Browsing by Author "Mustafa, Ghulam"

Now showing 1 - 13 of 13

Citation Count: Mustafa, Ghulam...et al. (2020). "A 6-point subdivision scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems", Mathematical Biosciences and Engineering, Vol. 17, No. 6, pp. 6659-6677.
A 6-point subdivision scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems
(2020) Mustafa, Ghulam; Baleanu, Dumitru; Ejaz, Syeda Tehmina; Anju, Kaweeta; Ahmadian, Ali; Salahshour, Soheil; Ferrara, Massimiliano; 56389
In this paper, we first present a 6-point binary interpolating subdivision scheme (BISS) which produces a C 2 continuous curve and 4th order of approximation. Then as an application of the scheme, we develop an iterative algorithm for the solution of 2nd order nonlinear singularly perturbed boundary value problems (NSPBVP). The convergence of an iterative algorithm has also been presented. The 2nd order NSPBVP arising from combustion, chemical reactor theory, nuclear engineering, control theory, elasticity, and fluid mechanics can be solved by an iterative algorithm with 4th order of approximation.
Citation Count: Mustafa, Ghulam...et al. (2020). "A Class of Refinement Schemes With Two Shape Control Parameters", IEEE Access, Vol. 8, pp. 98316-98329.
A Class of Refinement Schemes With Two Shape Control Parameters
(2020) Mustafa, Ghulam; Hameed, Rabia; Baleanu, Dumitru; Mahmood, Ayesha; 56389
A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.
Citation Count: Khan, Faheem...et al. (2020). "A Computational Method for Subdivision Depth of Ternary Schemes", Mathematics, Vol. 8, No. 5.
A Computational Method for Subdivision Depth of Ternary Schemes
(2020) Khan, Faheem; Mustafa, Ghulam; Shahzad, Aamir; Baleanu, Dumitru; M. Al-Qurashi, Maysaa; 56389
Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon atk-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.
Citation Count: Hameed, Rabia...et al. (2021). "A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes", Advances in Difference Equations, Vol. 2021, No. 1.
A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes
(2021) Hameed, Rabia; Mustafa, Ghulam; Baleanu, Dumitru; Chu, Yu-Ming; 56389
In this article, we present the continuity analysis of the 3D models produced by the tensor product scheme of (m + 1)-point binary refinement scheme. We use differences and divided differences of the bivariate refinement scheme to analyze its smoothness. The C-0, C(1 )and C-2 continuity of the general bivariate scheme is analyzed in our approach. This gives us some simple conditions in the form of arithmetic expressions and inequalities. These conditions require the mask and the complexity of the given refinement scheme to analyze its smoothness. Moreover, we perform several experiments by using these conditions on established schemes to verify the correctness of our approach. These experiments show that our results are easy to implement and are applicable for both interpolatory and approximating types of the schemes.
Citation Count: Hameed, Rabia...et al. (2020). "A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision Scheme", Mathematical Problems in Engineering, Vol. 2020.
A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision Scheme
(2020) Hameed, Rabia; Mustafa, Ghulam; Liaqat, Amina; Baleanu, Dumitru; Khan, Faheem; Al-Qurashi, Maysaa M.; Chu, Yu-Ming; 56389
In this article, we present a new subdivision scheme by using an interpolatory subdivision scheme and an approximating subdivision scheme. The construction of the subdivision scheme is based on translation of points of the 4-point interpolatory subdivision scheme to the new position according to three displacement vectors containing two shape parameters. We first study the characteristics of the new subdivision scheme analytically and then present numerical experiments to justify these analytical characteristics geometrically. We also extend the new derived scheme into its bivariate/tensor product version. This bivariate scheme is applicable on quadrilateral meshes to produce smooth limiting surfaces up toC3continuity.
Citation Count: Mustafa, Ghulam...et al. (2020). "A New Computational Approach to Estimate the Subdivision Depth of n-Ary Subdivision Scheme", IEEE Access, Vol. 8, pp. 187146-187155.
A New Computational Approach to Estimate the Subdivision Depth of n-Ary Subdivision Scheme
(2020) Mustafa, Ghulam; Shahzad, Aamir; Khan, Faheem; Baleanu, Dumitru; Chu, Yuming; 56389
The n-ary subdivision scheme has traditionally been designed to generate smooth curve and surface from control polygon. In this paper, we propose a new subdivision depth computation technique for n-ary subdivision scheme. The existing techniques do not ensure the computation of subdivision depth unless some strong condition is assumed on the mask of the scheme. But our technique relaxes the effect of strong condition assumed on the mask of the scheme by increasing the number of convolution steps. Consequently, a more precise subdivision depth technique for a given error tolerance is presented in this paper.
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: 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: Mustafa, Ghulam...et al. (2020). "The inequalities for the analysis of a class of ternary refinement schemes", AIMS Mathematics, Vol. 5, No. 6, pp. 7582-7604.
The inequalities for the analysis of a class of ternary refinement schemes
(2020) Mustafa, Ghulam; Ejaz, Syeda Tehmina; Baleanu, Dumitru; Chu, Yu-Ming; 56389
The ternary refinement schemes are the generalized version of the binary refinement schemes. This class of the schemes produce the smooth curves with the less number of refinement steps as compared to the binary class of schemes. In this paper, we present the inequalities for the analysis of a class of ternary refinement schemes. There are three simple algebraic expressions in each inequality. Further these algebraic expressions contain only the coefficients used in the refinement rules of the ternary schemes.
Citation Count: Ejaz, Syeda Tehmina...et al. (2020). "The numerical solution of fourth order nonlinear singularly perturbed boundary value problems via 10-point subdivision scheme based numerical algorithm", AIP Advances, Vol. 10, No. 9.
The numerical solution of fourth order nonlinear singularly perturbed boundary value problems via 10-point subdivision scheme based numerical algorithm
(2020) Ejaz, Syeda Tehmina; Baleanu, Dumitru; Mustafa, Ghulam; Malik, Safia; Chu, Yu-Ming; 56389
The subdivision scheme is used to illustrate smooth curves and surfaces. It is an algorithmic technique which takes a coarse polygon as an input and produces a refined polygon as an output. In this paper, a 10-point interpolating subdivision scheme is used to develop a numerical algorithm for the solution of fourth order nonlinear singularly perturbed boundary value problems (NSPBVPs). The studies of convergence and approximation order of the numerical algorithm are also presented. The solution of NSPBVPs is presented to see the efficiency of the algorithm.
Citation Count: Ejaz, Syeda Tehmina...et al. (2021). "The refinement-schemes-based unified algorithms for certain nth order linear and nonlinear differential equations with a set of constraints", Advances in Difference Equations, Vol. 2021, No. 1.
The refinement-schemes-based unified algorithms for certain nth order linear and nonlinear differential equations with a set of constraints
(2021) Ejaz, Syeda Tehmina; Mustafa, Ghulam; Baleanu, Dumitru; Chu, Yu-Ming; 56389
We first present a generalized class of binary interpolating refinement schemes and their properties. Then the refinement-schemes-based unified algorithms for the solution of certain nth order linear and nonlinear differential equations with a set of constraints are presented. Moreover, several algorithms based on the refinement schemes for solving differential equations are the special cases of our algorithms.