In this study, we construct a family of single root finding method of optimal order four and then... more In this study, we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously. Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation. Some non-linear equations are taken from physics, chemistry and engineering to present the performance and efficiency of the newly constructed method. Some real world applications are taken from fluid mechanics, i.e., fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations. These non-linear equations are then solved using newly developed simultaneous iterative schemes. Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance, using very less number of function evaluations in each step. Local convergence order of single root finding method is computed using CAS-Maple. Local computational order of convergence, CPU-time, absolute residuals errors are calculated to elaborate the efficiency, robustness and authentication of the iterative simultaneous method in its domain.
Mehran University Research Journal of Engineering and Technology
In this article, we first construct family of two-step optimal fourth order iterative methods for... more In this article, we first construct family of two-step optimal fourth order iterative methods for finding single root of non-linear equation. We then extend these methods for determining all the distinct as well as multiple roots of single variable non-linear equation simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in case of single root finding method and 6 for simultaneous determination of all distinct as well as multiple roots of a non-linear equation. The computational cost, basins of attraction, computational efficiency, log of residual fall and numerical test functions validate that the newly constructed methods are more efficient as compared to the existing methods in the literature.
This article addresses MHD nanofluid flow induced by stretched surface. Heat transport features a... more This article addresses MHD nanofluid flow induced by stretched surface. Heat transport features are elaborated by implementing double diffusive stratification. Chemically reactive species is implemented in order to explore the properties of nanofluid through Brownian motion and thermophoresis. Activation energy concept is utilized for nano liquid. Further zero mass flux is assumed at the sheet’s surface for better and high accuracy of the out-turn. Trasnformations are used to reconstruct the partial differential equations into ordinary differential equations. Homotopy analysis method is utilized to obtain the solution. Physical features like flow, heat and mass are elaborated through graphs. Thermal stratified parameter reduces the temperature as well as concentration profile. Also decay in concentration field is noticed for larger reaction rate parameter. Both temperature and concentration grows for Thermophoresis parameter. To check the heat transfer rate, graphical exposition of ...
In this paper, we present and analyse fourth order method for finding simultaneously multiple zer... more In this paper, we present and analyse fourth order method for finding simultaneously multiple zeros of polynomial equations. S. M. Ilič and L. Rančič modified cubically convergent Ehrlich Aberth method to fourth order for the simultaneous determination of simple zeros [5]. We generalize this method to the case of multiple zeros of complex polynomial equations. It is proved that the method has fourth order convergence. Numerical tests show its efficient computational behaviour in the case of multiple real/complex roots of polynomial equations.
In this work, we construct a new family of inverse iterative numerical technique for extracting a... more In this work, we construct a new family of inverse iterative numerical technique for extracting all roots of nonlinear equation simultaneously. Convergence analysis verifies that the proposed family of methods has local 10th-order convergence. Among the test models investigated are blood rheology, a fractional nonlinear equation model, fluid permeability in biogels, and beam localization models. In comparison to other methods, the family of inverse simultaneous iterative techniques gets initial estimations to exact roots within a given tolerance while using less function evaluations in each iterative step. Numerical results, basins of attraction for fractional nonlinear equation, residual graphs are presented in detail for the simultaneous iterative techniques. The newly developed simultaneous iterative techniques were thoroughly investigated and proven to be efficient, robust, and authentic in their domain.
In this paper, we present three-step quadrature based iterative method for solving non-linear equ... more In this paper, we present three-step quadrature based iterative method for solving non-linear equations. The convergence analysis of the method is discussed. It is established that the new method has convergence order eight. Numerical tests show that the new method is comparable with the well known existing methods and in many cases gives better results. Our results can be considered as an improvement and reflnement of the previously known results in the literature.
In this paper, we present a generalized Ceby?ev type inequality for absolutely continuous functio... more In this paper, we present a generalized Ceby?ev type inequality for absolutely continuous functions whose derivatives belong to L(subscript p) [a, b], p＞1. Applications for probabilty density functions are also given.
We establish here new three and four-step iterative methods of convergence order seven and fourte... more We establish here new three and four-step iterative methods of convergence order seven and fourteen to find roots of non-linear equations. The new developed iterative methods are variants of Newton's method that use different approximations of first derivatives in terms of previously known function values, thus improving efficiency indices of the methods. The seventh and fourteenth order iterative methods use four and five function values including one derivative of a function. So, the efficiency indices of these methods are , respectively. Numerical examples are given to show the performance of described methods.
We establish some new weighted Chebyshev type integral inequalities for functions whose first der... more We establish some new weighted Chebyshev type integral inequalities for functions whose first derivatives belong to L p (a,b), ∀1≤p<∞ .
Let K be a non-empty closed convex subset of an arbitrary Banach space E. Let T:K→K be uniformly ... more Let K be a non-empty closed convex subset of an arbitrary Banach space E. Let T:K→K be uniformly L Lipschitzian with F(T)≠ϕ. Let {k n }⊆[1,∞) be a sequence with lim n→∞ k n =1. For any x 0 ∈K and fixed u∈K, we define the sequence {x n } by x n+1 =α n u+β n x n +γ n T n x n , where {α n }, {β n }, {γ n } are real sequences in (0,1) satisfying some conditions. Then the sequence {x n } converges strongly to a fixed point of T.
We present a new three-step predictor corrector type iterative method for finding simple and real... more We present a new three-step predictor corrector type iterative method for finding simple and real roots of nonlinear equations in single variable. Experiments show that our new method is more efficient than the well known methods available in the literature. The comparison table is given which demonstrate that in most of the cases the new method perform better near the original root on both sides.
In this paper, we will improve and generalize inequality of Ostrowski type for mappings whose sec... more In this paper, we will improve and generalize inequality of Ostrowski type for mappings whose second derivatives belong to L 1 (a, b). Some well known inequalities can be derived as special cases. In addition, perturbed mid-point inequality and perturbed trapezoid inequality are also obtained. The obtained inequalities have immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid and midpoint formula. Applications to some special means are also investigated.
In account of the famous Čebyšev inequality, a rich theory has appeared in the literature. We est... more In account of the famous Čebyšev inequality, a rich theory has appeared in the literature. We establish some new weighted Čebyšev type integral inequalities. Our proofs are of independent interest and provide new estimates on these types of inequalities.
In this paper, we point out and analyse six two-step iterative methods for finding multiple as we... more In this paper, we point out and analyse six two-step iterative methods for finding multiple as well as distinct zeros of non-linear equations. We prove that the methods for multiple zeros as well as for distinct zeros have fourth order convergence. The methods calculate the multiple as well as distinct zeros with high accuracy. The numerical tests show their better performance in case of algebraic as well as non-algebraic equations.
In this study, we construct a family of single root finding method of optimal order four and then... more In this study, we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously. Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation. Some non-linear equations are taken from physics, chemistry and engineering to present the performance and efficiency of the newly constructed method. Some real world applications are taken from fluid mechanics, i.e., fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations. These non-linear equations are then solved using newly developed simultaneous iterative schemes. Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance, using very less number of function evaluations in each step. Local convergence order of single root finding method is computed using CAS-Maple. Local computational order of convergence, CPU-time, absolute residuals errors are calculated to elaborate the efficiency, robustness and authentication of the iterative simultaneous method in its domain.
Mehran University Research Journal of Engineering and Technology
In this article, we first construct family of two-step optimal fourth order iterative methods for... more In this article, we first construct family of two-step optimal fourth order iterative methods for finding single root of non-linear equation. We then extend these methods for determining all the distinct as well as multiple roots of single variable non-linear equation simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in case of single root finding method and 6 for simultaneous determination of all distinct as well as multiple roots of a non-linear equation. The computational cost, basins of attraction, computational efficiency, log of residual fall and numerical test functions validate that the newly constructed methods are more efficient as compared to the existing methods in the literature.
This article addresses MHD nanofluid flow induced by stretched surface. Heat transport features a... more This article addresses MHD nanofluid flow induced by stretched surface. Heat transport features are elaborated by implementing double diffusive stratification. Chemically reactive species is implemented in order to explore the properties of nanofluid through Brownian motion and thermophoresis. Activation energy concept is utilized for nano liquid. Further zero mass flux is assumed at the sheet’s surface for better and high accuracy of the out-turn. Trasnformations are used to reconstruct the partial differential equations into ordinary differential equations. Homotopy analysis method is utilized to obtain the solution. Physical features like flow, heat and mass are elaborated through graphs. Thermal stratified parameter reduces the temperature as well as concentration profile. Also decay in concentration field is noticed for larger reaction rate parameter. Both temperature and concentration grows for Thermophoresis parameter. To check the heat transfer rate, graphical exposition of ...
In this paper, we present and analyse fourth order method for finding simultaneously multiple zer... more In this paper, we present and analyse fourth order method for finding simultaneously multiple zeros of polynomial equations. S. M. Ilič and L. Rančič modified cubically convergent Ehrlich Aberth method to fourth order for the simultaneous determination of simple zeros [5]. We generalize this method to the case of multiple zeros of complex polynomial equations. It is proved that the method has fourth order convergence. Numerical tests show its efficient computational behaviour in the case of multiple real/complex roots of polynomial equations.
In this work, we construct a new family of inverse iterative numerical technique for extracting a... more In this work, we construct a new family of inverse iterative numerical technique for extracting all roots of nonlinear equation simultaneously. Convergence analysis verifies that the proposed family of methods has local 10th-order convergence. Among the test models investigated are blood rheology, a fractional nonlinear equation model, fluid permeability in biogels, and beam localization models. In comparison to other methods, the family of inverse simultaneous iterative techniques gets initial estimations to exact roots within a given tolerance while using less function evaluations in each iterative step. Numerical results, basins of attraction for fractional nonlinear equation, residual graphs are presented in detail for the simultaneous iterative techniques. The newly developed simultaneous iterative techniques were thoroughly investigated and proven to be efficient, robust, and authentic in their domain.
In this paper, we present three-step quadrature based iterative method for solving non-linear equ... more In this paper, we present three-step quadrature based iterative method for solving non-linear equations. The convergence analysis of the method is discussed. It is established that the new method has convergence order eight. Numerical tests show that the new method is comparable with the well known existing methods and in many cases gives better results. Our results can be considered as an improvement and reflnement of the previously known results in the literature.
In this paper, we present a generalized Ceby?ev type inequality for absolutely continuous functio... more In this paper, we present a generalized Ceby?ev type inequality for absolutely continuous functions whose derivatives belong to L(subscript p) [a, b], p＞1. Applications for probabilty density functions are also given.
We establish here new three and four-step iterative methods of convergence order seven and fourte... more We establish here new three and four-step iterative methods of convergence order seven and fourteen to find roots of non-linear equations. The new developed iterative methods are variants of Newton's method that use different approximations of first derivatives in terms of previously known function values, thus improving efficiency indices of the methods. The seventh and fourteenth order iterative methods use four and five function values including one derivative of a function. So, the efficiency indices of these methods are , respectively. Numerical examples are given to show the performance of described methods.
We establish some new weighted Chebyshev type integral inequalities for functions whose first der... more We establish some new weighted Chebyshev type integral inequalities for functions whose first derivatives belong to L p (a,b), ∀1≤p<∞ .
Let K be a non-empty closed convex subset of an arbitrary Banach space E. Let T:K→K be uniformly ... more Let K be a non-empty closed convex subset of an arbitrary Banach space E. Let T:K→K be uniformly L Lipschitzian with F(T)≠ϕ. Let {k n }⊆[1,∞) be a sequence with lim n→∞ k n =1. For any x 0 ∈K and fixed u∈K, we define the sequence {x n } by x n+1 =α n u+β n x n +γ n T n x n , where {α n }, {β n }, {γ n } are real sequences in (0,1) satisfying some conditions. Then the sequence {x n } converges strongly to a fixed point of T.
We present a new three-step predictor corrector type iterative method for finding simple and real... more We present a new three-step predictor corrector type iterative method for finding simple and real roots of nonlinear equations in single variable. Experiments show that our new method is more efficient than the well known methods available in the literature. The comparison table is given which demonstrate that in most of the cases the new method perform better near the original root on both sides.
In this paper, we will improve and generalize inequality of Ostrowski type for mappings whose sec... more In this paper, we will improve and generalize inequality of Ostrowski type for mappings whose second derivatives belong to L 1 (a, b). Some well known inequalities can be derived as special cases. In addition, perturbed mid-point inequality and perturbed trapezoid inequality are also obtained. The obtained inequalities have immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid and midpoint formula. Applications to some special means are also investigated.
In account of the famous Čebyšev inequality, a rich theory has appeared in the literature. We est... more In account of the famous Čebyšev inequality, a rich theory has appeared in the literature. We establish some new weighted Čebyšev type integral inequalities. Our proofs are of independent interest and provide new estimates on these types of inequalities.
In this paper, we point out and analyse six two-step iterative methods for finding multiple as we... more In this paper, we point out and analyse six two-step iterative methods for finding multiple as well as distinct zeros of non-linear equations. We prove that the methods for multiple zeros as well as for distinct zeros have fourth order convergence. The methods calculate the multiple as well as distinct zeros with high accuracy. The numerical tests show their better performance in case of algebraic as well as non-algebraic equations.
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