Malik Zaka Ullah

Nematicons in liquid crystals by extended trial equation method

This paper employs extended trial equation method to retrieve nematicons in liquid crystals from its governing equation. In addition, several other forms of solution naturally emerged from the integration algorithm. These are shock waves, singular solitons, snoidal waves, periodic singular waves, plane waves and others. These variety of solutions are being reported for the first time in the context of liquid crystals.

An accelerated iterative method for computing weighted Moore-Penrose inverse

The goal of this paper is to present an accelerated iterative method for computing weighted Moore-Penrose inverse. Analysis of convergence is included to show that the proposed scheme has sixth-order convergence. Using a proper initial matrix, a sequence of iterates will be produced, which is convergent to the weighted Moore-Penrose inverse. Numerical experiments are reported to show the efficiency of the new method.

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.

Resonant optical solitons with parabolic and dual-power laws by semi-inverse variational principle

Abstract This paper obtains bright optical soliton solutions from resonant nonlinear Schrödinger’s equation by the aid of semi-inverse variational principle. The two forms of nonlinear media studied are parabolic and dual-power law. The necessary constraints for the existence of these solitons are also presented.

An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs

An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs

Higher order multi-step Jarratt-like method for solving systems of nonlinear equations

This paper proposes a multi-step iterative method for solving systems of nonlinear equations with a local convergence order of 3m4, where m(2) is the number of steps. The multi-step iterative method includes two parts: the base method and the multi-step part. The base method involves two function evaluations, two Jacobian evaluations, one LU decomposition of a Jacobian, and two matrixvector multiplications. Every stage of the multi-step part involves the solution of two triangular linear systems and one matrixvector multiplication. The computational efficiency of the new method is better than those of previously proposed methods. The method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations.

An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs

In this research paper, we present higher-order quasilinearization methods for the boundary value problems as well as coupled boundary value problems. The construction of higher-order convergent methods depends on a decomposition method which is different from Adomain decomposition method (Motsa and Sibanda, 2013). The reported method is very general and can be extended to desired order of convergence for highly nonlinear differential equations and also computationally superior to proposed iterative method based on Adomain decomposition because our proposed iterative scheme avoids the calculations of Adomain polynomials and achieves the same computational order of convergence as authors have claimed in Motsa and Sibanda, 2013. In order to check the validity and computational performance, the constructed iterative schemes are also successfully applied to bifurcation problems to calculate the values of critical parameters. The numerical performance is also tested for one-dimension Bratu and Frank-Kamenetzkii equations.

Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

Solving systems of nonlinear equations when the nonlinearity is expensive

Construction of multi-step iterative method for solving system of nonlinear equations is considered, when the nonlinearity is expensive. The proposed method is divided into a base method and multi-step part. The convergence order of the base method is five, and each step of multi-step part adds additive-factor of five in the convergence order of the base method. The general formula of convergence order is 5 ( m - 2 ) where m ( ? 3 ) is the step number. For a single instance of the iterative method we only compute two Jacobian and inversion of one Jacobian is required. The direct inversion of Jacobian is avoided by computing LU factors. The computed LU factors are used in the multi-step part for solving five systems of linear equations that make the method computational efficient. The distinctive feature of the underlying multi-step iterative method is the single call to the computationally expensive nonlinear function and thus offers an increment of additive-factor of five in the convergence order per single call. The numerical simulations reveal that our proposed iterative method clearly shows better performance, where the computational cost of the involved nonlinear function is higher than the computational cost for solving five lower and upper triangular systems.

Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation

The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.