VFAST Transactions on Mathematics
https://vfast.org/journals/index.php/VTM
<p>The <strong><em>VFAST Transactions on Mathematics</em></strong> is a yearly peer-reviewed scientific journal published by the VFAST-Research Platform. It was established in 2013 and covers the areas of Pure and Applied Mathematics and its applications. <span>The goal of this journal is to provide an international platform to Mathematicians all over the world to promote, share, and discuss various new issues and development in the field of Mathematics and its application.</span></p><p>ISSN: 2309-0022 (Online), 2411-6343 (Print)</p><p>Editor in Chief: Dr. Muhammad Arif, Associate Professor, Abdul Wali Khan University Mardan</p><p><strong><span style="text-decoration: underline;"><span class="csspropertycolor" style="box-sizing: inherit; color: red; font-family: Consolas, Menlo, 'courier new', monospace; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff; 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src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJcAAAA1CAYAAABMS3mcAAAQIUlEQVR4Ae1dCVAVRxpGwUTjieCBqBBAc3jHCF6IQZREjCgG1xuTbFgjq0bd9VgUDR6gEg/wAlcUUeS+7xuPuJpNUkmlkspZqSRu5ahUUjm0cv5bX2OPPfPmzXtc2s+iq4aeN9090//XX//d/feBnZ2dHb0cE02x+2Mp5pWY9qsdg2ZzABwCl8CpVatWkd2ufbuorL6c8sry2692DFrMgfL6CoqO3cEIZhcXv4+9MKMgk1rzyizMIr2rNb9hS+8CFtnFOZRfXkDF1SVUVldOKAhU7OKaEsqvKKCcklyGmS3Jpc0rlFRcfFwjufYe2svIpUeEpjzDRwBOQWURldaVUcWFSuYDyKKqYiqtLaPyhgp2FVYVU25pngJkU75ji3ELKgqZ3Ofz0lmtDl0YSgGBATRpymTyn+FPwSHBFLltC6VknFEwyyrKZpXd1uQFucApNI12LSEXCAUQCiuLqPZKHaWkn6G1G9dRYNCT9KDHgzTAdQA59XEmR6fe1H+AC7l7uNPkKb60cs1Kik9MoOpLNVRSW8ZqtK2BaE1+AXTVpWr61/ZIcnvQvRFwgG5w9evXl54Lf57hgkppzXdkitNq5GLqvb6cVqx+kfr07WMImh6gHTt2pGcWPEPpeRmsiUCzAcLKBFZz84LmLvF0InXv0b3JuHCsQMrKi1WsAjc3H3c6XYvIxZq/0lyquVxL4RF/azZwHEDuT5sRwPojtlhbtQVY9596evavz5pgM2XKFDp37hx98MEH9McffxB31/93nQoLC2nhwoWmaZ7wo/KGSpvR7i0iF7TV2axz5OI6wAQIThTujxgxgp566ikKCgqiSZMmUY8ePSymiY6JZrVVW2C28htN/fTA6So5/fz86IcffuBcsuivW7dOlX6Q22DWh80qzpZeqzebXFD10XtuDTN1+g0zZ86k/Px8Va3UQ/Lq1au0Zs0a6tKliwpETspZwUGsr4L+nK2QChod/au/LF6gkik9I10PAvrtt98YVl9++aVu+Pvvv0+dOnVS3uXl5cneLzseTSYXgMPob/2mfyjCciLAX7JkCf3000+6IFl6eObMGd13+kwYbzMEAz4FlYX0SsJ+lSxXrlzRFf/3339XxXvnnXfMxrvvvvuUuCHzQ6iioVLqCtckcgE4mBY2RW1ShOTE6ty5M3300Ue6wDT14ezZs03e7+3jQ9WXa6UGE5qEj5g5LvDr6+vNQnDo0CGVrOg2mHO//PKLKu6Rfx9VzDgyajGryQViweiXdPqESkCAN2TIEHN4NPt5XNwt45vQ5IYuCGV9MORFRjCRr5KaUlqybLGC0fLlyw0xuHDhghIXWMbGxhrGLyoqUuKj/1XzqrwVzmpyoUbCkizWSNz7+PgYgtGSwDOpps1kTFwsa3ZkJRgKm2Nkb29vlfgpKSnk5uZG27Ztsyr+mDFjlG8kJB2WVntZTS7YWPym+ilCAUBHR0erwGhJpJiYGNU38d2y2jLp7D0gO6zvkdsjlfxGR0cbip6RkUGpqanM9FBTU0MlJSV09uxZQt/TyL399tvKN2Y+HcS6KjJWNqvIlVuWR/HHExSBeM3EKOdOuFmzZqm+jSkSaFGZmkYULqa0Jk+ZrOTVCJu8vDwlHsdT9Ldu3WqUnPr27cvS93PpJx0WvFwskgugQdV7DfFSgXH+/HlD4Vs7UAQe95i7hBWfCyKDD8L3dOzJcBo4cKAhBNnZ2So8tfJt2bLFMP2cOXOU9JibldFUY5FcKMSU8+q+z0MPPWQoeFsEnjihHkgsf345FVUXSzNFhMIFVtweFTI3xBCGnJwchRxaYuG3Jc115MgRJX1yWrKyikKGSsbzYEguaC3UxqDgIEUQCH758mVD4NoqEPOPvCC6de8u1UgJWhTzojx/q1evNoShpeTKzMxUvpWQdETKTr0hucBALI/p2atR1QO4nj17mgXtxx9/pNDQUEVoxPf09KSkpCTdNKh9WC3BCwR+WFgYff/997rxV6xYoYqbfO6UNDUW5DqXk6bkz1Kz1lJyiX22+EQ5R4yG5IKaP52eogCGwt+wYYNuwcNQKJJEe4+5RdE98sgjhvH1tOO1a9dUadb8Yw0zS3A1fDd91ucpyqYOdh1YHlFJjFx6erpKFi1eGzduNErORpU8zdHk45RT2ri48G5ioP22IblgNH05tnEdNBcEBax1mO7h4dzHNNDSpUtVz2F5h9Na4MeOHUsRERHkMkCtxb7++mvtp1TvCwqexTSrFMPwoiy2lq1rt64sj/7+/iZ5Fx9gTnX+/Pk0QCNz165dadGiRWRpwATNyLFOz8+QbnADohmSCx3mJWFqgogA8ft58+YpgkLgGzdu8CD69NNPWRgs1T///DOBMBwU+Dt37lTi4sbDw4NcXV2poaFB9Zz/6NPn9loxD08PNs+ZYWYZtbYmtfXvygtV5OnlqcjH82zka80sXl5eRtGVsMcee4x9BwOIoqoSyizSX0re1jIbvd8suaAN0N/yD3hCAQtk0HPi8hm95gBzYtyJNU7vfWJcnkb0AwMDlfy4DnSlsroyKcwRwKuktpQWCCsh3nzzTTHruvczZsxQ5AEe7u7uuvG0DxEXl9dQL2ZfMyrkuxVmSC4YBb0neCvCd+vWTSsjfffdd0o4hMVErJFbvPj2vNujjz5qFFU3TEzv5NybbXC4W+Bpv5tdkkOpWecUPKZOnaorg/iwOeQS511f2rCWNcdSdA00LYhFcvlM9FHA0pvu+eqrr5RwkOvw4cMidib36GfwWvf444+bhFt6IPbjHJ0cpSIXChj7Bwa7D1ZkfOuttwxFag65OH7w0S+W0YCKimdILiwInOx3ezoDwmgdluh26NA4QkK4pcnXF154QQFej6za92t/h4SEKOn7ufRnTZFWg9zN37CWHz5x28Cph5koU1PJJXYLFi5dJO28oiG5EIjlI0/PeVopTHNAoZ+AMFz9+/cXsWP306ZNo/j4eHZfXFysxEX8Tz75RBW/qqqKfH19za5gHTZsmJLezd2NyuvlmmMEbpjk9/XzVfI5fvx4lYziD2DDsYNvNG20fft2Ja6DgwPra8mqtSySC7P8/9y8QREIwustxcWSEREg9DXeffdd+vjjjyk4OFgJGzVqFMNVtLQjHebZvvjiC5XtBs+RXuvE74zzGcdWY8rU3+B5wcYMcaAzfPhwrSjsN0aLDzzwAPXu3Zu6d+9OHCNtZLYlXljblpx2qlX2mIIEbXWZbRbxQaj4g8fUKyUTExO1crPfsFWJBa93z+1Wly5dshh3/fr1Jt/55ptvVOnCng+Tan6RFxIIBgN0VmE22Ts4KHkGebDbpykO3Q6smRPx3L1vN2tVOJH5d2XzDckFlYsIUMFcuHHjxpnFZuXKlUo8Hh8+VD2mhkT3+uuvs6kkMR6/T0hIEKMq96dOnVK9P3b/HmlrLwoe2OWV5qs0GGTE5hVodiOHgVJ4eLhKXqSNO/SKPIZjC1rPkFyoCdiMMXLMSJWQWqKIIMGAWlFRQSdPnmTN3HvvvScGm9y/8cYblJaWRsnJyVRZWWkSLj5Ak8EJCJ+t6ZLQeMg1CAgG7Y++q8/E8aq8I/9OTk40YcIEtvspKiqKoK2nT5/OKqMoJ+6dnZ3ZLnZs55NdY3H5DckFITDUjdq5TQUM2v877dDxFwGHiQTElx1o5A8tALaa7Y6LIec+zio5RJn07js5dKLV61azQQKIKru8nFjwDcnFI9a8WmcCyI2bt6d47gTRRo8ercoDzpcA2DyPMvucEKio1ZdraO/BfWyzrMsAF5VMnFwg4MRJE2nLy1sbT76pLmFy8vfILKuYN4vkgkDYVh/x0t9VQJgb1bQF0bDOnAMPH5sZMHsgCmJL9wAdTTq26YFwmHjGqTbnc9PZQADNKGyMGK1DLlsjFS8Li+TiESFsb6feqkJ+ae3atuCS6p2ff/656psg15GTR6XtyHO8LPmcMGgysRaMX9xuhXAex9K7ZA1XkQuHv+VXFLI+AoQUBc0rz6djycdNChpapa0cVlGIGgv3gTMDVXsXeT7b/cbykgkHcAmcYmW458BedqodzyBqBNiH5RyoRehAh79oOjw+ePBgq/Prs88+MyGWk7MTYVkLX16CJgV55Plt9+UiGE5IBKcYuWA5x0ZO+NyKPnrsGNYJ5aoXo54JkyaYFPyyZctajWDYv6fVWPgNwyQuEB39lcXLl7B4PL/tfmO5yYID59KtstQ/6S5qx1bC8ZK8D9Bw7QKNGDXChACYxtBbqWot67CiNSAgwOS99h3tKS0njdA0Iw8gWFrueZN4eoRsf6ZfpncBF/MZwYGw6HTyZggaLGCG+uwpnuGRI0daNIiKhPvwww9NlkTzd3Xu0oUyCrLYqIprT8zbDRo8qJ1cwlwjx0ti3zy5Bg4ayDrS6NfwQsbapYi1qwwLGZPY2GaFNeHV1dVUW1tLBQUFFBkZSeLyGT1QvMd7sz6WaDyEnch/ur/hN/XeJf2zDuaxR97RxEgvgzHZjQXEqtSL/72kkIv1fWrLKDUz1arTBa0Fp4OdHUVu30JVl2pYZ503x/gdsTrCZkHGEm9MRN+8eZNwLhcuzCHy1bXffvstU+hDhw5VZEQFhMOeAmvxkzSeMbmQad+pU9hmVK7BeB8IW/5heXZx1bc6WyMwThcMjwhno1KcCs01JHw0wxFrbJdYkJ9vRrl48SJhEh57D7nDrh9vb2/28/r16wqR/vzzT/aMnw1hDY6SxrFMLmR82IhhbDkvM1MIFuS88gLWdCalnKAFSxaSh5eHApI5gTEFEvDkdNpzYA87xRkTsyATSIsL/bz6qw00b756h5G598n8nJ96I+74ycrKYuR5+OGHGVZ819PcuXNp06ZNLOz4cVP7osxymsmbdeRCYiyEO3n2JJXVV6iaLhADGzTxnyAwCMBUBnYiY+nvnoN7KfbAHjp0LJ5Ss86ytViw/MOsoCUq3gOiYQpk2PDbq0/NZNwiiWVIx8mVm5tLmzdvpl27djHyiIeS4IRGOPFQXhny3gp5sJ5c/GOLwhZTRUMFIwqaSq5x4IMguKB90CkHgXDhHs/EppWnQ3wYR2uv1NPGLabHY/Lv2qLPyYU+F5Ym8f2dr732mmJXhFw7duxgBMMf7fEItij3rTw3nVxI6GDvwJaFQAOV1jX+YyqQRCQMJ5roi+Fc22EWYOe+3SRufrVhQFUalZNLPGOe7/Pcv199QC+IhTMz7hXZ7VoqCCzDM2cH0f7DB9ja9qqL1ax5xOnGOEAOGotdZXlsmgnNHk4kxoaGxNNJtOy5ZdSrV697CVCVLJxc4r4CbJWDE1d+YBc1HEaPLS0TidI3T3PpCWDvYE+jx4ym+Yvm07qN6+nAkYN0NPkYm/xGnysqOorQpE70nUjoZ+i94157Jh6miybx119/ZSTCH3FX0/3338+eo/m8hzBoPXLdQ6C0WgGjOcTWOZzIjPnT0tJSwoYX7WEk0Fw4I1XUZvcAnu3kugcKsdUqQytj0U6uVgZU1oK+G/lqJ1c7uVqfAzgW6v+epXM8vT2fEwAAAABJRU5ErkJggg==" alt="" /></p><p><span>This work is licensed under a </span><a href="http://creativecommons.org/licenses/by/3.0/" rel="license" target="_blank">Creative Commons Attribution License</a> CC BY</p><p> </p>Study of second-order Hankel determinant for functions starlike with respect to symmetric points
https://vfast.org/journals/index.php/VTM/article/view/1421
In this paper, our aim is to determined the sharp bounds of coefficients, Fekete-Szegö, Zalcman inequalities for the family SS∗ tanh of starlike function with respect to symmetric points linked with tan hyperbolic function. We also estimate determinant of H2,2Khalil UllahAbbas Qadir2023-03-232023-03-2311Solution of Time-Fractional Third-Order Partial Differential Equations of One and Higher Dimensions
https://vfast.org/journals/index.php/VTM/article/view/1301
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.Safdar AliFozia HanifMuhammad IlyasRehan ShamsMuhammad RehanSyed Inayatullah2023-03-172023-03-1711An Innovative Soft Rough Dual Hesitant Fuzzy Sets and Dual Hesitant Fuzzy Soft Rough Sets
https://vfast.org/journals/index.php/VTM/article/view/1219
This article seeks to demonstrate the novel properties of soft rough dual hesitant fuzzy sets (DHFSRSs) and dual hesitant fuzzy soft rough sets (SRDHFSs). The fundamental characteristics of DHFSRSs and SRDHFSs are thoroughly<br />investigated. Additionally, we present a portrayal hypothesis for the DHFSRSs and SRDHFSs, which demonstrates that the level arrangements of the DHFSRSs and SRDHFSs can be used to characterize both the lower and upper DHFSRSs and SRDHFSs estimates in an identical manner.Tasawar AbbasRehan ZafarSana AnjumAmbreen AyubZamir Hussain2023-03-162023-03-1611The effect of oscillating streams on heat transfer in viscous magnetohydrodynamic MHD fluid flow
https://vfast.org/journals/index.php/VTM/article/view/1386
This study focuses on developing and proving exact solutions for equations of motion involving a fluid with finite conductivity, variable viscosities, and heat transfer in the presence of a transverse magnetic field. By utilizing a transformation variable, the governing equation is transformed into a assortment of simple ordinary differential equations, enabling accurate solutions to be achieved for the problem. The solutions demonstrate that the distribution of vorticity is proportional to the stream function, which is disturbed by oscillating (sine or cosine) or even and exponential streams. This study compares the profiles of steady fluid flow with changing viscosity and heat transfer travelling on a plane. The comparison is used to identify differences in the profiles of providing insight into the changes that occur in interesting factors. These findings have significant implications for understanding fluid dynamics in complex systems and may have important applications in fields such as engineering and physics.Afaque Ahmed BhuttoIftikhar AhmedSaeed Ahmed RajputSyed Asad Raza Shah2023-03-052023-03-0511The Hydrodynamics of Gravity-Driven Vessel Drainage of Third Order Fluid using Perturbation Method
https://vfast.org/journals/index.php/VTM/article/view/1313
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.Azam AliK. N. MemonSyed Feroz ShahMohsin AmurA. M. Siddiqui2022-12-312022-12-3111A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function
https://vfast.org/journals/index.php/VTM/article/view/1288
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.Sanaullah JamaliZubair Ahmed KalhoroAbdul Wasim ShaikhMuhammad Saleem ChandioSanaullah Dehraj2022-12-312022-12-3111A new three step derivative free method using weight function for numerical solution of non-linear equations arises in application problems
https://vfast.org/journals/index.php/VTM/article/view/1289
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.<br />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.Sanaullah JamaliZubair Ahmed KalhoroAbdul Wasim ShaikhMuhammad Saleem ChandioSanaullah Dehraj2022-12-312022-12-3111Improved Mid-Point Derivative based Closed Newton-Cotes Quadrature Rule
https://vfast.org/journals/index.php/VTM/article/view/1344
The main motivation for this work lies in the construction of new and efficient methods to improve the efficiency index of ‘Mid-Point Derivative Based Closed Newton-Cotes Quadrature Rules’. Proposed methods use the derivative values at the mid-points in each strip of integrations such as Mid-Point Derivative Based Closed Trapezoidal, Simpson One Third, Simpson Three eight and Bool’s Rule. The degree of precision and order of accuracy of proposed methods are higher than all the existing methods. Furthermore, error terms of the proposed methods are calculated by using the concept of precision. An extensive comparison of the proposed formulas, classical and mid-point quadrature rules for the number of function evaluation, error terms, coefficient of error terms and results obtained from some different problems are given. The comparisons illustrate that the new proposed Closed Newton-Cotes Rules are much superior to Classical Rules and Zhao and Li’s ‘Mid-Point Derivative Based Closed Newton-Cotes Quadrature’ schemes.Mir Sarfraz Khalil SaandShakeel RindZuabir AhmedAbdul WasimOwais Ali2022-12-312022-12-3111Analysis of Two-Level Complex Shifted Laplace Preconditioner and Deflation-Based Preconditioner for Helmholtz Equation
https://vfast.org/journals/index.php/VTM/article/view/1304
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.Rao Faisal RajputHanan ShiekhK. B. Amur2022-12-312022-12-3111The melting behavior of Paraffin RT-50 in a finned cylindrical surface
https://vfast.org/journals/index.php/VTM/article/view/1320
<p>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.</p>Waris AliAsif Ali ShaikhFeroz ShahSajjad Hussain2022-12-312022-12-3111Effect of Slip Condition on Unsteady Tank Drainage Flow of third Order Fluid
https://vfast.org/journals/index.php/VTM/article/view/1338
Unsteady, isothermal, and incompressible third-order fluid is considered in the tank, cylindrical coordinates are taken into account and the flow of the fluid is considered in the z-direction. The behavior of viscoelastic fluid is analyzed with slip conditions. Velocity profile, flow rate, and mathematical relation between time depth of mass and time efflux, etc. are calculated with the perturbation method. Perturbation is a suitable technique for nonlinear partial differential equations because it converts them into the linear form which is easy to solve. Different physical flow parameters are inspected graphically. A comparison of slip and no-slip is presented also Newtonian and third-order fluids are compared too.Muneer Ahmed MaharKamran Nazir MemonSyed Feroz ShahAzam Ali AmurA. M. Siddiqui2022-12-312022-12-3111Exact solution on the impact of slip condition for unsteady tank drainage ﬂow of Ellis ﬂuid
https://vfast.org/journals/index.php/VTM/article/view/1352
<p>In this paper, we look into the effect of slip condition on isothermal and incompressible Ellis fluid of an unsteady tank drainage flow. The non-linear PDE (partial differential equation) is solved exactly by applying the governing continuity and momentum equations, subject to the proper boundary condition, using the separation of variables approach. Unique situations this model put out by Ellis fluid is used to develop concepts like Newtonian, Power law model, and Bingham Plastic model solution. On setting the slip parameterexact solution for Ellis fluid flow is retrivred as well as Newtonian solution is bring back, which was done through Bernoulli's equation. Expressions for velocity field, pipe shear stress, volume flux, velocity average, depth of fluid in the tank at different times and also the relationship between length of the time be different with depth of the tank and the length of time required to complete the drainage is determined. Graphical representation is given of the effects of various development factors on the velocity field <em>Vz</em> and fluid depth <em>H(t</em>). The tank can empty faster for Ellis fluid compared to its special situations, according to the analogy of Ellis, Power law, Newtonian, and Binghan plastic fluids for the relation of depth with respect to time.<strong></strong></p>Naina Salar ShaikhKamran Nazir MemonMuhammad Suleman SialA. M. Siddiqui2022-12-312022-12-3111Optimum solutions of partial diﬀerential equation with initial condition using optimization techniques
https://vfast.org/journals/index.php/VTM/article/view/1170
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<br />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.Muhammad BilalShakoor MuhammadNekmat UllahFazal HananSubahan Ullah2022-12-172022-12-1711Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation
https://vfast.org/journals/index.php/VTM/article/view/1211
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.Liaquat Ali ZardariShakeel Ahmed KambohAbbas Ali GhotoDr. Kirshan Kumar LuhanaDr. Shah Zaman Nizamani2022-12-092022-12-0911Power Hamy Mean Operators for managing Cubic Linguistic Spherical Fuzzy Sets and their Applications
https://vfast.org/journals/index.php/VTM/article/view/1147
<p>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.</p><p><strong> </strong></p>Tasawar AbbasFaisal MumtazZamir HussainRehan Zafar2022-11-282022-11-2811Assessment of the COVID-19 Pandemic's Impact on Gasoline Prices in Pakistan
https://vfast.org/journals/index.php/VTM/article/view/1153
<p>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,<br />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.</p>Muhammad BilalMuhammad AamirSaleem AbdullahNoor MahmoodUmair KhalilNida KhalidMaqbool AhmedMuhammad NaeemShakoor MuhammadLaiba Sultan Dar2022-11-172022-11-1711Analysis of Entropy-dependent solitons in ETG-driven magneto plasma by Variational Iteration Method
https://vfast.org/journals/index.php/VTM/article/view/1079
<p>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. The linear dispersion relation and KdV equation are created after the entropy drift is taken into account in the ETG mode. 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 impact of inhomogeneity drift and the magnetic ﬁeld on the conﬁguration 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 in magnetically restrained plasmas, solitary waves can be observed. The graphical ﬁndings are also provided using Maple-18.</p>Tasawar AbbasEhsan Ul HaqAmbreen AyubAsadullah Dawood2022-11-142022-11-1411Using Crank-Nicolson’s scheme to discretize the Laplacian in a polar gird system with symmetric and asymmetric lines
https://vfast.org/journals/index.php/VTM/article/view/1151
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<br />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.Rabnawaz MallahWajid Ahmed SiyalSaira AslamMuhammad Suleman SialInayatullah Soomro2022-11-122022-11-1211Analytical Solution of Transient Flow of Fractional Oldroyd-B Fluid between Oscillating Cylinders
https://vfast.org/journals/index.php/VTM/article/view/1184
<p>This paper investigates the fractional Oldroyd-B ﬂuid ﬂow across two interminable coaxial cylinders, where the ﬂuid’s motion is generated by the oscillatory movement of cylinders and the oscillating pressure gradient. The proﬁle of velocity and shear stress of the ﬂow is derived with the assistance of Caputo fractional derivative utilizing an analytical technique, ﬁnite Hankel transform and Laplace transforms. The semi-analytical solution is then displayed as generalized functions of G and R satisfying fundamental constitutive equations and all initial and boundary conditions. To validate the results, some limitations have been imposed on the determined equations and the results have been contrasted against previous results. Moreover, the inﬂuence of various parameters on the ﬂow of fractionalized Oldroyd-B ﬂuid is investigated and depicted graphically.</p>Khadija ShaikhFozia ShaikhRahim Bux KhokharK.N. Memon2022-11-122022-11-1211Delta Perturbation Method for Couette-Poiseuille flows in Third grade fluids
https://vfast.org/journals/index.php/VTM/article/view/1179
<p>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</p>Kamran Nazir MemonAhsan MushtaqueFozia ShaikhAA GhotoA. M. Siddiqui2022-11-122022-11-1211Unsteady Flow of an MHD Tangent Hyperbolic Nanoﬂuid Over a Stretching Sheet
https://vfast.org/journals/index.php/VTM/article/view/1215
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.Muhammad Asif JamalM. FaizanAhmed FaridFozia ShaikhFozia hanif2022-06-302022-06-3011Image Driven Isotropic Diffusivity and Complementary Regularization Approach for Image Denoising Problem
https://vfast.org/journals/index.php/VTM/article/view/1186
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<br />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.Memoona PirzadaKhuda Bux AmurMuzaffar Bashir ArainRajab Ali Malookani2022-06-302022-06-3011Algebra Based Encryption and Decryption Algorithms
https://vfast.org/journals/index.php/VTM/article/view/1188
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<br />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<br />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.<br />3Shakeel Ahmed KambohSuhail Aslam KhaskheliAbbas Ali GhotoSunny Kumar AassoriMuzaffar Bashir Arain2022-06-302022-06-30113D Finite Difference Formulation and Simulation of EHD Ion-Drag Model
https://vfast.org/journals/index.php/VTM/article/view/1189
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<br />simulation package COMSOL Multiphysics. The comparison reveals that the numerical simulation results obtained by both the methods are appreciably close<br />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.Shakeel Ahmed KambohSakina KambohAbbas GhotoFozia ShaikhNawab ahmedKamran Nazir Memon2022-06-302022-06-3011Exact solution of some nonlinear PDE$ {s} $ with conformable fractional derivative
https://vfast.org/journals/index.php/VTM/article/view/1028
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.Alireza Mohammadpour2022-04-222022-04-2211Upper Bound of the Third Hankel Determinant for a Subclass of Analytic Functions Subordinate to Cosine Function
https://vfast.org/journals/index.php/VTM/article/view/392
<p>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.</p>Khurshid AhmadSerkan AraciMirajul HaqBilal Khan2022-04-162022-04-1611Upper Bound of the Third Hankel Determinant for a Subclass of Multivalent Functions Associated with the Bernoulli Lemniscate
https://vfast.org/journals/index.php/VTM/article/view/1022
<p>Let RL; SL and CL represent the families of multivalent bounded turning, multivalent starlike, and multivalent convex functions that are subordinated with Bernoulli lemniscate in the open unit disk E = {z : |z|<1}. In this particular paper, our goal is to find the upper bounds of Hankel’s third order determinant for the above-mentioned families.</p>Khalil UllahJihad YounisKhurshid AhmadA ManickamBilal KhanMirajul Haq2021-12-312021-12-3111A Study of Fourth Hankel Determinant of Certain Analytic Function
https://vfast.org/journals/index.php/VTM/article/view/1019
<p>The main motive of this paper is to nd an upper bound of the fourth Hankel determinant H 4;1 (f) for a subclass S; with hyperbolic domain.</p>Neelam KhanNazar KhanBasem Aref FrasinMirajul HaqBilal Khan2021-12-312021-12-3111Structures of Digraphs Arizing from Lambert Type Maps
https://vfast.org/journals/index.php/VTM/article/view/1021
<p>The Well-known function W e<sup>W</sup> is called Lambert Map. This map has been viewed by many researchers for finding the approximate solutions of exponential function especially in numerical analysis. Later, it has been incorporated in number theory for finding integral solutions of exponential congruences under a fixed modulus. Instead of We<sup>W</sup>, we use the function W<sup>2 </sup>e <sup>W</sup> and call this function as Discrete Lambert Type Function (DLTF). In this work, we produce graphs using DLTF, and discuss their structures. We show that the digraphs over DLTF satisfy many structures as these have been followed for using We<sup>W</sup>. It would be of great interest, if these results could be generalized for We<sup>W</sup>for all integers n, in the future.</p>Tayyiba SabahatSufyan AsifAsif Abd ur Rehman2021-12-312021-12-3111Upper Bound Sequences of Rotationally Symmetric Triangular Prism Constructed as Halin Graph Using Local Fractional Metric Dimension
https://vfast.org/journals/index.php/VTM/article/view/1020
<p>In this paper, we consider rotationally symmetric traingular planar network with possible planar symmetries. We ﬁnd local fractional metric dimension of planar symmetries. The objective is to search sequences of local fractional metric dimension of triangular prism planar networks by joining diﬀerent copies. We propose and prove generalized formulas of all sequences for local fractinal metric<br />dimension over triangular prism.</p>Mamoona FarooqAsif Abd ul RehmanM. Khalid MahmoodDaud Ahmad2021-12-312021-12-3111On Fixed Points of Digraphs Over Lambert Type Map
https://vfast.org/journals/index.php/VTM/article/view/1023
<p>Define f(y)=y<sup>2</sup>h<sup>y</sup> where h belongs to (Z/mZ), the Discrete Lambert Type Map (DLTM). For a set of vertices and edges over DLTM, diagraphs are obtained in which the vertices are from a whole range of residues modulo a fixed integer s, and edges are obtained when f (y) ≡ v(mods<sup>K</sup>) is solvable in Y and in terms of Diophantine equation as well. In this paper we proposed new results for for fixed point of digraphs over DLTM.</p>Tayyiba SabahatSufyan AsifAsif Abd ur Rehman2021-12-312021-12-3111Upper Bound of the Third Hankel Determinant for a Subclass of Analytic Functions Connected with Sine Functions
https://vfast.org/journals/index.php/VTM/article/view/273
Let RL sin and SL represents the families of multivalent bounded<br />turning and multivalent starlike functions that are subordinate with sine function in the open unit disk E = fz : jzj < 1g : For these families our aim is to nd the bounds of Hankel determinant of order three. Further, the estimate of third order Hankel determinant for the family SL sin in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions. Also we discuss H (3) determinant for the above mentioned families. 2 sin<br /><br />Khalil UllahQaiser KhanKhurshid AhmadA. ManickamMirajul HaqBilal Khan2020-12-312020-12-3111A Study of Third Hankel Determinant for Certain Subclasses of Analytic Function
https://vfast.org/journals/index.php/VTM/article/view/366
<p>Recently the Hankel determinant problems got attractions of many well-known authors. Third Hankel determinant problems were determined for different subclasses of analytic functions. Here in our present investigation, we define certain new subclasses of analytic functions and then we obtain the upper bonds for the third Hankel determinant.</p>Hifsa BibiBilal KhanKhurshid AhmadG. MurugusundaramoorthyJihad YounisMirajul Haq2020-12-312020-12-3111Implementation of Discrete Search Optimization Algorithms in Parallel and Their Performance Analysis
https://vfast.org/journals/index.php/VTM/article/view/576
The present paper discusses the implementation of the discrete search optimization techniques on a parallel platform (SGI-Altix 450 shared memory 64 processors system). We show that the combination of Asynchronous Parallel Iterative Deepening and Parallel Window Search technique tends to give more challenging speedups, memory consumption, and efficiency with less resource consumption as compared to the rest of the techniques. In general 5 techniques are compared with Parallel Asynchronous Window Search technique among which includes Depth First Search, Parallel Iterative Deepening, Parallel A*, Parallel Window Search and Parallel Asynchronous Iterative Deepening A*.Muhammad Hanif DuradMohammad ZulqurnainAnila UsmanIdrees Ahmad2020-11-182020-11-1811Stability Analysis of SEIVHR Epidemic Model with Saturated Incidence Rate
https://vfast.org/journals/index.php/VTM/article/view/303
The aim of this paper, to analyze stability analysis of SEIVHR epidemic model with a generalized non-linear incidence rate that spread in the host population horizontally. First, we formulate the model and find its basic reproduction number. Two equilibrium exists, namely; the disease-free and endemic equilibrium. The disease-free equilibrium is stable both locally and globally when the threshold quantity less than unity. The endemic equilibrium is locally and globally asymptotically stable when the threshold exceeds unity. Finally, we show some numerical results for the proposed model.Muhammad Ali KhanSehra .2020-03-182020-03-1811More Modular Arithmetic in Five Regular Partitions by Jacobi Triple Product Formula
https://vfast.org/journals/index.php/VTM/article/view/426
In a paper, Calkin et al., Divisibility properties of the 5-regular and 13-regular partition functions, Integers 8 (2008), [#A60], authors have discussed interesting properties for 5-regular partition functions of integers. In the continuation of this paper, we have obtained and conjectured various interesting results. In this note, we use nothing more than Jacobi's triple product identity to obtain results for 5-Regular Partitions that are stronger than those obtained by Calkin and his collaborators. The motivation for this paper is an observation that some generating functions of 5-Regular partitions are congruent to functions related to the Ramanujan's Q-series.Aneela .M. Khalid MahmoodDaud AhmadShahbaz Ali2019-12-312019-12-3111Parallelization of Encryption Algorithms using MPI
https://vfast.org/journals/index.php/VTM/article/view/577
<em>Encryption is a basic technique to achieve data confidentiality. A number of algorithms including RSA, DES, Blowfish, and AES have been parallelized using MPI and have been employed for practical file encryption. Unified performance analysis of all these algorithms has been presented using two ECB and counter modes. On basis of experimental results, some guidelines have been suggested for the end-user to select the appropriate algorithm for achieving enhanced speed up</em>Muhammad Hanif DuradAhmad RazaAli AsadMuhammad Naveed Akhtar2019-12-312019-12-3111Unified Computational Analysis of Conventional Numerical Methods for Time Dependant Heat Equation
https://vfast.org/journals/index.php/VTM/article/view/579
<em>The objective of this paper is to perform a unified error and computation time analysis of conventional numerical methods for solving the heat equation. The numerical techniques employed include Forward Difference, Backward Difference, Crank Nicolson, Alternate Directions Scheme, and DuFort-Frankel methods for the time-dependent heat equation. The heat equation has been implemented to 1-D, 2-D, and 3-D problems, both for the Neumann and Dirichlet boundary conditions. On basis of experimental results, theoretical justifications have been provided regarding the performance of each method.</em>Muhammad Naveed AkhtarMuhammad Hanif DuradAnila UsmanIrfan ul Haq2019-12-312019-12-3111Extension of Sufficient Conditions for P–Valent Starlike and Convex Function
https://vfast.org/journals/index.php/VTM/article/view/418
In this article, we consider a subclass of p-valent analytic functions which generalize the classes of p-valent starlike and p-valent convex functions of complex order. For this class, certain simple sufficiency criteria are obtained and further some applications to the generalized Alexander integral operator is also given. Several known results appear as special cases of our workM ArifMuhammad AyazMuhammad Asad2019-04-092019-04-0911Probabilistic Picture Graph
https://vfast.org/journals/index.php/VTM/article/view/394
The probabilistic graphs is an ecient mathematical tool to deal with uncertain real life problems. Probabilistic picture graph is an extension of probabilistic graphs corresponding to a given joint probability distribution. It can work very eciently in uncertain scenarios which involve more answers to these type, yes and no. In this work. we redene and introduce the idea of probabilistic graphs and probabilistic picture graphs. Some types of probabilistic picture graphs such as a strong probabilistic picture graph, complete probabilistic picture graph and complement of probabilistic picture graph are introduced and some properties are also described. The six operations namely cartesian product, composition, union, direct product, lexicographical product and strong product on probabilistic picture graph are also dened. Finally, we describe the utility of probabilistic graphs by revisiting application in a joint venture.Sidra YousafAsif Abd ur Rehman2018-12-172018-12-1711