparative study it is observed that the proposed model performs well as compared to well existing methods. PubDate: 2022-06-30 Issue No:Vol. 10 (2022)

Authors:Shakeel Ahmed Kamboh, Suhail Aslam Khaskheli, Abbas Ali Ghoto, Sunny Kumar Aassori, Muzaffar Bashir Arain Abstract: In this study an encryption/decryption algorithm is proposed and developed. The developed algorithm is based on the idea that any given plain text can be encrypted like a block cipher with a combination of three encryption keys k1, K2, ...KN that use any value between N = 1, 2, 3, n, . Then the cipher values can be used to make the blocks of alphabets containing only A, B, C, D, E, F, G, H, I, J, each block is separated by a space. The steps of algorithm could also be reversible for decryption of the cipher text. A MATLAB code is written to implement the algorithm and tested different input messages.Secret message consists of website link and bank account details. The specialty of the algorithm is that it can be ﬂexibly used to encrypt and decrypt the secret messages containing not only English alphabets but also those messages containing the numbers, punctuations, elementary mathematics operations and the special characters. The performance of the algorithm is evaluated in terms of computational time, memory usage. From the analysis it is found that the proposed algorithm is faster in terms of execution time as compared to the modern algorithm which makes the algorithm computationally secure. The proposed research particularly contributes as the addition of knowledge in the ﬁeld of cryptography and generally to the information security; consequently can be beneﬁcial to the society. 3 PubDate: 2022-06-30 Issue No:Vol. 10 (2022)

Authors:Kamran Nazir Memon, Shakeel Ahmed Kamboh, Sakina Kamboh, Abbas Ghoto, Fozia Shaikh, Nawab ahmed Abstract: This paper presents the simulation of electrohydrodynamically driven micropump obtained by using 3D ﬁnite difference method. EHD governing equations are discretized and then explicitly deﬁned for output parameters. A 3D prototype of ion-drag micropump with symmetric electrodes is modeled and simulated for the velocity, the pressure, electric potential and electric ﬁeld. The objective of this study was to evaluate the results obtained by ﬁnite difference method (FDM) with the results obtained by a ﬁnite element method (FEM) based simulation package COMSOL Multiphysics. The comparison reveals that the numerical simulation results obtained by both the methods are appreciably close to each other. The simulation results are also compared with the existing ex- perimental data and it was found that there are not high discrepancies between simulation and experimental results. The paper concludes that in case of regular geometries of ion-drag micropump the FDM is easy to implement and provides more control on different parameters involved in the simulation as compared to built-in ﬁnite element method based package. PubDate: 2022-06-30 Issue No:Vol. 10 (2022)

Authors:Alireza Mohammadpour Abstract: In this article, the reduced differential transform method is improved to solve some nonlinear conformable time fractional partial differential equations. The solutions obtained by the purposed methd to solve the Korteweg-de Vries equation, k(m,n) for m=n=2, and Wu-Zhang (2+1)-dimensional dispersive long wave equation, show that the method is very effective. PubDate: 2022-04-22 Issue No:Vol. 1 (2022)