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Noor, Muhammad Aslam,Noor, Khalida Inayat,Mohyud-Din, Syed Tauseef,Shaikh, Noor Ahmed The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper, we apply a new decomposition method for solving initial and boundary value problems, which is due to Noor and Noor [18]. The analytical results are calculated in terms of convergent series with easily computable components. The diagonal Pade approximants are applied to make the work more concise and for the better understanding of the solution behavior. The proposed technique is tested on boundary layer problem; Thomas-Fermi, Blasius and sixth-order singularly perturbed Boussinesq equations. Numerical results reveal the complete reliability of the suggested scheme. This new decomposition method can be viewed as an alternative of Adomian decomposition method and homotopy perturbation methods.
VARIANTS OF NEWTON'S METHOD USING FIFTH-ORDER QUADRATURE FORMULAS: REVISITED
Noor, Muhammad Aslam,Waseem, Muhammad The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results.
VARIATIONAL DECOMPOSITION METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS
Noor, Muhammad Aslam,Mohyud-Din, Syed Tauseef The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper, we implement a relatively new analytical technique by combining the traditional variational iteration method and the decomposition method which is called as the variational decomposition method (VDM) for solving the sixth-order boundary value problems. The proposed technique is in fact the modification of variatioanal iteration method by coupling it with the so-called Adomian's polynomials. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Comparisons are made to verify the reliability and accuracy of the proposed algorithm. Several examples are given to check the efficiency of the proposed algorithm. We have also considered an example where the VDM is not reliable.
Variants of Newton's method using fifth-order quadrature formulas: Revisited
Muhammad Aslam Noor,Muhammad Waseem 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results. In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results.
SOME PROPERTIES OF NONCONVEX FUNCTIONS
MUHAMMAD ASLAM NOOR 경남대학교 수학교육과 2018 Nonlinear Functional Analysis and Applications Vol.23 No.3
In this paper, we introduce and study a new class of convex functions with respect to an arbitrary function, which is called the k-convex function. These functions are nonconvex functions and include the convex function and φ-convex convex as special cases. We study some properties of k-convex functions. It is shown that the minimum of k-convex functions on the k-convex sets can be characterized by a class of variational inequalities, which is called the k-directional variational inequalities. Some open problems are also suggested for future research.
GENERAL BICONVEX FUNCTIONS AND BIVARIATIONAL-LIKE INEQUALITIES
MUHAMMAD ASLAM NOOR 경남대학교 수학교육과 2022 Nonlinear Functional Analysis and Applications Vol.27 No.1
In this paper, we consider and introduce some new concepts of the biconvex functions involving an arbitrary bifunction and function. Some new relationships among various concepts of biconvex functions have been established. We have shown that the optimality conditions for the general biconvex functions can be characterized by a class of bivariational-like inequalities. Auxiliary principle technique is used to propose proximal point methods for solving general bivariational-like inequalities. We also discussed the conversance criteria for the suggested methods under pseudo-monotonicity. Our method of proof is very simple compared with methods. Several special cases are discussed as applications of our main concepts and results. It is a challenging problem to explore the applications of the general bivariational-like inequalities in pure and applied sciences.
Regularized mixed quasi equilibrium problems
MUHAMMAD ASLAM NOOR 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
In this paper, we introduce and study a new class of equilibrium problems, known as regularized mixed quasi equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving equilibrium problems and variational inequalities involving the convex sets.
RESOLVENT EQUATIONS TECHNIQUE FOR VARIATIONAL INEQUALITIES
Noor, Muhammad-Aslam 한국전산응용수학회 1997 Journal of applied mathematics & informatics Vol.4 No.2
In this paper we establish the equivalence between the general resolvent equations and variational inequalities. This equiva-lence is used to suggest and analyze a number of iterative algorithms for solving variational inclusions. We also study the convergence criteria of the iterative algorithms. Our results include several pre-viously known results as special cases.
NEW INEQUALITIES FOR GENERALIZED LOG h-CONVEX FUNCTIONS
Muhammad Aslam Noor,Khalida Inayat Noor,Farhat Safdar 한국전산응용수학회 2018 Journal of applied mathematics & informatics Vol.36 No.3
In the paper, we introduce some new classes of generalized log$h$-convex functions in the first sense and in the second sense. We establish Hermite-Hadamard type inequality for different classes of generalized convex functions. It is shown that the classes of generalized log $h$-convex functions in both senses include several new and known classes of log $h$ convex functions. Several special cases are also discussed. Results proved in this paper can be viewed as a new contributions in this area of research.
WELL-POSED VARIATIONAL INEQUALITIES
Muhammad, Aslam-Noor 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.11 No.1
In this paper, we introduce the concept of well-posedness for general variational inequalities and obtain some results under pseudomonotonicity. It is well known that monotonicity implies pseudomonotonicity, but the converse is not true. In this respect, our results represent an improvement and refinement of the previous known results. Since the general variational inequalities include (quasi) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems.