Browsing by Author "Ghaffar, Abdul"

Now showing 1 - 20 of 21

Citation Count: Ghaffar, Abdul...et al. (2019). "A new class of 2m-point binary non-stationary subdivision schemes", Advances in Difference Equations, Vol. 2019, No. 1.
A new class of 2m-point binary non-stationary subdivision schemes
(Springer Open, 2019) Ghaffar, Abdul; Ullah, Zafar; Bari, Mehwish; Nisar, Kottakkaran Sooppy; Al-Qurashi, Maysaa M.; Baleanu, Dumitru; 56389
A new class of 2m-point non-stationary subdivision schemes (SSs) is presented, including some of their important properties, such as continuity, curvature, torsion monotonicity, and convexity preservation. The multivariate analysis of subdivision schemes is extended to a class of non-stationary schemes which are asymptotically equivalent to converging stationary or non-stationary schemes. A comparison between the proposed schemes, their stationary counterparts and some existing non-stationary schemes has been depicted through examples. It is observed that the proposed SSs give better approximation and more effective results.
Citation Count: Ghaffar, A...et al. (2019). "A New Class of 2Q-Point Nonstationary Subdivision Schemes and Their Applications",Mathematics, Vol. 7, No. 7.
A New Class of 2Q-Point Nonstationary Subdivision Schemes and Their Applications
(MDPI AG, 2019) Ghaffar, Abdul; Bari, Mehwish; Ullah, Zafar; Iqbal, Mudassar; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
The main objective of this study is to introduce a new class of 2q-point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs
Citation Count: Ghaffar, Abdul...et al. (2020). "A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order", Advances in Difference Equations, Vol. 2020, No. 1.
A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order
(2020) Ghaffar, Abdul; Ali, Ayyaz; Ahmed, Sarfaraz; Akram, Saima; Junjua, Moin-ud-Din; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; 56389
We investigate some solitary wave results of time fractional evolution equations. By employing the extended rationalexp((-psi '/psi)(eta))-expansion method, a few different results including kink, singular-kink, multiple soliton, and periodic wave solutions are formally generated. It is worth mentioning that the solutions obtained are more general with more parameters. The exact solutions are constructed in the form of exponential, trigonometric, rational, and hyperbolic functions. With the choice of proper values of parameters, graphs to some of the obtained solutions are drawn. On comparing some special cases, our solutions are in good agreement with the results published previously and the remaining are new.
Citation Count: Shahzad, Aamir...et al. (2020). "A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes", Symmetry-Basel, vol. 12, No. 1.
A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes
(2020) Shahzad, Aamir; Khan, Faheem; Ghaffar, Abdul; Mustafa, Ghulam; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.
Citation Count: Ashraf, Pakeeza...et al. (2021). "A shape-preserving variant of Lane-Riesenfeld algorithm", AIMS Mathematics, Vol. 6, No. 3, pp. 2152-2170.
A shape-preserving variant of Lane-Riesenfeld algorithm
(2021) Ashraf, Pakeeza; Mustafa, Ghulam; Khan, Husna A.; Baleanu, Dumitru; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; 56389
This paper introduces a family of shape-preserving binary approximating subdivision schemes by applying a shape-preserving variant on the Lane-Riesenfeld algorithm. Using the symbols of subdivision schemes, we determine convergence and smoothness, Holder continuity, and support size of the limit curves. Furthermore, these schemes produce monotonic and convex curves under the certain conditions imposed on the initial data.
Citation Count: Mustafa, Ghulam...et al. (2020). "A subdivision-based approach for singularly perturbed boundary value problem", Advances in Difference Equations, Vol. 2020, No. 1.
A subdivision-based approach for singularly perturbed boundary value problem
(2020) Mustafa, Ghulam; Ejaz, Syeda Tehmina; Baleanu, Dumitru; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; 56389
A numerical approach for solving second order singularly perturbed boundary value problems (SPBVPs) is introduced in this paper. This approach is based on the basis function of a 6-point interpolatory subdivision scheme. The numerical results along with the convergence, comparison and error estimation of the proposed approach are also presented.
Citation Count: Ashraf, P...et al. (2020). "Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations",Mathematics, Vol. 8, No. 3.
Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme
(MDPI AG, 2020) Pakeeza, Ashraf; Bushra, Nawaz,; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; Ghaffar, Abdul; Ahmed Khan, Muhammad Aqeel; Akram, Saima; 56389
Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.
Citation Count: Bari, Mehwish...et al. (2021). "Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces", Advances in Difference Equations, Vol. 2021, No. 1.
Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces
(2021) Bari, Mehwish; Mustafa, Ghulam; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
Subdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having C1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence >= 3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409-420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.
Citation Count: Ghaffar, Abdul...et al. (2019). "Construction and Application of Nine-Tic B-Spline Tensor Product SS", Mathematics, Vol. 7, No. 8.
Construction and Application of Nine-Tic B-Spline Tensor Product SS
(MDPI, 2019) Ghaffar, Abdul; Iqbal, Mudassar; Bari, Mehwish; Hussain, Sardar Muhammad; Manzoor, Raheela; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
In this paper, we propose and analyze a tensor product of nine-tic B-spline subdivision scheme (SS) to reduce the execution time needed to compute the subdivision process of quad meshes. We discuss some essential features of the proposed SS such as continuity, polynomial generation, joint spectral radius, holder regularity and limit stencil. Some results of the SS using surface modeling with the help of computer programming are shown.
Citation Count: Ali, F.A.M...et al. (2020). "Construction of Cubic Timmer Triangular Patches and Its Application in Scattered Data Interpolation", Mathematics, Vol. 8, No. 2.
Construction of Cubic Timmer Triangular Patches and Its Application in Scattered Data Interpolation
(MDPI AG, 2020) Ali, Fatin Amani Mohd; Karim, Samsul Ariffin Abdul; Saaban, Azizan; Hasan, Mohammad Khatim; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown's validation.
Citation Count: Karim, S.A.A...et al. (2020). "Construction of New Cubic Bézier-Like Triangular Patches With Application in Scattered Data İnterpolation", Advances in Difference Equations, Vol. 2020, No. 1.
Construction of New Cubic Bézier-Like Triangular Patches With Application in Scattered Data İnterpolation
(Springer, 2020) Karim, Samsul Ariffin Abdul; Saaban, Azizan; Skala, Vaclav; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
This paper discusses the functional scattered data interpolation to interpolate the general scattered data. Compared with the previous works, we construct a new cubic Bézier-like triangular basis function controlled by three shape parameters. This is an advantage compared with the existing schemes since it gives more flexibility for the shape design in geometric modeling. By choosing some suitable value of the parameters, this new triangular basis is reduced to the cubic Ball and cubic Bézier triangular patches, respectively. In order to apply the proposed bases to general scattered data, firstly the data is triangulated using Delaunay triangulation. Then the sufficient condition for C1 continuity using cubic precision method on each adjacent triangle is implemented. Finally, the interpolation scheme is constructed based on a convex combination between three local schemes of the cubic Bézier-like triangular patches. The detail comparison in terms of maximum error and coefficient of determination r2 with some existing meshfree methods i.e. radial basis function (RBF) such as linear, thin plate spline (TPS), Gaussian, and multiquadric are presented. From graphical results, the proposed scheme gives more visually pleasing interpolating surfaces compared with all RBF methods. Based on error analysis, for all four functions, the proposed scheme is better than RBFs except for data from the third function. Overall, the proposed scheme gives r2 value between 0.99920443 and 0.99999994. This is very good for surface fitting for a large scattered data set.
Citation Count: Ghaffar, Abdul...et al. (2019). "Family of odd point non-stationary subdivision schemes and their applications", Advances in Difference Equations.
Family of odd point non-stationary subdivision schemes and their applications
(Springer Open, 2019) Ghaffar, Abdul; Ullah, Zafar; Bari, Mehwish; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
The (2s-1)-point non-stationary binary subdivision schemes (SSs) for curve design are introduced for any integer s2. The Lagrange polynomials are used to construct a new family of schemes that can reproduce polynomials of degree (2s-2). The usefulness of the schemes is illustrated in the examples. Moreover, the new schemes are the non-stationary counterparts of the stationary schemes of (Daniel and Shunmugaraj in 3rd International Conference on Geometric Modeling and Imaging, pp.3-8, 2008; Hassan and Dodgson in Curve and Surface Fitting: Sant-Malo 2002, pp.199-208, 2003; Hormann and Sabin in Comput. Aided Geom. Des. 25:41-52, 2008; Mustafa et al. in Lobachevskii J. Math. 30(2):138-145, 2009; Siddiqi and Ahmad in Appl. Math. Lett. 20:707-711, 2007; Siddiqi and Rehan in Appl. Math. Comput. 216:970-982, 2010; Siddiqi and Rehan in Eur. J. Sci. Res. 32(4):553-561, 2009). Furthermore, it is concluded that the basic shapes in terms of limiting curves produced by the proposed schemes with fewer initial control points have less tendency to depart from their tangent as well as their osculating plane compared to the limiting curves produced by existing non-stationary subdivision schemes.
Citation Count: Hussain, Sardar Muhammad...et al. (2020). "Generalized 5-Point Approximating Subdivision Scheme of Varying Arity", Mathematics, Vol. 8, No. 4.
Generalized 5-Point Approximating Subdivision Scheme of Varying Arity
(2020) Hussain, Sardar Muhammad; Rehman, Aziz Ur; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; Ghaffar, Abdul; Abdul Karim, Samsul Ariffin; 56389
The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Holder regularity, limit stencils, error bound, and shape of limit curves. The efficiency of the scheme is also depicted with assuming different values of shape parameter along with its application.
Citation Count: Rashid, Maliha...et al. (2020). "Multidimensional fixed points in generalized distance spaces", ADVANCES IN DIFFERENCE EQUATIONS, Vol. 2020, No. 1.
Multidimensional fixed points in generalized distance spaces
(2020) Rashid, Maliha; Bibi, Rabia; Kalsoom, Amna; Baleanu, Dumitru; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; 56389
The main aim of this paper is to study distance spaces, to provide some useful remarks with examples regarding distance spaces, and to establish multiple fixed point results for aC-distance space in the presence of different contractive conditions. This concept allows us to reduce the multidimensional case to a one-dimensional case.
Citation Count: Manzoor, Raheela...et al. (2020). "Numerical Analysis of Fluid Forces for Flow Past a Square Rod with Detached Dual Control Rods at Various Gap Spacing", Symmetry-Basel, Vol. 12, No. 1.
Numerical Analysis of Fluid Forces for Flow Past a Square Rod with Detached Dual Control Rods at Various Gap Spacing
(2020) Manzoor, Raheela; Ghaffar, Abdul; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; 56389
A two-dimensional numerical study was conducted for flow past a square rod in the presence of two control rods. One is placed vertically in the upstream direction and the second one is placed horizontally in the downstream direction of the square rod. The influence of gap spacing was studied by taking g(1) = 1-5 and g(2) = 0.5-5 (where g(1) is the gap between the upstream control rod and the main rod, and g(2) is the space between the main rod and the downstream control rod) at Re = 160. The simulation results were obtained in the form of vorticity contour, drag and lift coefficients, Strouhal number, and force statistics. Under the effect of gap spacing, three different flow modes were found and named according to their behavior. It was found that the mean drag coefficient showed decreasing behavior by increasing the value of g(2) continually at a fixed value of g(1). The largest value of Cdmean was found at (g(1), g(2)) = (1, 1) and the greatest percentage reduction in Cdmean was obtained at (g(1), g(2)) = (1, 3), which is 139.72%. The effect of thrust was also noticed for all selected values of g(1) and g(2). Furthermore, it was noticed that the Strouhal number and the root mean square values of the drag and lift coefficients smaller values than the single rod values, except for the Clrms value of (g(1), g(2)) = (1, 3) and (1, 4).
Citation Count: Khalid, Aasma...et al. (2019). "Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells", Mathematics, Vol. 7, No. 6.
Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells
(MDPI, 2019) Khalid, Aasma; Naeem, Muhammad Nawaz; Ullah, Zafar; Ghaffar, Abdul; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; (Al-Qurashi, Maysaa M.; 56389
This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution.
Citation Count: Akram, Saima...et al. (2020). "Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation", Frontiers in Physics, Vol. 8.
Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation
(2020) Akram, Saima; Nawaz, Allah; Yasmin, Nusrat; Ghaffar, Abdul; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; 56389
In this paper, the periodic solutions of a certain one-dimensional differential equation are investigated for the first order cubic non-autonomous equation. The method used here is the bifurcation of periodic solutions from a fine focus z = 0. We aimed to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. For classes C3, 8, C4, 3, C7, 5, C7, 6, eight periodic multiplicities have been found. To investigate the multiplicity >9, the formula for the focal value was not available in the literature. We also succeeded in constructing the formula for η10. By implementing our newly developed formula, we are able to get multiplicity ten for classes C7, 3, C9, 1, which is the highest known to date. A perturbation method has been properly established for making the maximal number of limit cycles for each class. Some examples are also presented to show the implementation of the newly developed method. By considering all of these facts, it can be concluded that the presented methods are new, authentic, and novel.
Citation Count: Harim, Noor Adilla...et al. (2020). "Positivity preserving interpolation by using rational quartic spline", AIMS Mathematics, Vol. 5, No. 4, pp. 3762-3782.
Positivity preserving interpolation by using rational quartic spline
(2020) Harim, Noor Adilla; Karim, Samsul Ariffin Abdul; Othman, Mahmod; Saaban, Azizan; Ghaffar, Abdul; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
In this study, a new scheme for positivity preserving interpolation is proposed by using C-1 rational quartic spline of (quartic/quadratic) with three parameters. The sufficient condition for the positivity rational quartic interpolant is derived on one parameter meanwhile the other two are free parameters for shape modification. These conditions will guarantee to provide positive interpolating curve everywhere. We tested the proposed positive preserving scheme with four positive data and compared the results with other established schemes. Based on the graphical and numerical results, we found that the proposed scheme is better than existing schemes, since it has extra free parameter to control the positive interpolating curve.
Citation Count: Ashraf, Pakeeza...et al. (2020). "Shape-Preserving Properties of a Relaxed Four-Point Interpolating Subdivision Scheme", Mathematics, Vol. 8, No. 5.
Shape-Preserving Properties of a Relaxed Four-Point Interpolating Subdivision Scheme
(2020) Ashraf, Pakeeza; Ghaffar, Abdul; Baleanu, Dumitru; Sehar, Irem; Nisar, Kottakkaran Sooppy; Khan, Faheem; 56389
In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.
Citation Count: Khalid, Aasma...et al.(2021). "Solutions of BVPs arising in hydrodynamic and magnetohydro-dynamic stability theory using polynomial and non-polynomial splines", Alexandria Engineering Journal, Vol. 60, No. 1, pp. 941-953.
Solutions of BVPs arising in hydrodynamic and magnetohydro-dynamic stability theory using polynomial and non-polynomial splines
(2021) Khalid, Aasma; Ghaffar, Abdul; Naeem, M. Nawaz; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
This paper describes the exceptionally precise results of 6th-order and 8th-order nonlinear boundary-value problems(BVPs). Cubic-Nonpolynomial spline(CNPS) and Cubic-polynomial spline(CNPS) are utilized to solve such types of BVPs. We develop the class of numerical techniques for a particular selection of the factors that are associated with nonpolynomial and polynomial splines. Using the developed class of numerical techniques, the problem is reduced to a new system that consists of 2nd-order BVPs only. The end conditions associated with the BVPs are determined. For each problem, the results obtained by CNPS and CPS is compared with the exact solution. The absolute error(AE) for every iteration is calculated. To show that the suitable responses established by using CNPS and CPS have a higher level of preciseness, the absolute errors of the CNPS and CPS have been compared with different techniques such as DTM, ADM, Parametric septic splines, Variational-iteration method(VIM), Daftardar Jafari strategy, MDM, Cubic B-Spline, Homotopy method(HM), Quintic and Sextic B-spline and observed to be more accurate. Graphs that describe the graphical comparison of CNPS and CPS at n=10 are also included in this paper.